This volume presents a collection of articles devoted to representations of algebras and related topics. Dististinguishe

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*English*
*Pages 378
[375]*
*Year 2013*

- Author / Uploaded
- David J. Benson
- David J. Benson
- Henning Krause
- Andrzej Skowronski (eds.)

**Commentary**- no title page

*Table of contents : Content: Infinite dimensional tilting theory / Lidia Angeleri Hügel --A survey of modules of constant Jordan type and vector bundles on projective space / David J. Benson --On representation-finite algebras and beyond / Klaus Bongartz --Quiver Hecke algebras and categorification / Jonathan Brundan --Ordered exchange graphs / Thomas Brüstle and Dong Yang --Introduction to Donaldson-Thomas invariants / Sergey Mozgovoy --Cluster algebras and singular supports of perverse sheaves / Hiraku Nakajima --Representations and cohomology of finite group schemes / Julia Pevtsova --Superdecomposable pure-injective modules / Mike Prest --Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves / Jan Šťovíček.*

Preface

The Workshop and International Conference on Representations of Algebras (ICRA 2012) took place in Bielefeld, Germany, during August 8–17, 2012. The aim of this book is to present some of the lectures to a wider audience. The meeting consisted of a workshop (four days) with six lecture series and a conference (five days) with invited plenary lectures and contributed talks in seven parallel sessions; it attracted 268 mathematicians from 31 countries. The meeting was the 15th in a series of international conferences which started 1974 in Ottawa, and it marked a generation change with some new organisational structure, modernising the concept implemented by the first generation of representation theorists over the last 35 years. The scientific organisers of the meeting were C. Bessenrodt, R. Farnsteiner, D. Happel, S. König, H. Krause, M. Reineke, J. Schröer. The external advisors were D. Benson, W. Crawley-Boevey, B. Keller, A. Kleshchev. The ICRA Award 2012 (for outstanding work by young mathematicians in the field of representations of finite-dimensional algebras) was given to Jan Šˇtovíˇcek for his key contributions to infinite-dimensional tilting theory and to approximation theory as well as for his work on representability of functors and existence of adjoints in triangulated categories. This book contains ten expository survey articles. The participants of the meeting were not all specialists in the area and so the speakers aimed to make their talks as selfcontained as possible. In fact, a characteristic feature of modern representation theory of algebras is the highly complex interaction with other branches of mathematics. Several of the powerful technologies developed within the field are contributing to other areas. In the opposite direction, new problems, ideas and points of view are coming into the subject from previously unrelated areas. These aspects are reflected in the papers presented here. The study of derived categories and tilting is one of the most successful methodologies of algebra representation theory. Infinite-dimensional tilting modules are discussed in the paper of Angeleri Hügel. They occur when studying torsion pairs in module categories, when looking for complements to partial tilting modules, or in connection with the Homological Conjectures. They share many properties with classical tilting modules, but they also give rise to interesting new phenomena as they are intimately related with localisation, both at the level of module categories and of derived categories. The last few years have seen the emergence of a new branch of representation theory for finite group schemes, namely modules of constant Jordan type. These are the analogues, among finitely generated modules, of vector bundles among coherent sheaves. Benson’s article gives an introduction to this topic in the special case of elementary abelian p-groups, and explains how such modules give rise to vector bundles on projective space. The theory of Chern classes then feeds back information into the

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representation theory. For the more general context of finite group schemes, see the article of Pevtsova. The aim of the article by Bongartz is to survey old and new results in the theory of representation-finite and minimal representation-infinite algebras over an algebraically closed field. The cornerstones of the article are the existence of multiplicative bases and coverings with good properties, and applications of the ray-categories attached to distributive categories. The author presents several essential results of the classical representation theory of finite-dimensional algebras in a more general form and proposes simplified proofs for them. The article also contains several important applications of the presented theory: the sharper version of the second Brauer–Thrall conjecture, the author’s recent result that there is no gap in the length of indecomposable modules of finite dimension over any algebra, the author’s criterion for finite representation type, a new proof of the classification of representation-finite selfinjective algebras, as well as the theorem that every indecomposable module over a representation-finite algebra admits a basis such that all arrows in its quiver representation are represented by matrices having only 0 and 1 entries. The paper by Brundan is a brief introduction to the quiver Hecke algebras discovered in 2008 by Khovanov, Lauda and Rouquier. They are certain Hecke algebras attached to symmetrisable Cartan matrices. It appears that Khovanov and Lauda came upon these algebras from an investigation of endomorphisms of Soergel bimodules (and related bimodules which arise from cohomology of partial flag varieties), while Rouquier’s motivation was a close analysis of Lusztig’s construction of canonical bases in terms of perverse sheaves on certain quiver varieties. The article explains these constructions in some detail, emphasising their application to the categorification of quantum groups. The article of Brüstle and Yang gives a survey on ordered exchange graphs arising in cluster theory. The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters. Through the recent categorifications of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated with objects such as a finite-dimensional algebra or a differential graded algebra concentrated in non-positive degrees. These constructions often come from variations of the concept of tilting, the vertices of the exchange graph being torsion pairs, t-structures, silting objects, support -tilting modules and so on. All these exchange graphs stemming from representation theory have the additional feature that they are the Hasse quiver of a partial order which is naturally defined for the objects. In this sense, the exchange graphs arising in cluster theory can be considered as a generalisation or as a completion of the poset of tilting modules which has been studied by Happel and Unger. The goal of the article is to axiomatise the thus obtained structure of an ordered exchange graph, to present the various constructions of ordered exchange graphs and to relate them among each other. The aim of the lectures of Mozgovoy was to introduce refined Donaldson–Thomas invariants for the categories of modules over Jacobian algebras associated to quivers with potentials. These invariants were first studied in the context of 3-Calabi–Yau manifolds, but Kontsevich and Soibelman developed a framework that allows to shift the study of these invariants from 3-Calabi–Yau varieties to other sources of 3-Calabi–

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Yau categories. One such source, closely related to representation theory, consists of quivers with potentials. The article follows closely these lectures. So the author defines refined Donaldson–Thomas invariants, computes them in some simple cases, and then discusses their basic properties, including integrality, positivity, and wallcrossing phenomena. The article by Nakajima contains an approach to the Geiß–Leclerc–Schröer conjecture on the cluster algebra structure in the coordinate ring of a unipotent subgroup and the dual canonical base. This is related with the author’s recent approach to the theory of cluster algebras based on perverse sheaves on graded quiver varieties. The new proposed idea is to use the singular support of a perverse sheaf, which is a langrangian subvariety in the cotangent bundle of the space of quiver representations. The author surveys links between the related conjectures and approaches, and proposes two new conjectures which give links between the theory developed by Geiß, Leclerc and Schröer and perverse sheaves via singular support. Pevtsova’s article is an introduction to the representation theory and cohomology of finite group schemes, concentrating on the theory of -points and …-support, and leading into a discussion of constant Jordan type and related classes of modules in this general context. This nicely complements the article of Benson. Superdecomposable pure-injective modules are discussed in the article by Prest. They form a particular class of infinite-dimensional modules and their existence reflects complexity in the category of finite-dimensional representations. For a finitedimensional algebra, the evidence to date is consistent with existence of superdecomposable pure-injectives being equivalent to having non-domestic representation type. The paper describes this interplay between finite and infinite-dimensional modules in terms of pointed modules. Methods for producing superdecomposable pure-injectives are presented and some details are given in the context of tubular algebras. A categorical framework for the theory of approximations and cotorsion pairs is presented in the paper by Št’ovíˇcek. His aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalising previously known results. The discussion includes motivating applications with the emphasis on constructing monoidal model structures for the derived category of quasi-coherent sheaves of modules over a scheme. The meeting received substantial support from the German Research Foundation (DFG) through the following grants: Collaborative Research Centre “Spectral Structures and Topological Methods in Mathematics” (SFB 701) and Priority Programme “Representation Theory” (SPP 1388). We are most grateful for their assistance. We also wish to express our thanks to the authors and referees of the papers in this volume and to Manfred Karbe of the European Mathematical Society Publishing House for his help in preparing it for publication. Aberdeen, Bielefeld, and Toru´n, October 2013

Dave Benson Henning Krause Andrzej Skowro´nski

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Infinite dimensional tilting theory Lidia Angeleri Hügel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 A survey of modules of constant Jordan type and vector bundles on projective space David J. Benson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 On representation-finite algebras and beyond Klaus Bongartz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Quiver Hecke algebras and categorification Jonathan Brundan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Ordered exchange graphs Thomas Brüstle and Dong Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Introduction to Donaldson–Thomas invariants Sergey Mozgovoy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Cluster algebras and singular supports of perverse sheaves Hiraku Nakajima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Representations and cohomology of finite group schemes Julia Pevtsova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Superdecomposable pure-injective modules Mike Prest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves Jan Št’ovíˇcek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Infinite dimensional tilting theory Lidia Angeleri Hügel

1 Introduction Tilting theory is a research area that has developed from representation theory of finite dimensional algebras [27], [54], [74] with applications going far beyond this context: tilting nowadays plays an important role in various branches of mathematics, ranging from Lie theory to combinatorics, algebraic geometry and topology. Classical tilting modules are required to be finite dimensional. This survey focusses on tilting modules that need not be finitely generated, as first defined in [40], [2]. The aim is twofold. We first explain the main tools of infinite dimensional tilting theory and exhibit a number of examples. Then we discuss the interaction with localization theory, and we employ it to classify large tilting modules over several rings. We will see that such classification results also yield a classification of certain categories of finitely generated modules. Moreover, they lead to a classification of Gabriel localizations of the module category, or of an associated geometric category. Large tilting modules occur in many contexts. For example, the representation type of a hereditary algebra is governed by the behaviour of certain infinite dimensional tilting modules (Theorem 3.1). Also the finitistic dimension of a noetherian ring is determined by a tilting module which is not finitely generated in general (Theorem 5.4). Large tilting modules further arise when looking for complements to partial tilting modules of projective dimension greater than one, or when computing intersections of tilting classes given by finite dimensional tilting modules (Sections 5.1 and 5.3). Finally, over a commutative ring, every non-trivial tilting module is large (Section 3.2). In fact, given a ring R, many important subcategories of mod-R can be studied by using tilting modules. Here mod-R denotes the category of modules admitting a projective resolution with finitely generated projectives, which is just the category of finitely generated modules when R is noetherian. More precisely, for any set mod-R consisting of modules of bounded projective dimension, there is a tilting module T , not necessarily finitely generated, whose tilting class T ? D fM 2 Mod-R j ExtiR .T; M / D 0 for all i > 0g coincides with ? D fM 2 Mod-R j ExtiR .S; M / D 0 for all i > 0; S 2 g. Conversely, for any tilting module T , no matter whether finitely generated or not, the tilting class T ? is determined by a set mod-R, in the sense that T ? D ? (Corollary 5.1 and Theorem 6.1). These results rely on work of Eklof and Trlifaj on the existence of approximations, on the relationship between tilting and approximation theory first discovered by Auslander and Reiten in the classical setup (Theorems 4.4 and 4.6), and on papers by Bazzoni, Eklof, Herbera, Sˇtovíˇcek, and Trlifaj which also make use of some sophisticated set-theoretic techniques.

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As a consequence, one obtains a bijection between the resolving subcategories of mod-R consisting of modules of bounded projective dimension and the tilting classes in Mod-R. For an Artin algebra, one also gets a bijection between the torsion pairs in mod-R whose torsion class contains all indecomposable injectives and the tilting classes T ? in Mod-R where T is a tilting module of projective dimension at most one (Corollary 6.4 and Theorem 6.8). The results above show that large tilting modules share many properties with the classical tilting modules from representation theory. There is an important difference, however. If T is a classical tilting module over a ring R, and S is the endomorphism ring of T , then the derived categories of R and S are equivalent as triangulated categories. This is a fundamental result due to Happel. A generalization for large tilting modules holds true under a mild assumption, as recently shown in papers by Bazzoni, Mantese, Tonolo, Chen, Xi, and Yang. But instead of an equivalence, one has that the derived category D.Mod-R/ is a quotient of D.Mod-S /. When T has projective dimension one, D.Mod-R/ is even a recollement of D.Mod-S / and D.Mod-Sz/, where Sz is a localization of S (Theorem 6.11). Tilting functors given by large tilting modules thus give rise to new phenomena and induce localizations of derived categories. Actually, localization already plays a role at the level of module categories. A typical example of a large tilting module is provided by the Z-module T D Q ˚ Q=Z. Following the same pattern, one can use localization techniques to construct tilting modules in many contexts. Indeed, every injective ring epimorphism R ! S with nice homological properties gives rise to a tilting R-module of the form S ˚ S=R (Theorem 7.1). Over certain rings, tilting modules of this shape provide a classification of all tilting modules up to equivalence. Hereby, we say that two tilting modules are equivalent if they induce the same tilting class. Such identification is justified by the fact that the tilting class T ? determines the additive closure Add T of a tilting module T . Furthermore, recall that in this way, when we classify tilting modules, we are also classifying resolving subcategories of mod-R, and in case we restrict to projective dimension one, torsion pairs in mod-R. The interaction between tilting and localization and its role in connection with classification problems is best illustrated by the example of a Dedekind domain R. The tilting modules over R are constructed as above from ring epimorphisms, or more precisely, from universal localizations of the ring R, which in turn are in bijection with the recollements of the unbounded derived category D.Mod-R/. In other words, tilting modules are parametrized by the subsets P Max-Spec R of the maximal spectrum of R. Hereby, the two extreme cases yield the trivial tilting module R when P D ;, and the tilting module Q ˚ Q=R when P D Max-Spec R (Section 8.1). This result extends to Prüfer domains and to arbitrary commutative noetherian rings. Over the latter, tilting modules can be classified in terms of sequences of specialization closed subsets of the Zariski prime spectrum. In both cases, tilting modules of projective dimension one correspond to categorical localizations of Mod-R in the sense of Gabriel (Sections 8.3 and 8.4).

Infinite dimensional tilting theory

3

But a similar situation also occurs when R is the Kronecker algebra (Section 8.2). Actually, S if we replace the maximal spectrum by the index set X of the tubular family t D x2X Ux and restrict our attention to infinite dimensional tilting modules, we get a complete analogy to the Dedekind case. Indeed, the large tilting modules over R are parametrized by the subsets of X. Here are the two extreme cases: P D ; yields the tilting module L corresponding to the resolving subcategory of mod-R formed by the preprojectives, and P D X the tilting module W corresponding to the resolving subcategory of mod-R formed by preprojective and regular modules. Moreover, all tilting modules are constructed from universal localization, with the only exception of the module L, that is, of the set P D ;. This analogy also allows a geometric interpretation of tilting. In fact, regarding X as an exceptional curve in the sense of [68], it turns out that large tilting modules correspond to Gabriel localizations of the category QcohX of quasi-coherent sheaves on X, the same result as in the commutative case when replacing Mod-R by QcohX. For arbitrary tame hereditary algebras, the classification of large tilting modules is more complicated due to the possible presence of finite dimensional direct summands from non-homogeneous tubes. Infinite dimensional tilting modules are parametrized by pairs .Y; P / where Y determines the finite dimensional part, and P is a subset of X. The infinite dimensional part is obtained as above from universal localization and from the module L. And again, tilting modules correspond to Gabriel localizations of the category QcohX (Section 8.5). Finally, these results lead to a classification of large tilting sheaves in the category QcohX on an exceptional curve X. When X is of domestic type, the classification is essentially the same as for large tilting modules over tame hereditary algebras. For X of tubular type there are many more tilting sheaves. Indeed, for each rational slope we find the same tilting sheaves as in the domestic case, and in addition there is one tilting sheaf for each irrational slope. Since every large tilting sheaf has a slope, this yields a complete classification (Section 8.6). The paper is organized as follows. In Section 2 we collect basic notation and definitions. In Section 3 we exhibit first examples of large tilting modules, e.g. over Z and over hereditary Artin algebras. Further examples are presented in Section 5, after reviewing the relationship between tilting and approximation theory in Section 4. Section 6 is devoted to the interplay between finitely generated modules and large tilting modules. In Section 7 we explain the fundamental construction of tilting modules from ring epimorphisms or universal localizations. Finally, in Section 8 we illustrate the classification results for Dedekind domains, commutative noetherian rings, Prüfer domains, tame hereditary algebras and the category of quasi-coherent sheaves on an exceptional curve. Many of the results presented here, in particular in Sections 1–6, are treated in detail in [52]. We also refer to the survey articles [88], [93], [94].

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2 Large tilting modules 2.1 Notation. Throughout this note, let R be a ring (associative, with 1), Mod-R (respectively, R-Mod) the category of all right (respectively, left) R-modules, and mod-R (respectively, R-mod) the full subcategory of all modules M admitting a projective resolution ! PkC1 ! Pk ! ! P1 ! P0 ! M ! 0 where all Pi are finitely generated. The modules M 2 mod-R are sometimes called of type FP1 . They have the property that the functor ExtiR .M; / commutes with direct limits for any i 0 (see e.g. [52, 3.1.6]). Of course, mod-R is just the category of finitely presented (respectively, generated) modules when R is right coherent (respectively, noetherian). If R is a k-algebra over a commutative ring k, we denote by D D Homk .; I / the duality with respect to an injective cogenerator I of Mod-k. When R is an Artin algebra, we take the usual duality D. For an arbitrary ring R, we can choose k D Z and D D HomZ .; Q=Z/. Given a class of modules M Mod-R, we denote by Add M the class of all modules isomorphic to a direct summand of a direct sum of modules of M, and by Prod M the class of all modules isomorphic to a direct summand of a direct product of modules of M. The class of all modules isomorphic to a direct summand of a finite direct sum of modules of M is denoted by add M. Moreover, we write lim M for the ! class of all modules isomorphic to a direct limit of modules of M. We set Mo D fX 2 Mod-R j HomR .M; X / D 0 for all M 2 Mg; M? D fX 2 Mod-R j ExtiR .M; X / D 0 for all i > 0 and M 2 Mg; and we define o M and ? M correspondingly. When M D fM g, we just write Add M , Prod M , M o , M ? , : : : . All these classes are regarded as strictly full subcategories of Mod-R. We denote by pdim M and idim M the projective and injective dimension of a module M , respectively, and for M Mod-R we write pdimM D supf pdimM j M 2 Mg: Further, we set P D fM 2 Mod-R j pdimM < 1g, and P 0 and all sets I ; (T3) there exists a long exact sequence 0 ! RR ! T0 ! ! Tr ! 0 with Ti 2 Add T for each 0 i r. The class T ? is then called the tilting class induced by T . Further, T and T ? are called n-tilting when pdim T n. The tilting class determines the additive closure of T : two tilting modules T and T 0 induce the same tilting class T ? D T 0 ? if and only if Add T D Add T 0 . In this case we say that T and T 0 are equivalent. Moreover, we say that a tilting module is large if it is not equivalent to any tilting module in mod-R. Dually, a module C is called a cotilting module provided it satisfies the following conditions: (C1) idim C < 1; (C2) ExtiR .C I ; C / D 0 for each i > 0 and all sets I ; (C3) there exists an injective cogenerator Q and a long exact sequence 0 ! Cr ! ! C0 ! Q ! 0 with Ci 2 Prod C for each 0 i r. The class ? C is then called the cotilting class induced by C . Again, C and C ? are called n-cotilting when idim C n. Two cotilting modules C and C 0 are equivalent if ? C D ? C 0 , or equivalently, Prod C D Prod C 0 . A cotilting module is said to be large if it is not equivalent to any cotilting module in mod-R. We will see in Proposition 6.2 that the dual D.T / of a tilting module is always a cotilting module. However, not all cotilting modules arise in this way in general, see Remark 6.3. Furthermore, the dual of a cotilting module need not be a tilting module, cf. Section 3. 2.3 Faithful torsion pairs. Recall that a pair of classes .T ; F / in Mod-R (or more generally, in an abelian category A) is a torsion pair if T D o F , and F D T o . Every class M of modules (or of objects in A) generates a torsion pair .T ; F / by setting F D Mo and T D o .Mo /. Similarly, the torsion pair cogenerated by M is given by

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T D o M and F D .o M/o . We say that a torsion pair .T ; F / is split if every short exact sequence 0 ! T ! M ! F ! 0 with T 2 T and F 2 F splits. Finally, a torsion pair .T ; F / in Mod-R, or in mod-R, is called faithful if R 2 F , and it is called cofaithful if T contains an injective cogenerator of Mod-R. 1-tilting modules generate cofaithful torsion pairs, and 1-cotilting modules cogenerate faithful torsion pairs. Proposition 2.1 ([40], [36]). (1) A module T is a 1-tilting module if and only if T ? coincides with the class Gen T of all modules isomorphic to a quotient of a direct sum of copies of T . Then .Gen T; T o / is a cofaithful torsion pair in Mod-R. (2) A module C is a 1-cotilting module if and only if ? C coincides with the class Cogen C of all modules isomorphic to a submodule of a direct product of copies of C . Then .o C; Cogen C / is a faithful torsion pair in Mod-R. Torsion pairs as in statement (1) or (2) above are called tilting, respectively cotilting torsion pairs. They will be characterized in Corollaries 4.7 and 6.6, and in Theorem 6.8. For an extension of Proposition 2.1 to tilting or cotilting modules of arbitrary projective, respective injective, dimension, we refer to [18].

3 First examples 3.1 Classical tilting modules. If T 2 mod-R, then the functors ExtiR .T; / commute with direct sums, so condition (T2) in Definition 2.2 is equivalent to (T20 ) ExtiR .T; T / D 0 for each i > 0. Moreover, it is easy to show that condition (T3) can be replaced by (T30 ) There exists a long exact sequence 0 ! RR ! T0 ! ! Tr ! 0 with Ti 2 add T for each 0 i r. We thus recover the original definition of tilting module from [27], [54], [74]. Observe that a tilting module belongs to mod-R whenever it is finitely generated, see e.g. [32, 4.7]. 3.2 Tilting modules over commutative rings. Over a commutative ring R, every finitely generated tilting module is projective. This was already observed in [39], [77]. It can also be derived from a more general statement on the unbounded derived category D.Mod-R/. Indeed, using that over a commutative ring vanishing of HomD.Mod-R/ .X; Y Œn/ is determined locally, one can show that every compact exceptional object X 2 D.Mod-R/ is projective up to shift, cf. [7]. So, all tilting modules are either equivalent to R or large. A classification of tilting and cotilting modules over commutative noetherian rings will be given in Section 8.3.

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3.3 A tilting and cotilting abelian group. Let us focus on the special case R D Z. A complete classification of the tilting Z-modules will be given in Section 8.1. We start by considering the Z-module Q ˚ Q=Z: It is a tilting and cotilting module. Indeed, the short exact sequence 0 ! Z ! Q ! Q=Z ! 0 gives condition (T3) in Definition 2.2, and the remaining conditions follow from the fact that Q ˚ Q=Z is an injective cogenerator of Mod-Z. The tilting class generated by Q ˚ Q=Z is the class D of divisible groups, i.e. the class of Z-modules M such that M D rM for all 0 6D r 2 Z. The corresponding torsion-free class R is the class of reduced groups, and the tilting torsion pair .D; R/ is a split torsion pair. The cotilting class cogenerated by Q ˚ Q=Z is Mod-Z. The dual module D.Q˚Q=Z/ D HomZ .Q˚Q=Z; Q=Z/ is a cotilting Z-module. Using Proposition 6.2, one proves that its cotilting class is the class of all torsion-free groups, i.e. the class of Z-modules M such that for 0 6D r 2 Z and 0 6D m 2 M always rm 6D 0. So, the cotilting torsion pair induced by D.Q ˚ Q=Z/ is the classical torsion pair whose torsion class is formed by the torsion groups, i.e. by the Z-modules M such that for all m 2 M there is 0 6D r 2 Z with rm D 0. Observe that D.Q ˚ Q=Z/ is not tilting. Indeed, condition (T2) fails because the two direct summands Q and D.Q=Z/ satisfy Ext1Z .Q; D.Q=Z/.N/ / 6D 0, see [45, V.2]. 3.4 Two large tilting modules over hereditary algebras. Let R be a (connected) hereditary Artin algebra of infinite representation type. The Auslander–Reiten-quiver of R is of the form

… p

…

… …

t

q

where p is the preprojective component, q is the preinjective component, and t consists of a family of regular components. Let us consider the following torsion pairs in Mod-R: (1) [80], [79]. The torsion pair .D; R/ cogenerated by t is a cofaithful torsion pair R

… p

…

t

D … …

q

induced by a large tilting module W 2 Mod-R. The shape of W in the tame case is described below, for the wild case we refer to [70, Section 4].

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By the Auslander–Reiten formula Gen W D D D o t D t? . The modules in D are called divisible. (2) The torsion pair .T ; F / generated by t is a faithful torsion pair induced by the cotilting module D.R W/, where R W 2 R-Mod is the left version of W. This follows from Proposition 6.2. The modules in F are called torsion-free, the modules in T are called torsion. The class Cogen D.R W/ D F D to D ? t consists of all direct limits of preprojective modules, and T D Gen t, see [6, 5.4] or Lemma 6.7. (3) [71], [61] The torsion pair .L; P / cogenerated by p is a cofaithful torsion pair P …

p

L

…

… t

…

q

induced by a large tilting module L. The class Gen L D L D o p D p? consists of the modules without indecomposable preprojective summands. The class P is the class of preprojective modules from [80, 2.7]: a module X belongs to P if and only if every non-zero submodule of X has a direct summand from p. The module L is called Lukas tilting module. It is constructed as follows. One defines inductively a chain in add p starting with A0 D R, and choosing at every step an embedding An AnC1 2 add p such that AnC1 =An S 2 add p and HomR .AnC1 ; n R/ D 0. This is possible by [80, 2.5]. Now set L0 D n0 An and take the short exact sequence 0 ! R ! L0 ! L1 ! 0. The module L D L0 ˚ L1 certainly satisfies (T1) and (T3), it is add p-filtered according to the definition in Section 4.2, and one verifies that it is a tilting module with the desired tilting class, see [70], [71] and [61, 3.3]. Notice that L has no finite-dimensional direct summands. Further, Add L D Add L0 by [71, 3.2], and when R is the Kronecker algebra or of wild representation type, then even Add L D Add M for any non-zero direct summand M of L, see [71, 3.1] and [70, 6.1]. (4) The torsion pair .Q; C / generated by q is a faithful torsion pair induced by the cotilting module D.R L/, where R L 2 R-Mod is the left version of L. This follows again from Proposition 6.2. The class Cogen D.R L/ D C D qo D ? q consists of the modules without indecomposable preinjective summands, and Q D Add q by [80, 3.3]. Notice that the tilting modules L and W determine the representation type of R: Theorem 3.1 ([80], 3.7–3.9, [6], Theorem 18). Let R be a (connected) hereditary Artin algebra. The following statements are equivalent. (1) R is of tame representation type.

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(2) The torsion pair .Q; C / splits. (3) The module L is noetherian over its endomorphism ring. (4) The class Add W is closed under direct products. Assume now that R is of tame representation type. Then W is a cotilting module equivalent to D.R L/, that is, Cogen W D C , see [79]. Moreover, the module W has an indecomposable decomposition L W D G ˚ fall Prüfer modules S1 g: Here, G denotes the generic module, that is, the unique indecomposable infinite dimensional module, up to isomorphism, having finite length over its endomorphism ring (which is a division ring), or in other words, G is the unique indecomposable torsion-free divisible module, see [80, 5.3 and p.408]. Further, for each quasi-simple module S 2 t, we denote by Sm the module of regular length m on the ray S D S1 S2 Sm SmC1 and let S1 D lim Sm be the corresponding Prüfer module. The adic module S1 ! determined by S is defined dually as the inverse limit along the coray ending at S . With similar arguments as in Section 3.3 one can now verify that D.R W/ is not a tilting module. Indeed, D.R W/ is isomorphic to the direct product of G and of all adic modules S1 , and condition (T2) fails as Ext1R .G; S1 .N/ / 6D 0 by [76]. For more details on tilting and cotilting modules over tame hereditary algebras we refer to Example 6.9 and to Sections 8.2 and 8.5. Examples of tilting modules of projective dimension bigger than one are discussed in Section 5, after providing some background on the connection between tilting and approximation theory.

4 Tilting and approximations In this section, we review some fundamental results from approximation theory, and we describe tilting classes in terms of the existence of certain approximations. 4.1 Resolving and coresolving subcategories. Recall that a full subcategory ModR (or modR) is resolving if it satisfies the following conditions: (R1) contains all (respectively, all finitely generated) projective modules, (R2) is closed under direct summands and extensions, (R3) is closed under kernels of epimorphisms.

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Coresolving subcategories of ModR (or of modR) are defined by the dual conditions. For example, mod-R is resolving, cf. [4, 1.1], and P and P 0 and all A 2 A. So pdimA D n implies X 2 A? . In fact, the properties of .A; B/ established above characterize the cotorsion pairs induced by tilting modules. Theorem 4.6 ([2], [92]). Let B Mod-R and A D ? B. The following statements are equivalent. (1) B is a tilting class. (2) .A; B/ is a complete hereditary cotorsion pair such that A P and A \ B is closed under coproducts. (3) .A; B/ is a hereditary cotorsion pair such that A P and B is closed under coproducts. Proof (Sketch). As verified above, (1) implies that A P and that A \ B and B are closed under coproducts. For (2) ) (1), observe first that since A P and A is closed under coproducts, the projective dimensions attained on A are bounded by some n 2 N. Moreover, the completeness of .A; B/ allows an iteration of left B-approximations yielding a long exact sequence f0

f1

0 ! R ! B0 ! B1 !

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13

with Bi 2 B and AiC1 D Coker fi 2 A for all i . Since R 2 A, we even have Bi 2 A \ B. But also An 2 A \ B, because ExtiR .A; An / Š ExtiCn R .A; R/ D 0 for all A 2 A and i > 0. The module T D B0 ˚ ˚ Bn1 ˚ An then satisfies Add T A \ B, and one checks that it is a tilting module with tilting class B. For (3) ) (2) one has to prove completeness. In fact, by refining set-theoretic methods originally developed by Eklof, Fuchs, Hill, and Shelah, it is proved in [92] that A is deconstructible, i.e. the modules in A are filtered by “small” modules from A. Here “small” means that the modules admit a projective resolution consisting of projectives with a generating set of bounded cardinality, and the bound can be chosen as the least infinite cardinal such that every right ideal of R has a generating set of cardinality at most . Now the “small” modules from A form a set, and it follows from Lemma 4.1 that the cotorsion pair .A; B/ is generated by this set. So .A; B/ is complete by Theorem 4.4. There is a dual characterization of cotilting classes, see [2, 4.2], and [84, 2.4]. Notice that these results generalize classical results for finitely generated tilting and cotilting modules over Artin algebras due to Auslander and Reiten [15]. We obtain the following consequence for 1-tilting modules. Corollary 4.7 ([13]). The tilting torsion pairs are precisely the torsion pairs .T ; F / in Mod-R such that for every R-module M (or equivalently, for M D R) there is a short exact sequence 0 ! M ! B ! A ! 0 with B 2 T and A 2 ? T . For a dual version of this result, see [13, 2.5] or Corollary 6.6.

5 Further examples Here is an immediate application of Theorem 4.6. Corollary 5.1. Let be a set of modules with pdim n. If ? is closed under coproducts ( for instance, if mod-R), then it is an n-tilting class. 5.1 Complements. As a consequence, we obtain the existence of complements to partial tilting modules. Notice that this fails when restricting the attention to finitely generated modules, as shown by the example in [78]. We say that a module M 2 Mod-R is a partial tilting module if it satisfies conditions (T1) and (T2). Dually, M is a partial cotilting module if it satisfies conditions (C1) and (C2).

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Theorem 5.2 ([3]). (1) Let M 2 Mod-R be a partial tilting module. There is N 2 Mod-R such that T D M ˚ N is a tilting module with tilting class T ? D M ? if and only if M ? is closed under coproducts. (2) Let M be a pure-injective partial cotilting module. There is N 2 Mod-R such that C D M ˚ N is a cotilting module with cotilting class ? C D ? M if and only if ? M is closed under products. Proof (Sketch of the if-part). (1) The class B D M ? is a tilting class by Corollary 5.1. A tilting module T with B D T ? is obtained by taking a sequence 0 ! R ! N ! A ! 0 with N 2 B and A 2 A and setting T D M ˚ N . Statement (2) is proven dually by employing Theorem 4.4 (2). In particular, it follows that every partial tilting module M 2 mod-R has a complement N as in 5.2 (1). Over an Artin algebra R, also the dual result holds true. Indeed, every M 2 mod-R is pure-injective, and using the Auslander–Reiten formula, it is shown in [65, 6.4] that ? M is closed under products. Corollary 5.3 ([65]). If R is an Artin algebra, then every partial cotilting module M 2 mod-R has a complement N as in 5.2 (2). In general, however, the complement N to a partial (co)tilting module M 2 mod-R will be infinite dimensional. For the case of a hereditary Artin algebra, see also [62]. 5.2 A tilting module determining the finitistic dimension. In this section, let R be a right noetherian ring. The big and the little finitistic dimension of R are defined as Findim R D pdim P and findim R D pdim P 0. A contour .v; w/ is essential if v and w are not interlaced and deep if vE is deep. It is clear that the ‘kernel’ of the natural presentation PQ ! AE is the smallest stable equivalence relation such that each minimal zero-path is equivalent to 0 and v is equivalent to w for each essential contour. Since the spaces A.x; y/ and As .x; y/ always have the same finite dimension we get that A is isomorphic to As iff there is a presentation such that all minimal zero-paths and all the differences v w coming from an essential contour .v; w/ are annihilated by .

3 The structure of mild k-categories 3.1 The main results. In this section we explain the main results from [3] and their proofs, but in their generalized and simplified versions made possible by [11], [14], [30], [41]. Theorem 3.1. Let A be a distributive category such that the ray-category AE is mild. Then we have: a) A is standard if the characteristic is not 2. b) A has always a filtered multiplicative basis.

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In particular by this and the next theorem there are only finitely many isomorphism classes of representation-finite algebras in each dimension. This answers a question asked by Gabriel in [36]. Theorem 3.2. Let A be a distributive category. a) A is mild resp. locally representation-finite iff As is so. In that case the Auslander– Reiten quivers are isomorphic. b) If A is mild and if it has a faithful indecomposable it is standard. In particular a minimal representation-infinite algebra is standard. The central statement is part a) of Theorem 3.1. To describe its proof let Q be the E If W kQ ! A and 0 W kQ ! A are two presentations common quiver of A and A. E One gets all of A, the rays of .˛/ and of 0 .˛/ coincide and we call this ray ˛. presentations just by all choices of elements in the various rays corresponding to the arrows of Q. As explained at the end of the last paragraph we have to find a presentation that annihilates all minimal zero-paths and all the differences v w coming from essential contours .v; w/. Starting from an arbitrary presentation we will achieve this in three steps explained in Sections 3.3 to 3.5. Step 1: The minimal zero-paths have length 2 and two different ones have no arrow in common, so that by changing the choice for one arrow in each minimal zero-path one can annihilate all of them. Step 2: There are only three types of non-deep essential contours allowed and they are pairwise disjoint. In char k ¤ 2 there is for each such contour a new choice of one arrow occurring in the contour – but not in a minimal zero-path – such that v w is annihilated. Step 3: For each arrow ˛ we multiply the element .˛/ 2 ˛E chosen before by an appropriate non-zero scalar to annihilate the differences v w for all contours. The existence of these scalars is equivalent to the vanishing of a certain cohomology group. The proof of step 3 given on three pages in [3, 8.3–8.6] is very elegant whereas the proofs of the first two steps require a careful local analysis of AE affording some endurance which is only at the very end rewarded by the nice structural results one obtains. The main working tool for the proofs of the first two steps is described in the next section: the cleaving diagrams due to Bautista, Larrión and Salmerón, see Section 3 in [3]. 3.2 Cleaving diagrams. A diagram D in a ray-category P is just a covariant functor F W D ! P from another ray-category D to P respecting zero-morphisms. Then F is called cleaving iff it satisfies the following two conditions and their duals: a) F D 0 iff D 0; b) If ˛ D.x; y/ is irreducible and F W F x ! F z factors through F ˛ then factors already through ˛. These conditions are very easy to verify. For any diagram F W D ! P the restriction F W P -Mod ! D-Mod has a left adjoint F and if F is cleaving, any M in D-Mod is a direct summand of F F M

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[3, Sections 3.2, 3.8]. It follows that P is not locally representation-finite resp. satisfies BT 2 if D does so. In this article D will always be given by its quiver QD , that has no oriented cycles, and some relations. Two paths between the same points give always the same morphism, and zero relations are written down explicitly. The diagram F W D ! P is then defined by drawing the quiver of D with relations and by writing the morphism F ˛ in P close to each arrow ˛. For example let D be the ray-category with the natural numbers 0; 1; : : : as objects and with arrows 2n ! 2n C 1 and 2n C 1 2n C 2 for all n or let Dk be the full subcategory of D supported by the natural numbers k where k 1. Then a cleaving functor from D resp Dk to P is called an infinite zigzag resp. a zigzag of length k. @ 1

2 @ R @

@ 3

@ R @

@

@ R @

@ @ R @

- - - - - - - - - -

Figure 2

A functor from D resp. Dk to P is just an infinite resp. finite sequence of morphisms .1 ; 2 ; : : :/ in P such that 2i and 2iC1 always have common domain and 2i1 and 2i common codomain. The functor is cleaving iff none of the i factors through one of its ‘neighbored’ morphisms. The domain of 1 is called the start of the zigzag. A crown in P is a zigzag that becomes periodic after n steps, i.e., one has 1 D nC1 for some even n 4. A zigzag is called low if none of the i is a profound morphism. The ray-category P is zigzag-finite resp. weakly zigzag-finite if in each point only finitely many zigzags resp. low zigzags start. By König’s graph theorem this means that there is no infinite zigzag in P resp. no infinite low zigzag. For instance any mild ray-category is weakly zigzag-finite. In [3, Section 3] the cleaving functors are defined for k-linear categories and applied in that context. There the definition and the verification that a given functor is cleaving are much more difficult. Therefore it is important that the whole proof of Theorem 3.1 can be done at the elementary combinatorial level whereas the transfer from AE to A is postponed to Theorem 3.2 which is also proved by elementary means. The possibility to proceed like that is already mentioned in [3, Section 3.8 c)], but there it would require some non-elementary results from [15], [17] and the proof of Theorem 3.2 even depends on BT 2. 3.3 Zero relations and critical paths. There are two types of minimal zero-paths v D ˛n ˛n1 : : : ˛2 ˛1 in A. Either v is annihilated by each presentation or not in which case we call v a critical path. Theorem 3.3. Let AE be mild. then we have: a) (Structure theorem for critical paths) Each critical path has length 2.

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b) (Disjointness theorem for critical paths) Two critical paths sharing an arrow are equal. The proof of this theorem is a relatively easy and short application of the technique of cleaving diagrams. It is sketched in [39, Section 13.11] and given with full details in [3, Section 4]. Note that for a critical path ˛2 ˛1 the first ray ˛E1 is not cotransit, the second not transit. Now for each critical path v the initially given presentation can be changed at one arrow of the path to annihilate v and this can be done for all critical paths at once because of the disjointness theorem. We end up with a presentation annihilating all zero-paths. 3.4 Commutativity relations and non-deep contours. For a contour C we denote by P .C / the full subcategory of AE supported by the points occurring as starting or ending points of arrows in v or w and by Q.C / the quiver of P .C / which is in general not a subquiver of Q. Figure 3 describes three (families of) ray-categories by quiver and relations. Each of these contains a non-deep contour .v; w/ and vE is always bitransit. For obvious reasons the contours C as well as the categories P .C / are called penny-farthings, dumb-bells and diamonds respectively. -q *q HH HH jq qp xn1H pp A pp p ˛An A p A ppq p AUq x D y D x 0 pp 6 pp pp ˛1 n 1; ˛1 ˛n D 0 pp v D ˛n : : : ˛1 ; w D 2 pq x1 q 0 D ˛e.i / : : : ˛1 ˛n : : : ˛i C1 YH H e W f1; : : : ; n 1g ! f1; : : : ; ng HH H q q e non-decreasing

p -p? ? xl yl v D ; w D r D 0 D s minfr; sg D 3; maxfr; sg 5

rz > Z } Z Z ZZ Z˛ Z Z ~r y Z x r Z = Z > Z ıZ ˇ Z ~ r Z t

v D ˇı, w D ˛, D 0, ˛ D Figure 3

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Theorem 3.4. Let A be a distributive category such that AE is mild. Then we have: a) (Structure theorem for non-deep contours) Any non-deep essential contour of A is equal or reverse to a dumb-bell, a penny-farthing or a diamond. b) (Disjointness theorem for non-deep contours) Two essential non-deep contours sharing a point are equal or reverse. The proof of part a) given in [3, Sections 5 to 7] is complicated especially for the case of diamonds. There is a simpler proof in [14, Section 2] using the following obvious strategy. For a non-deep essential contour C D .v; w/ we choose paths v D v1 : : : vn and w D w1 : : : wm from x to y. Up to duality it suffices to consider the case that vE is transit. We also choose a path p D p1 p2 : : : pr such that D pE generates the E y/ and we use the abbreviations ˛ D vE1 , ˇ D w Å1 , D vE2 : : : vEn and radical of A.y; Åm . Then the contour induces in AE the diagram shown in Figure 4. ıDw Å2 : : : w : ˛ @ R @ PP * P P q ˇ ı P

: ˛ @ R? @ PP * P q ˇ P ı P Figure 4

The proof of part a) is just a careful analysis of the fact that the obvious subdiagram z 5 or some diagrams deduced from it cannot be cleaving if AE is mild. Part b) of type D is proved in [14] and originally in [3] under the stronger assumption that the contours share an arrow. By the definition of a non-deep contour C D .v; w/ and because vE is bitransit there is always an invertible morphism C with .v/ D C .w/ and in case of a penny2 id C bC .C / because of .C /4 D 0. To get the new farthing C we have C D aC 0 presentation wanted in step 2 we set 0 .C / D C .C / for each dumb-bell C , b 0 .˛C / D C .˛C / for each diamond and finally 0 .C / D .aid C 2a .C /.C / for each penny-farthing. Here we use the fact that 2 is invertible in k. All these choices are independent of each other and we set 0 . / D . / for the remaining arrows. Then all v w coming from non-deep essential contours are annihilated by 0 and so are all zero-paths since we have only changed arrows that are bitransit. In a penny-farthing where ˛1 ˛n is a zero-path one can find in all characteristics a new presentation annihilating v w by setting 0 .˛1 / D .˛1 /C1 . All critical paths are still annihilated because ˛1 does not lie on a critical path. Thus the only penny-farthings that cause serious trouble in characteristic 2 are those where ˛1 ˛n is a critical path.

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3.5 Contour functions and cohomology. Let A be a distributive category with rayE A contour function is a map c that assigns to each contour .v; w/ of category P D A. A a non-zero scalar such that c.u; w/ D c.u; v/c.v; w/ and c.sv; sw/ D c.v; w/ D c.vt; wt/ hold whenever this makes sense. A contour function is called exact if there is a function e from the set of arrows of QP to k such that c.v; w/ D ı.e/ WD e.˛n /e.˛n1 / : : : e.˛1 /.e.ˇm /e.ˇm1 / : : : e.ˇ1 //1 where v and w are the paths ˛n : : : ˛1 and ˇm : : : ˇ1 . The set C.P / of all contour functions is an abelian group under pointwise multiplication and the set E.P / of all exact contour functions is a subgroup. As shown in [3, Section 8.1], the quotient H.P / is isomorphic to the second cohomolgy group E k / defined in the next section, but this is irrelevant for us. H 2 .A; Now let be a presentation annihilating all zero paths and non-deep contours. Then there is for each contour .v; w/ a uniquely determined non-zero scalar c.v; w/ such that .v/ D c.v; w/.w/ and this defines a contour function. Namely for a non-deep contour we have c.v; w/ D 1 and for a deep contour .v/ and .w/ are both generators of the one-dimensional socle of the bimodule A.x; y/. By the next theorem – called Roiter’s vanishing theorem in [3] – we have c D ı.e/ for some function e W Q1 ! k and then the new presentation 0 with 0 .˛/ D e.˛/1 .˛/ induces the wanted isomorphism k AE ' A. If AE is mild and there are four arrows starting or ending in a point of the quiver then E Thus we can assume in step 3 all compositions of irreducible morphisms vanish in A. that at most three arrows start or end in a point. Theorem 3.5. If P is a weakly zigzag-finite ray-category such that at most three arrows start or end at a point of its quiver then we have H.P / D 0. I give some details of the proof for two reasons: First, it is in my opinion the most ingenious single argument of the article on multiplicative bases, and second, the above statement is more general than the vanishing theorem proven in [3] only for zigzag-finite ray-categories. This generalization is due to Geiß who observed in his unpublished ‘Diplomarbeit’ that the proof of [3] still works for weakly zigzag-finite ray-categories. The reader should be warned that we have to change the definitions of [3] slightly to keep the arguments valid. If the quiver of P is finite and contains no oriented cycle, the proof of the vanishing theorem is easy and it is given in [5]. One considers a source in the quiver and proceeds by induction. This should also work in the general situation, but – to cite A. V. Roiter – the question is: Induction on what? Well, here are the definitions needed to create a kind of source or sink in a quiver with oriented cycles. A tackle of length n with start in y is just a low zigzag z D .1 ; 2 ; : : : ; n / with start in y whose last morphism n is irreducible. The efficiency e.z/ of the tackle is the word .d.1 /; d.2 /; : : : ; d.n // and we order these words lexicographically. The tackle is efficient if its efficiency is maximal among the efficiencies

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of the tackles starting in y. If P is mild only finitely many tackles start in a fixed point so that there is always an efficient tackle as soon as there is one. The key lemma reads as follows: Lemma 3.6. Let P be a ray-category and let z D .1 ; 2 ; : : : ; n / be an efficient tackle starting in y. Choose paths ri with rEi D i . Suppose that .v D qrn p; w D wm w 0 w1 / is an essential contour with paths q, p, w 0 and arrows w1 ; wm . Then p resp. q has length 0 if n is odd resp. even. Proof. We consider the case n D 5. This will make clear how to treat the general case. Suppose that pE is not an identity. Then the sequence z5 D .Er1 ; : : : ; rE4 ; qErE5 ; w Åm / contains no profound morphism and it cannot be a zigzag because then its efficiency is greater than that of z. Since v and w are not interlaced there exists a non-trivial path q3 such that qE3 rE4 D qErE5 . E w Å1 / not containing a profound Next consider the sequence z4 D .Er1 ; : : : ; rE3 ; rE4 p; morphism. Again this cannot be a tackle because then its efficiency is too big. Since v4 D q3 r4 p is interlaced with v it is not interlaced with w and so there is a non-trivial path p2 with rE3 pE2 D rE4 p. E Åm / and z2 D .Er1 ; rE2 pE2 ; w Å1 / and we find Similarly we look at z3 D .Er1 ; : : : ; qE3 rE3 ; w non-trivial paths q1 and pE0 with qE1 rE2 D qE3 rE3 and rE1 pE0 D rE2 pE2 so that finally the sequence z1 D .E q1 rE1 ; w Åm / is a tackle with larger efficiency than z. This contradiction ends the proof of the lemma.

p0

@ w 1 @ R @ p

p2 ) 9 @ @ r1 @ r2 r4 r5 @ r3 ? @ @ J @ @ J R @ R @ XXX H J XXX HH J XXX H q1 q XX H q3 XXX HH J XXX HJ X H ^ J j H z X

w0

? wm

Figure 5

As already said the foregoing lemma is just a slight generalization of [3, 8.4]. The point is that the domain of w1 behaves like a source in the remaining part of the proof of the vanishing theorem. We do not repeat the arguments, but only explain why they are still correct. The Section 8.5 remains valid for a weakly zigzag-finite ray-category as is easy to see and we look now at the proof of Lemma 8.6 in [3] thereby finishing the proof of Theorem 3.5. The only thing to be modified is the start of the induction.

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If there is no tackle in our new sense any arrow ˛ starting in y induces a profound morphism ˛E and we still can take a D 0 in the proof of Lemma 8.6. 3.6 The neighborhoods of non-deep contours. To go on with the proofs we need to know how a penny-farthing is related to the whole ray-category. This is analyzed in [11] using some partial results from [3]. A short complete proof is given in Section 4.2 of [14]. Theorem 3.7. Let C be a penny-farthing in a mild ray-category P . Suppose that P .x0 ; y/ ¤ 0 for some y not in C . Then we have n D 2 and we are in one of the following three situations: a) There is an arrow ˇ W x0 ! b and this is the only arrow outside C ending or starting in C . We have ˇ D 0, ıˇ D 0 for all arrows ı starting in b and y D b. b) There is an arrow W x1 ! c and this is the only arrow outside C ending or starting in C . We have ˛1 D 0, ı D 0 for all arrows starting in c and y D c. c) There is an arrow ˇ W x0 ! b as well as an arrow W x1 ! c. These are the only two arrows outside C ending or starting in C . We have 0 D ˇ, 0 D ˇ˛2 , 0 D ıˇ for all arrows starting in b, 0 D ˛1 , 0 D for all arrows starting in c and y D b ¤ c. In all three cases there are no additional arrows ending in b or c. Analogous results hold for dumb-bells and diamonds and this leads in [14] to the following result: Theorem 3.8. A minimal representation-infinite ray-category has no non-deep contour. 3.7 The case of characteristic 2. We have already seen at Riedtmann’s example that in characteristic 2 the linearizations of the ray-category and the stem-category are not always isomorphic and we know that this trouble is caused by certain penny-farthings. So let P be the set of all penny-farthings in A such that ˛E1 E˛En ¤ 0. For any subset N E The of P we define a base category AEN having the same objects and morphisms as A. composition of two morphisms E and E also coincides with the composition in AE except for the case where the domains and codomains all belong to the same penny-farthing E such that A is the product contained in N . In this case the composition is the ray of A and A in the stem-category. Thus AEN is a mixture of ray- and stem-categories. Theorem 3.9. Suppose the characteristic of k is 2. Let A be a distributive category such that AE is mild. Then we have with the above notations: a) The composition defined above in k AEN is associative. The category k AEN is E but it is not standard provided N is not empty. distributive with ray-category A,

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b) Each distributive category B with BÅ ' AE is isomorphic to k AEN for some subset N of P . Two such categories are isomorphic iff the subsets are conjugate under E the automorphism group of A. The proof of the first part uses some of the information contained in Theorem 3.7 and the proof of the second part runs parallel to the proof for characteristic different from 2. This is explained very well in [3, Section 9]. One uses again the vanishing of a certain cohomology group. 3.8 The proof of Theorem 3.2. We give some details because the proof of these results in [3, Section 9.7] is too complicated and the one in [39, Section 13.17] contains a minor error. Let A be a distributive category such that AE is mild. First we show that A is standard if it has a faithful indecomposable. As the proof of Theorem 3.1 shows this is clear E So let P be if there is no penny-farthing P as in Figure 3 with ˛Æ1 E˛Æn ¤ 0 in A. such a penny-farthing. If A.y; x0 / D 0 D A.x0 ; y/ holds for all y … P then A.x0 ; / is projective-injective, whence the only candidate for a faithful indecomposable, and annihilated by ˛1 ˛n . Thus, up to duality, we can assume A.x0 ; y/ ¤ 0 for some y … P . Then we are in one of the three situations described in Theorem 3.7. By a well-known result [62] the points b and c can always be separated into a receiver and an emitter and the quiver of A and AE is separated into a connected component containing x0 and at most two other components. Since there is a faithful indecomposable these components are actually empty and we are in one of the three situations described in Theorem 3.7. As shown in [11] by a direct calculation the Auslander–Reiten quivers of A and As coincide in all three cases and there is no faithful indecomposable. Next, let A be finite and representation-finite. Using the correspondence of Proposition 2.3 between ideals of A and AE we know by induction that AE is mild. If it is minimal representation-infinite it has a faithful indecomposable and so we have A ' k AE by the above, a contradiction. Thus AE is also representation-finite. Reversely, let AE be finite and representation-finite. Again by induction A is mild. If it is minimal representation-infinite it has a faithful indecomposable and we end up E again with the contradiction that A ' k A. We have shown for finite distributive categories that A is representation-finite iff AE is so. This implies easily that A is locally representation-finite iff AE is so. The analogous statement for mildness follows from the correspondence between the ideals E Finally, the Auslander–Reiten quivers are isomorphic because for each of A and A. penny-farthing one has either a projective-injective corresponding to the point x0 or one is in one of the three cases from Theorem 3.7. In the second case the Auslander– Reiten quivers of A and its standard-form are glued together by the same rules from the isomorphic Auslander–Reiten quivers obtained by separating all occurring points b or c into emitters and receivers. In the first case one knows how the Auslander–Reiten sequences with a projective-injective in the middle look like.

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4 The topology of a ray-category 4.1 The simplicial complex and the universal covering of a base category. The following material is from [3, Sections 1.10, 10]. For each base category B one defines a simplicial complex S B by taking the set of objects as S0 B and the set of n-tuples .n ; : : : ; 2 ; 1 / of composable morphisms with n : : : 2 1 ¤ 0 as Sn B. The face operators are defined in the usual way by dropping a morphism at the ends or by composing two in between and the degeneracy operators are defined by inserting identities (see [3, Section 1.10]). Let Cn B be the free abelian group with basis Sn B and define the differential dn W Cn B ! Cn1 B by the alternating sum of the appropriate face operators, i.e. by dn .n ; : : : ; 2 ; 1 / D .n ; : : : ; 2 / .n ; : : : ; 2 1 / C C .1/n .n1 ; : : : ; 2 ; 1 /: Then one obtains a chain complex C B whose homology groups are denoted by Hn B whereas H n .B; Z/ is the n-th cohomology group of Hom.C B; Z/ for any abelian group Z. A functor F W B 0 ! B between base categories is a covering if it satisfies the following conditions a), b) and the dual of b). Condition a) says that F D 0 is equivalent to D 0. Condition b) means that any point x in B can be lifted to a point x 0 and any W x ! y can be lifted to a unique 0 W x 0 ! y 0 . It follows that is irreducible iff F is, whence a covering induces a covering between the quivers. The covering W Bz ! B is ‘the’ universal covering if for any covering F W B 0 ! B any x in B with liftings x 0 in B 0 and xQ in Bz there is exactly one functor G W Bz ! B 0 with D F G and G xQ D x 0 . Then G is again a covering and even an automorphism for F D . The group of all these automorphisms is called the fundamental group …B of B and B is simply connected if this group is trivial. It is easy to see that one obtains the universal covering by the following construction. A walk w D ˛n : : : ˛1 of length n from x to y is a formal composition of arrows ˇ in QB and formal inverses ˇ 1 such that the domains and codomains fit together well and x is the domain of ˛1 , y the codomain of ˛n . Two walks v, w can be composed to the walk wv if the end of v is the start of w. The homotopy is the smallest equivalence relation on the set of all walks such that:1) ˛˛ 1 idy and ˛ 1 ˛ idx for all ˛ W x ! y, 2) v ' w and v 1 ' w 1 for all paths v; w mapped to the same non-zero morphism under the canonical presentation PQB ! B and 3) v w implies uv uw resp. vu wu whenever these compositions are defined. Now the points of the universal covering are the homotopy classes of walks with a fixed start x and the fundamental group consists of the homotopy classes with start and end in x. The multiplication is induced by the composition of walks. Since QB is connected this construction is essentially independent of the chosen base point x. The following ‘Hurewicz-isomorphism’ from [3, 10.4] is useful:

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Proposition 4.1. Let P be a connected ray-category with fundamental group … and universal covering Pz . Then …=Œ…; … is isomorphic to H1 P . In particular one has H1 Pz D 0. The elementary definitions above are familiar from algebraic topology, but for an arbitrary base category the construction of the universal covering following these lines is an impossible task because it involves the word problem for groups as shown in [23]. However, the reader can easily verify that in Riedtmann’s example B2 is simply connected whereas B1 admits the universal covering shown in Figure 6. Here the horizontal arrows are mapped onto the loop and the relations in the universal covering are the lifted ones.

q q q

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Figure 6

4.2 The main results for ray-categories. A ray-category P is called interval-finite if the quiver QP of P is directed and if there are only finitely many paths between any two points of P . For a point x in an interval-finite ray-category we denote by Px the set of all y ¤ x such that P .x; y/ ¤ 0. We order this set by y z iff the only non-zero morphism from x to z factors through y. Furthermore we denote by P x the full subquiver of QP consisting of the points y where no path ending in x starts. We say that x is separating if each connected component of P x contains at most one connected component of the Hasse-diagram of Px . Theorem 4.2 ([17], [8]). Let P be a weakly zigzag-finite interval-finite ray-category. a) H1 P D 0 holds iff H1 C D 0 holds for all finite convex subcategories C of P . In that case all objects are separating. b) If P is finite and all objects are separating then H1 P D 0 holds. The next result [30] of Fischbacher is of central importance. Theorem 4.3. Let W Pz ! P be the universal covering of a zigzag-finite ray-category P . Then we have: a) The fundamental group …P is free. b) H 2 .P; Z/ D 0 for all abelian groups Z. c) Pz is interval-finite.

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By this result and Section 5 one can always use covering theory instead of cleaving diagrams in the proof of the multiplicative basis theorem, but that does not abbreviate the arguments. Fischbacher’s proof of a) and b) is by an induction based on his ‘reduction-lemma’ which shows for a ray-category P with at least one contour – among other things – the existence of some arrows such that P and the quotient of P by the ideal generated by these arrows have the same fundamental group. The proof of the reduction lemma uses only the key Lemma 3.6 so that parts a) and b) remain valid if P is only weakly zigzag-finite. The proof of part c) is based on Theorem 4.2 and it does no longer work in the weakly zigzag-finite case. Nevertheless, we have the following result from [13] which plays an essential role in the proofs of BT 0 and of BT 2 in our sharper version. Theorem 4.4. Let P be a mild ray-category. Then the statements a), b) and c) of Theorem 4.3 are true. For the proof we can assume that P is minimal representation-infinite and that it contains an infinite zigzag Z and a profound morphism that is not irreducible. Any profound morphism occurs in Z infinitely many times because otherwise P = still contains the infinite zigzag consisting of the end of Z where no longer occurs. But if a zigzag contains three times the same morphism, one can construct a crown C in the obvious way. Thus P is finite. Now one proves the following crucial result [13]. Proposition 4.5. Let P be a minimal representation-infinite ray-category containing a crown and a profound morphism that is not irreducible. Then there is a profound morphism not occurring in an essential contour. The proof of the proposition takes 15 pages and it is at the moment the most complicated one mentioned in this survey so that it should be replaced by a better argument. However, the proof is similar to the proof for the existence of a multiplicative basis: the main problem is to find a finite strategy and this consists of a rather boring local part and a rather nice global part. All this is explained well – I hope – in Section 2.2 of [13]. The theorem is then an easy consequence of the proposition. Namely, it follows directly from the construction of the universal coverings that the fundamental groups of P and P = coincide as well as the quivers of Pz and P = and also H 2 .P; Z/ embeds into H 2 .P =; Z/. The following fact can be proven with the above proposition and the finiteness criterion.

e

Proposition 4.6. Let P be a minimal representation-infinite ray-category. Then there is always a profound morphism not occurring in an essential contour. Back in 1983 I tried to prove this directly thereby obtaining a proof for the existence of interval-finite universal coverings based on the representation-finite case. I am still wondering whether there is such a direct proof.

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5 Covering theory 5.1 Coverings of translation quivers and k-linear covering functors. As far as I know the place where coverings are used for the first time in the representation theory of algebras is the paper ‘Group representations without groups’ [35] by Gabriel and Riedtmann. They consider coverings of the ordinary quiver, but soon after Riedtmann started her work on representation-finite selfinjective algebras in [66] by looking at coverings of the (stable) Auslander–Reiten quiver. This point of view was further developed by Gabriel in [15] where all the following material comes from. My contribution to that article – stated clearly in the introduction of [15] – was only to improve some results and to work together on Section 6 on simply connected algebras. A translation quiver .; / is a pair consisting of a locally-finite quiver without loops and double-arrows and a bijection W X ! Y between two subsets of 0 such that for all x in X there is an arrow ˛ W y ! x in iff there is an arrow ˛ W x ! y. We denote the set of these y as . x/C D x and call the vertices in 0 n X projective, in 0 n Y injective. The full subquiver supported by x, x , x is called a mesh and the mesh-category k./ Pis the quotient of the path category k by the ideal generated by all mesh relations ˛ W y!x ˛ .˛/ for x in X . The translation quiver is stable if X and Y coincide with 0 . The most important examples of (stable) translation quivers are (stable) Auslander–Reiten quivers. For any oriented tree T one has a stable translation quiver ZT . The underlying set is Z T and the translation is given by .z; x/ D .z 1; x/. There is an arrow .z; x/ ! .z 0 ; y/ iff either z D z 0 and there is an arrow x ! y in T or z D z 0 1 and there is an arrow y ! x in T . Of course ZT does not depend on the orientation of T but only on the underlying graph. A covering of translation quivers is a map f W 0 ! between the quivers with 0 f D f and such that x 0 is projective resp. injective iff f x 0 is so. It is clear how z ! , the fundamental group … and simply to define a universal covering W connected translation quivers. There is the following construction of the universal y by covering. Given a connected translation quiver .; / one defines a new quiver y adding a new arrow x W x ! x for each x in X . A walk in is a formal composition of old and new arrows and their formal inverses such that the occurring starts and ends fit together well and the composition of walks is defined in the obvious way. The homotopy is the smallest equivalence relation stable under left or right multiplication with the same walk and under ‘inversion’ and such that ˛˛ 1 is equivalent to an identity for each arrow ˛ – old or new – or each formal inverse and such that x is equivalent to ˛.˛/ for each arrow ˛ ending in a non-projective vertex x. The fundamental group is isomorphic to the set of all homotopy classes of walks starting and ending in x endowed with the multiplication induced by the composition of walks. The universal covering z has the homotopy classes with start in x as its points. The arrows, the translation z ! are all defined in a natural way. and the covering W In contrast to the case of base categories the universal covering has now always good properties which is due to the fact that the homotopy relation is homogeneous provided one gives the new arrows x the degree 2. There is a morphism of translation-quivers

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z to ZA2 which makes it possible to argue by induction. It follows in particular from z has no oriented that all paths between two fixed points have the same length and so cycles. A k-linear functor F W M ! N between two locally bounded k-linear categories is called a covering functor if it induces for all m in M and n in N isomorphisms M M M.m; m0 / ' N.F m; n/; M.m0 ; m/ ' N.n; F m/: F m0 Dn

F m0 Dn

For instance any covering f W 0 ! of translation quivers induces a covering functor k.f / W k. 0 / ! k./ provided both mesh categories are locally bounded. A locally bounded category C is an Auslander-category if it is isomorphic to the full subcategory ind A of the indecomposables over some locally representation-finite category A or – equivalently – if C has global dimension at most 2 and any projective p in C admits an exact sequence 0 ! p ! i0 ! i1 where the ik are projective and injective [1, Section VI.5]. One has the following results. Theorem 5.1 ([15]). Let A be a locally representation-finite category with Auslander– Reiten quiver A . Then there is a covering functor zA / ! ind A: F W k. This is a version of Riedtmann’s theorem from [66] (Satz 2.3). Theorem 5.2 ([15], Proposition 3.5). Let F W C ! D be a covering functor. Then C is an Auslander category iff D is. Combining these two results we see that for any locally representation-finite category A the full subcategory Aos of k.A / consisting of the projective points is locally representation-finite. We call this the old standard form of A. 5.2 Galois coverings. Around 1980 it was clear to many people that it would be good to have a covering theory that is induced by some group action and independent of the Auslander–Reiten quiver. There are several more or less equivalent ways to obtain such a theory (see e.g. [44], [45], [46], [24], [82]), but I follow Gabriel who presents in [37, Section 3] on 8 pages more than we need. This theory was generalized by Dowbor and Skowro´nski in [25]. Let G be a group of k-linear automorphisms of a locally bounded category A. We assume that G acts freely on A, i.e., ga ¤ a for g ¤ 1, and locally bounded, i.e., for given a, b in A there are only finitely many g with A.a; gb/ ¤ 0. Then there is a quotient F W A ! A=G and this is a covering functor. In fact, a covering functor E W A ! B with Eg D g for all g in G induces an isomorphism A=G ' B iff E is surjective on the objects and G acts transitively on all fibres. Such an E is called a Galois covering. Of course, the action of G on A induces an action m 7! mg on A-mod by scalar extension. This action is free if m 6' mg holds for all g ¤ 1 and m ¤ 0.

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Theorem 5.3 ([37]). Let A be a locally bounded connected category and G a group of automorphisms acting locally bounded on A and free on A and A-mod. Denote by F the quotient and by F W A-mod ! A=G-mod the left adjoint to the restriction. a) F is exact. It preserves dimensions and indecomposability. b) A is locally representation-finite iff A=G is so. In that case F induces a Galois covering ind A ! ind.A=G/ and a covering A ! A=G between the Auslander–Reiten quivers. Here F is simply defined by ‘taking direct sums of vector spaces and linear maps’ so that the theorem is very helpful. But the problem is to find for a given B a category A and a group G with B ' A=G such that first the theorem applies and second A-mod has better properties than B-mod. Now the preceding theorem can be applied to the universal covering W Pz ! P of a ray-category if the fundamental group G is free. Namely G acts freely and locally bounded on k Pz and freely on Pz -mod because G is torsion-free. Thus we see that P is locally representation-finite iff Pz is locally representation-finite. This is very useful once we know that the Auslander–Reiten quiver of k Pz or large portions thereof can be easily determined. This will be guaranteed in many cases by the next results. 5.3 Coverings of ray-categories and of Auslander–Reiten quivers. Let A be a directed Schurian algebra. For any a in the quiver Q of A let Qa be the full subquiver consisting of the points b such that there is no path from b to a. The point a is separating if the supports of different indecomposable direct summands of the radical of A.a; / lie in different connected components of Qa. If Pz is an interval-finite universal covering of a ray-category, then each finite convex subcategory A of k Pz satisfies these assumptions. As a slight generalization of the separation criterion due to Bautista–Larrión one has: Theorem 5.4 ([2], [8]). Let A be a (connected) Schurian directed algebra such that each point is separating. a) If A is representation-finite it has a finite simply connected Auslander–Reiten quiver. b) If A is minimal representation-infinite A has a simply connected component consisting of the 1 -orbits of all indecomposable projectives. Next we prove amongst other things that the two definitions of the standard form coincide for a locally representation-finite category. This was shown for the first time in [17]. Theorem 5.5. Let A be a locally representation-finite category with associated raycategory P having universal covering W Pz ! P . Then we have: a) k Pz is simply connected.

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b) induces a universal covering 0 W k Pz ! kP . Furthermore, …P is isomorphic to …kP . c) kP is isomorphic to the full subcategory of k.kP / formed by the projective points. d) As and Aos are isomorphic. Proof. a) Recall that P and therefore also Pz are supposed to be connected. We know that A and its standard category As D kP have the same Auslander–Reiten quiver from Theorem 3.2. Furthermore Pz is interval-finite and the fundamental group …P is free by Theorem 4.3. Finally, Pz and P are locally representation-finite by Theorem 5.3 and each finite connected convex subcategory C of Pz has a simply connected Auslander– Reiten quiver by the last theorem. Now one shows with the usual arguments that the Auslander–Reiten quiver of a connected locally representation-finite category is connected. Thus k Pz is connected. To show that it is even simply connected let x; y be two points in k Pz and v; w two paths from x to y consisting of irreducible maps. Then there are only finitely many modules occurring in v and w and there is a finite convex connected subcategory C of Pz such that all these modules are in fact C -modules and the two paths are two paths formed by irreducible maps in C -mod. But C -mod has a simply connected Auslander–Reiten quiver and so the two paths have the same length. This implies that y z . It passes through only k Pz is interval-finite. Now, let w be any closed walk in kP finitely many points of k Pz which lie in a finite convex subset that consists only of C -modules for another finite connected convex subcategory of Pz . The original walk is yC and so it is null-homotopic. Here one uses the fact that in a finite simply a walk in connected translation quiver any walk w can be contracted to a point within the quiver y corresponding to the convex hull of the points occurring in w. b) Of course induces a Galois covering k W k Pz ! kP that in turn induces a covering 0 W k Pz ! kP by Theorem 5.3. This is isomorphic to the universal covering zkP ! kP because z is simply connected. Now any g in …P gives rise to an W kP automorphism of 0 . The resulting homomorphism …P ! …kP is bijective, because …P acts simply transitive on the fibres of 0 . c) We have k.k Pz / ' ind k Pz by Theorem 5.1. The quotient k.k Pz / ! k.kP / induces a quotient between the full subcategories formed by the projective points and this is the wanted isomorphism. d) This follows from c) because A and As have the same Auslander–Reiten quiver.

6 Two classification results 6.1 The structure of large indecomposables over simply connected representationfinite algebras. In this section we study the indecomposables over a representationfinite algebra with simply connected Auslander Reiten quiver. The support A of any

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indecomposable U is then a convex subcategory by [7], whence again simply connected, and U is a sincere A-module. So it suffices to classify the representation-finite algebras with simply connected Auslander–Reiten quiver having a sincere indecomposable together with all these modules. Call these algebras ssc in the sequel and denote by n.A/ the number of points of an algebra. In [6] I published a list of 24 infinite families of algebras together with some modules that contains all ssc algebras and their sincere indecomposables up to duality and up to some exceptional algebras with n.A/ 72. At the end of 1982 I had determined by computer also the exceptional algebras and their sincere indecomposables and verified that they occur only for n.A/ 13. I never published these results because the exceptional algebras are not very important and my results consisted in unreadable computer-lists only. It is however very remarkable that there are only finitely many ways to construct an indecomposable over an ssc algebra and the only way I know how to prove this is to do the classification of the large ssc algebras. The complete list of these algebras as given e.g. in [39, Section 10.6] does not really matter. But the next result that one can verify by a look at the list plays an important role later on in the proofs of the finiteness criterion and of BT 2. In a certain sense the truth of the BT 2 conjecture is equivalent to the first property of ssc algebras mentioned in the next result. Theorem 6.1. Let A be an ssc algebra having an indecomposable of dimension n 1000. Then A contains a line with at least n6 points. Moreover, A is the union of at most three lines. The simple method to obtain the 24 families is the inductive procedure to construct ssc algebras via one-point extensions as described in [15, Section 6]. The results [63], [54], [55] of Nazarova, Roiter and Kleiner on representations of partially ordered sets play an essential role and the proof is a nice interplay between Auslander–Reiten theory and results of the Kiev-school similar as in [71]. The 24 families as well as the bound 13 were later verified by Ringel in the last chapter of the book [72] by a different method and without computer. This method was also used by Dräxler to confirm my numerical results about the exceptional algebras in [27]. Finally two of my students produced pictures showing the zoo of exceptional algebras in [75]. Dräxler used his numerical results to derive the following interesting fact [26]: Theorem 6.2. Any indecomposable over an ssc algebra admits a basis such that all arrows are represented by matrices with only 0 and 1 as entries. We also need the next result from [10] which is proven by elementary algebraic geometry. Theorem 6.3. Any indecomposable over an ssc algebra is accessible.

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6.2 The critical algebras and tame concealed algebras. A Schurian directed algebra A D k AE with H1 AE D 0 is called critical if A is not representation-finite, but any proper convex subalgebra is. By Theorem 5.4, any critical algebra has a simply connected component C in its Auslander–Reiten quiver containing all indecomposable projectives. It follows easily that C is a full subquiver of ZT for some tree T . z n , n 4, or Theorem 6.4. The only trees occurring for critical algebras are of type D Ezn , 6 n 8. I announced this theorem at Luminy in 1982 and the proof as well as the possible z n appeared in [9]. The quite technical method for the proof is the same algebras for D as in Section 6.1. In 1983 I also determined by computer the critical algebras for the types Ezn , n D 6; 7; 8 using my previously obtained results for the ssc algebras. In particular I never used tilting theory as claimed in [78, page 247] In parallel work [47] Happel and Vossieck studied the representation-infinite algebras B having a preprojective component and such that the quotient B=BeB is representation-finite for each non-zero idempotent e. Using a theorem of Ovsienko [65] they derived the analogue of Theorem 6.4, namely that with the exception of a generalized Kronecker-algebra B is the endomorphism algebra of a preprojective tilting module over an extended Dynkin-quiver. Furthermore they classified all these so-called tame concealed algebras with the help of a computer by certain frames which give a complete description by quiver and relations. For types different from Azn the algebras studied by Happel and Vossieck are obviously critical and the two classes even coincide just because the numbers of isomorphism classes coincide. In fact, this is also clear for theoretical reasons if one knows that for critical algebras only extended Dynkin-quivers occur. As far as I know there is no simple proof for that even though von Höhne has verified this by hand in [80]. We state for later use: Theorem 6.5. The critical algebras are given by the frames of Happel and Vossieck. We do not reproduce the frames that can be found in the original articles and many other places, e.g. in [39]. We will refer to the list as the BHV-list. Besides Theorem 6.2 this list is the only result in this survey that depends on computer calculations. We leave it up to the reader whether this makes it trustworthy or not. The only place where the correctness of this list matters is in the finiteness criterion given later on. In the other ‘applications’, e.g. in the proofs of Section 3, one uses only the fact that some very special members of the list like certain tree-algebras or fully commutative quivers are not representation-finite. This can always be checked easily and it was known since a long time. Later on we also use the following non-trivial fact from [12] which is again proven with geometric methods like degenerations of modules. Theorem 6.6. Over a tame concealed algebra any indecomposable is accessible.

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7 Some applications 7.1 On indecomposables over representation-finite algebras. By Theorem 3.2 any representation-finite algebra with a faithful indecomposable U is standard. Let P D AE be the ray-category and W Pz ! P the universal covering inducing the Galois covering F W k Pz ! kP ' A. By Theorem 5.3 there is a k Pz -module Uz with F Uz ' U . Thus there are up to Galois covering only finitely many ways to construct an indecomposable over a representation-finite algebra. In particular we obtain from Theorems 6.3 and 6.2: Theorem 7.1. Any indecomposable over a representation-finite algebra is accessible and it admits a basis such that all arrows are represented by matrices containing only 0 and 1 as entries. For further applications the following facts are useful. Here an infinite line L in the universal covering Pz of a ray-category is called periodic if the stabilizer GL within the fundamental group is not trivial. The periodic length of L is then the number of GL -orbits on L (see [25]). Lemma 7.2. Let W Pz ! P be the universal covering of a ray-category with d elements such that Pz is interval-finite and weakly zigzag-finite. a) Let q ! z1 ! ! z t1 ! z t x0 q 0 be a line L of length l 0 0 in Pz with .z1 / ¤ .x / and .q/ D .q / and such that v D q ! z1 ! ! z t1 ! z t is a path. Let g be the element in the fundamental group with gq D q 0 . Then adding on the right end the path gv one gets another convex subcategory in Pz . Continuing that way in both directions one obtains an infinite periodic line L0 of periodic length l 1 at most. b) Any line of length 2d contains a ‘subline’ L that can be prolonged to an infinite periodic line of length at most 2d . c) If P is minimal representation-infinite and Pz contains an infinite periodic line then P is a zero-relation algebra. Proof. a) That one gets a convex subcategory by adding gv is an easy consequence of the fact that all points of Pz are separating by Theorem 4.2. Then one dualizes and adds gw on the right end where w ¤ v is the second subpath of L ending in the sink z t . This procedure goes on forever to the right side and after that one goes to the left. By construction g is in the stabilizer of the obtained line L0 and the periodic length is d 1 at most. b) Take a line L of length 2d and mark all sources q1 ; q2 ; : : : ; qr . Thus L looks like q1 ! qi ! qr !

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P and we get 2d jr D1 dim P .qj ; /. So there are three sources mapped onto the same point in P . After renumbering we can assume they are q1 , qi , qr . For two of them the situation of part a) occurs and we are done. c) Let L be a periodic line in Pz containing the source q. For each natural number n 1 let Ln be the subline of L of length n starting in q and going to the right side. Then there is an indecomposable representation Un of Pz with support Ln and dimension n. The pushdowns F Un are all indecomposable and annihilated by all contours. Infinitely many of them are faithful because P is minimal representationinfinite. Thus P is defined by zero-relations. Using parts of the lemma, the detailed structure of the large ssc algebras and k-linear covering functors I proved in [6]: Theorem 7.3. Let A be a basic representation-finite algebra of dimension d. Let u.A/ be the number of indecomposable A-modules and let u.d / be the supremum of the u.A/ when A runs through all representation-finite algebras of dimension d . Then we have: a) An indecomposable A-module has at most dimension maxf2d; 1000g. b) There is a constant C such that for all d 4 one has 2 22d C7 C Cd:

p d

u.d / 9d 6

The mere existence of upper bounds was proved in [52] by methods from model theory. To make these bounds concrete was back in 1981 one of my motivations for the classification of the ssc algebras. 7.2 A criterion for finite representation type. Given a finite dimensional algebra A as a subalgebra of some endomorphism algebra or by generators and relations it might be impossible to find a quiver Q and an admissible ideal I such that A and kQ=I are Morita equivalent. So we assume in our criterion right from the beginning that A D kQ=I holds and that we know the dimension d of A. Then it is easy to find out whether A is distributive and to determine the ray-category AE and one has the following criterion for finite representation type. Theorem 7.4. Let A D kQ=I be a connected distributive algebra of dimension d given by a quiver and an admissible ideal. Let W Pz ! P be the universal covering E Then A is representation-finite iff it satisfies the following two conditions: of P D A. a) P contains no zigzag of length 2d . b) Pz contains no algebra of the BHV-list as a full convex subcategory. Proof. By Theorem 3.2 we know that A is representation-finite iff P is so. For the easy implication let P be representation-finite. A zigzag Z of length 2d in P would contain one non-invertible morphism at least three times. Thus we find a

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crown in P which is impossible. So P is zigzag-finite. By Theorems 4.3 and 5.3 one gets that Pz is locally representation-finite and so it contains no algebra of the BHV-list as a convex subcategory. Here we do not need any classification but only the fact that all algebras in the BHV-list are representation-infinite. Reversely, P is zigzag-finite whence P is representation-finite iff Pz is locally representation-finite by Theorems 4.3 and 5.3 again. Observe that Pz is interval-finite and satisfies H1 Pz D 0. If Pz is not locally representation-finite there are two cases possible. First assume that there is a finite convex subcategory B which is not representation-finite. Then one finds also a critical convex subcategory C by removing certain sinks or sources of B. Then C is an algebra of the BHV-list by Theorem 6.5 which is a contradiction. If all finite convex subcategories are representation-finite then there is a point x in Pz such that there are infinitely many indecomposables U with U.x/ ¤ 0. The supports of these modules can get arbitrarily large. By Theorem 6.1 the convex support of an indecomposable of dimension at least 12d C 1000 contains a line of length 2d . Thus Pz contains an infinite line by Lemma 7.2, whence a zigzag Z. Its push-down F Z is a zigzag in P which is again a contradiction. Note that for this implication we need both classification results. I announced a criterion similar to the one above in 1982 at Luminy. Of course I had to make more assumptions because a lot of theorems entering the proof above were not yet proven at that time. However also the original criterion [8] needed more than the first Jans condition and preprojective tilting. Fischbacher used the criterion in [31] for the classification of all maximal representation-finite and minimal representation-infinite algebras with three points. If the reader has the energy to apply the criterion to some cases where the fundamental group is free in two generators and the algebra is not defined by zero-relations he will appreciate very soon that one has to look at convex subcategories of Pz only. A detailed example is given in [8]. The criterion leads to an algorithm [23] that decides in ‘polynomial’ dependence of the dimension of A whether A is representation-finite or not even though the number of indecomposables can grow exponentially by Theorem 7.3. We end this section with the following statement whose proof is left to the reader. The field k is here arbitrary. Theorem 7.5. Let P be a ray-category. a) If P is minimal representation-infinite it is finite. b) If kP is locally representation-finite for one field it is so for all fields. In that case the Auslander–Reiten quivers, the dimension-vectors of the indecomposables etc. are independent of k.

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7.3 The proofs of Brauer–Thrall 0 and Brauer–Thrall 2. Now we prove the sharper version of BT 0 proposed by Ringel. Theorem 7.6. Let A be a basic finite-dimensional algebra. If there is an indecomposable of dimension n there is also an accessible module of that dimension. Proof. For representation-finite algebras all indecomposables are accessible by Theorem 7.1 and for non-distributive algebras the theorem holds by Ringel’s result from Section 2.2. Thus we can assume that A is distributive and minimal representationinfinite whence standard by Theorem 3.2. The universal covering Pz is not locally representation-finite and interval-finite with free fundamental group by Theorem 4.4. If we find a critical algebra B as a convex subcategory any indecomposable B-module is accessible by Theorem 6.6. Their images under F provide accessible modules in all dimensions. If all finite subcategories of Pz are representation-finite there are arbitrarily large indecomposables over a representation-finite convex subcategory. These indecomposables are accessible by Theorem 7.1 and again their images under F give accessible modules in all dimensions. Observe that this proof uses none of the two classification results. Moreover Theorem 1.1 (but not Theorem 7.6) remains valid if k is a splitting field for A, i.e., if all simple A-modules of finite dimension have trivial endomorphism algebra k. Our version of Brauer–Thrall 2 reads as follows: Theorem 7.7. Let A be a basic representation-infinite algebra of dimension d . Then there is a natural number e maxf30; 4d g and pairwise non-isomorphic e-dimensional indecomposables Ui , i 2 k , such that for any n 1 there exist pairwise non-isomorphic indecomposables Un;i having a chain of n C 1 submodules such that all successive quotients are isomorphic to Ui . Proof. We can assume that A is minimal representation-infinite. If A is not distributive there are two idempotents e; f and two linearly independent elements v; w in eAf annihilated by the radical of A. For any natural number n and any element i in k we consider the morphism .n; i / W .Ae/n ! .Af /n given by the matrix vEn C w.iEn C Nn / and its cokernel U.n; i / of dimension n.dim Af 1/. Here En is the n n-identity matrix and Nn a nilpotent n n Jordanblock. A simple direct calculation shows that the U.n; i / are pairwise non-isomorphic indecomposables admitting an exact sequence 0 ! U.n 1; i / ! U.n; i / ! U.1; i/ ! 0. If A is distributive and minimal representation-infinite it is standard by Theorem 3.2 and we call its ray-category P . By Theorems 4.4 and 5.3 the universal covering Pz is interval-finite with free fundamental group and Pz is not locally representation-finite. Thus Pz contains a critical convex subcategory C or all finite convex subcategories are representation-finite whence Pz contains a line of length 2d by Theorem 6.1.

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First let C be a convex subcategory of Pz which is tame concealed of type Ezn . The wanted families of modules are then the push-downs of the obvious P1 .k/-families of indecomposable C -modules with dimension-vector nı where ı is the null-root of C . The dimension of each Ui is the sum of the components of the null-root whence smaller than 30. Next assume that there is a periodic line L in Pz of periodic length e 2d . Then A is a zero-relation algebra by Lemma 7.2. To produce the wanted modules one can invoke the theory of Dowbor and Skowro´nski [25] or one can observe with Ringel [74] that A is special biserial. One always finds the wanted modules as appropriate band-modules with dim Ui 2d . By part b) of Lemma 7.2 we can assume from now on that Pz does not contain a line of length 2d . The only case not yet settled is when Pz contains a tame concealed algebra C of type z Dn as a convex subcategory. Any such subcategory contains a line of length n 1 so that we can assume n 1 2d . Then the push-downs of the obvious P1 .k/-families of C -modules produce indecomposables with dim Ui 4d . The proofs given in [4], [18], [29] of the usual weaker form of BT 2 do not need Theorem 4.4 or the classification of the critical algebras, but the classification of the large ssc algebras is always needed. With a little more work one finds in all three cases occurring in the last proof a natural P1 .k/-family of modules. 7.4 Finite representation type is open. In the important early paper [34] Gabriel introduces for fixed natural numbers d and n the varieties algd of d -dimensional unital associative algebras and algd modn of pairs consisting of a d -dimensional algebra A and an n-dimensional A-module M . These varieties are equipped with natural actions of Gld and Gln and the projection W algd modn ! algd has the following property. Proposition 7.8. The image of a closed Gln -stable subset of algd modn is closed. From this Gabriel derives with the help of some semi-continuity properties of fibres: Proposition 7.9. For any n the set U.n/ of all d -dimensional algebras having only finitely many isomorphism classes of n-dimensional modules is open. Finally one gets: Theorem 7.10. The set find of all representation-finite algebras of dimension d is open in algd . For the proof Gabriel observes that find is the intersection of all U.n/ by BT 2 and a countable union of constructible sets by Auslander’s homological characterization of representation-finite algebras. Then one has the surprising fact that the intersection as well as the union stop at a finite level which implies that there are constants C1 and C2 such that all representation-infinite algebras of dimension d have infinitely many isomorphism classes of indecomposables of dimension C1 and all representation-finite

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algebras of dimension d have at most C2 isomorphism classes of indecomposables. Of course these results are now surpassed by Theorem 7.7 and Theorem 7.3. Theorem 7.10 is used in [3] for the proof of part a) of Theorem 3.2. Geiß has combined in [42] Gabriel’s arguments with Drozd’s theorem on tame and wild algebras [28], [21], [40] to show. Theorem 7.11. Any deformation of a tame algebra is tame. This result is very useful because a lot of interesting algebras with unknown module structure degenerate to special biserial algebras which are always tame [43], [83]. For instance this is used in the recent interesting results of Brüstle, de la Peña and Skowro´nski on tame strongly simply connected algebras [19]. Unfortunately, one does not get a description of the indecomposables with this method. The theorems of Gabriel and Geiß have been analyzed and generalized in [20], [53], but the question whether tame type is open is not yet answered. 7.5 The ‘classification’ of representation-finite and minimal representation-infinite algebras. For representation-finite algebras there is no classification in the strict sense known. We have already seen several times that representation-finite algebras behave worse than minimal representation-infinite algebras. So one might ask whether these algebras can be classified. If we look at a distributive algebra A then we have by Theorems 3.2 and 4.4 that A ' kP for some ray-category P with an interval-finite universal covering Pz that is not locally representation-finite. If all finite convex subcategories are representation-finite then A is a zero-relation algebra by Lemma 7.2 and in fact even a special biserial algebra as observed in [74] where Ringel classifies the minimal representation-infinite ones among these and where he analyzes their module categories. On the other hand the case where k Pz contains a tame concealed algebra of type Ezn with n D 6; 7; 8 leads to algebras with at most 9 points and thus to a finite classification problem. However Fischbacher’s list in [31] shows that this classification will probably end up with an unreadable list. By the way, ‘mild can be wild’, i.e., there are a lot of wild algebras that are minimal representation-infinite so that in general one will not get the structure of the modules. 7.6 The classification of representation-finite selfinjective algebras. The classification of the blocks with cyclic defect and their indecomposable modules due to Dade, Janusz and Kupisch [22], [51], [57], [33] is an early highlight of the representation theory of algebras. Also the detailed study of representation-finite symmetric algebras undertaken by Kupisch in [56], [58] is quite impressing, but his results formulated in terms of Cartan numbers are difficult to understand and far from a complete classification.

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The situation changed after the invention of almost split sequences when Riedtmann discovered in [66] that the stable Auslander–Reiten quiver of a representation-finite algebra has a simple structure (see also Todorov’s work in [79]). For a selfinjective algebra only the missing projective-injectives have to be inserted and the possible configurations of these points were first studied purely combinatorially by Riedtmann, but later on Hughes–Waschbüsch [49] as well as Bretscher–Riedtmann–Läser [16] found independently two direct constructions and the second group of authors classified that way the configurations and so also the representation-finite selfinjective standard algebras. However, Riedtmann had observed earlier that for some configurations and only in characteristic 2 there is also an exceptional non-standard algebra. The published proofs that this is the only ‘accident’ are complicated. In the approach by Riedtmann, Bretscher and Läser it is contained in the articles [67], [69], [16], in the approach by Kupisch, Waschbüsch and Scherzler in the articles [56], [58], [60], [81]. Some of these are long and difficult to read. Here, to illustrate the strategy of Section 3 we will analyze which critical paths and non-deep contours occur for representation-finite selfinjective algebras by using the classification of the standard-algebras. It turns out that the difficult steps 1 and 2 of Section 3 will never occur except for one case where we deal with one pennyfarthing glued to a Brauer-quiver algebra. Thus one can get a complete proof of the classification of the representation-finite selfinjective algebras by reading [66], [16] and Sections 8.3–8.6 of [3]. Let T be a Dynkin-diagram. For any subset C of ZT we denote by ZTC the simply connected translation quiver obtained from ZT by adding for each c 2 C a new point c which is projective and injective and arrows c ! c ! 1 c. We call C a configuration of T if ZTC is the Auslander–Reiten quiver of a locally representationfinite selfinjective category A D A.T; C / which is then uniquely determined because of k.ZTC / ' ind A.T; C / (see Theorem 5.1). The period e of a configuration is the smallest natural number such that e stabilizes C . We set m.An / D n, m.Dn / D 2n 3, m.E6 / D 11, m.E7 / D 17, m.E8 / D 29. The following crucial result follows easily from the well-known properties of the additive functions in k.ZT / given in [36]. Lemma 7.12. Let C be a configuration of T . Let S be a simple A-module with projective covering P .S/ D c and injective hull I.S / D d . Then we have: a) Hom.P .S /; I.S // D k and the only profound morphisms in ind A.T; C / are the non-zero elements in Hom.P .S /; I.S //. b) d D m.T / c, whence C is stable under m.T / and e divides m.T /. c) k.ZTC /.c ; r c / D k.ZTC /.c ; s c / D 0 for all r > m.T /, s m.T / and non-trivial automorphisms of T .

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Proof. For any locally bounded category the profound morphisms in ind A are of the form P .S/ ! S ! I.S/. The rest follows from the shape of the additive functions. Details can be found in [16, Proposition 1]. Now let A be a representation-finite selfinjective algebra with Auslander–Reiten zA is isomorquiver A . By Riedtmann’s structure theorem [66] its universal covering phic to some ZTC for a uniquely determined Dynkin-diagram T called the tree class of A and a configuration C of T . The standard-form As is then the full subcategory of projectives in k.ZTC =…/ where … is an admissible subgroup of Aut ZT leaving C invariant and one obtains that way all standard-algebras up to isomorphism. The group … is the fundamental group of AE and its structure is determined in [67, 3.3] and [16, Section 1.6]. Lemma 7.13. The fundamental groups are always cyclic and we are in one of the following two situations: a) … D h r i for some r m.T / and some ( possibly trivial ) automorphism of T . Then we call … small. b) … D h se i for some s 1 such that se < m.T /. Then we call … large. This occurs only for the types An and Dn . An analogue of the next lemma is proven for the old definition of ‘standard’ in [16]. We include a proof because we want to know which non-deep contours occur and whether all morphisms are bitransit. Kupisch calls an algebra regular if all morphisms are bitransit. There is no critical path in a regular algebra. Lemma 7.14. If the fundamental group of AE is small any morphism is bitransit and there is no non-deep contour. Therefore A is standard. Proof. Lemma 7.12 implies that all non-invertible endomorphisms of an indecomposable projective are annihilated by all irreducible morphisms. Thus all morphisms are bitransit and there is no non-deep contour. An application of step 3 in the proof of Theorem 3.1 shows that A is standard. For the case of large fundamental groups we need more information. By part b) of Lemma 7.12 the period e of a configuration always divides m.T /. Thus to describe all possible configurations means to classify all standard algebras with fundamental group h m.T / i. This is done in the Sections 6–8 of the article [16] whose Sections 2–4 can be proven much shorter and without using tilting theory. For type An one gets the well-known Brauer-quiver algebras. A Brauer-quiver Q is a finite connected quiver such that: i) Each arrow belongs to a simple oriented cycle, ii) Each vertex belongs to exactly two cycles and iii) Two cycles meet in one vertex at most. It follows that the cycles can be divided into two types say ˛ and ˇ such that cycles meeting in a vertex belong to different types. Each arrow belongs only to one cycle so that we can speak of ˛-arrows and ˇ-arrows. The relations that a Brauer-quiver

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algebra has to satisfy are i) the compositions of two arrows of different types is 0, and ii) ˛ ax D ˇ bx holds for any point x belonging to an ˛-cycle of length ax and a ˇ-cycle of length bx . These relations do not define an admissible ideal since there are always loops. For examples we refer to [16, Section 6]. The algebras of type Dn are obtained by a kind of glueing of two or three Brauerquiver algebras. In the first case the fundamental groups are small. Thus we only sketch the construction of [16, Section 7.2] for the glueing of three Brauer-quiver algebras and for n 5. So let 5 n D n1 C n2 C n3 be given and let Pi , i D 1; 2; 3, be Brauer-quiver algebras with ni vertices including a distinguished vertex xi lying on a ˇ-loop. To get the quiver of the algebra D.P1 ; P2 ; P3 / one separates the points xi in Pi into an ˛-sink xi and an ˛-source xiC and one adds arrows i W xi ! xiC replacing the loops. where the indices are taken modulo 3 so that one Then one identifies xiC with xiC1 obtains a cycle of length 3 containing the -arrows. This is the only triangle such that the quiver remains connected after removing the arrows of the triangle (and no point). Now for e < m.T / the algebra D.P1 ; P2 ; P3 / has an additional symmetry induced by e . This is only possible for P1 D P2 D P3 D P , n D 3m, e D 2m 1 and for the rotation of the -cycle as additional symmetry. We do not give in detail the relations that D.P1 ; P2 ; P3 / has to satisfy, but we remark that respects the relations on D.P; P; P / and we denote by N.P / the quotient. Then the quiver of N.P / is obtained from P D P1 by adding a loop in the point x1 and it follows from the relations of D.P; P; P / that N.P / has a penny-farthing with loop as the only non-deep contour and the ˛-path of length 2 through x1 as the only critical path. From this one gets by an easy direct argument that in characteristic 2 the linearization of the stem-category of N.P / is the only non-standard algebra in accordance with [81], [69]. For the smallest possible Brauer-quiver algebra with two points one finds as D.P; P; P / the algebra whose quiver is shown in Figure 7. q S S

q S 7 S

S w S q S S S

S

7

S

S wq S

q

S w S q 7

Figure 7

It admits the threefold rotation as a symmetry and the quotient N.P / is the algebra A.1/ we started with in Figure 1. Our round-trip through the land of mild categories has come to an end. We conclude with the proof that the N.P / from above are the only non-standard algebras.

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Theorem 7.15. Let A ¤ k be a connected representation-finite selfinjective algebra with standard-form As . Then we are in one of the following two situations: a) As is not isomorphic to some N.P / for a Brauer-quiver P with at most 2 vertices. Then all arrows are bitransit and there is no non-deep contour. Consequently A is standard. b) As is isomorphic to some N.P / for a Brauer-quiver P with at most 2 vertices. Then there is exactly one critical path and one non-deep contour. A is isomorphic to the linearization of its ray-category or of its stem-category and these two are isomorphic iff the characteristic is not 2. Proof. Part b) follows from the above discussion. For part a) we work with raycategories. So let C be a configuration for the tree class An with period e and ef D n: Denote by Ps the ray-category of the full subcategory of projectives in k.ZTC =h se i/. Then Pf is the ray-category of a Brauer-quiver algebra and we have a f -fold Galois covering W Pf ! P1 . We claim that all irreducible morphisms – whence all morphisms – in P1 are bitransit and that P1 has no non-deep contour. Now up to duality any such contour involves three irreducible morphisms ; ı; with ¤ ı and ı ¤ 0 and ¤ 0 (see Figure 4). This situation can be lifted to Pf where it is impossible. Next let W x ! y be an irreducible morphism in P1 with x ¤ y and let D m : : : 2 1 be a generator of the ‘radical’of P1 .y; y/ written as a product of irreducible morphisms. Assume that ¤ 0. Lifting x and the irreducible morphisms one obtains upstairs a non-zero product mC1 m : : : 1 with .1 / D . Thus all i belong to a cycle of the Brauer-quiver say with r points and one has r : : : m : : : 1 ¤ 0 for some irreducible morphisms j and also m : : : 1 r : : : mC1 ¤ 0. We have .1 r : : : mC2 / D i with i 1 whence .1 / and .mC1 / are two irreducible morphisms not annihilated by . So they coincide as seen above. This gives D .mC1 m : : : 1 / D .m : : : 1 / and is cotransit. Dually one gets that is transit. Now for any s we have a Galois covering Ps ! P1 . Any non-bitransit morphism in Ps would produce one in P1 which is impossible. For non-deep contours the same argument works. Finally, let C be a configuration for the tree class D3m with period e D 2m 1. Using conventions analogous to the above we know that P1 is N.P / with only one non-deep contour which is a penny-farthing with loop in x and only two non-bitransit irreducible morphisms ˛1 starting in x and ˛n ending in x. We only have to look at the 2-fold Galois covering W P2 ! P1 that preserves non-bitransit irreducible morphisms and non-deep contours. One checks easily that the preimages of the ˛i are bitransit and that there is no non-deep contour in P2 that is mapped to the penny-farthing.

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Quiver Hecke algebras and categorification Jonathan Brundan

1 Introduction Over a century ago Frobenius and Schur understood that the complex representation theory of the symmetric groups Sn for all n 0 categorifies the graded Hopf algebra of symmetric functions Sym. The language being used here is much more recent! It means the following. Let Rep.CSn / denote the category of finite dimensional CSn modules and S./ be the irreducible Specht module indexed by partition ` n. The isomorphism classes fŒS./ j ` ng give a distinguished basis for Grothendieck group ŒRep.CSn / of this semisimple category. Given an Sm -module V and an Sn S module W , we can form their induction product V B W WD IndSmCn V W . This m Sn operation descends to the Grothendieck groups to give a multiplication making ŒRep.CS/ WD

M ŒRep.CSn / n0 S

into a graded algebra. Moreover the restriction functors ResSmCn for all m; n 0 m Sn induce a comultiplication on ŒRep.CS/, making it into a graded Hopf algebra. The categorificiation theorem asserts that it is isomorphic as a graded Hopf algebra to Sym, the canonical isomorphism sending ŒS./ to the Schur function s 2 Sym. There have been many variations and generalizations of this result since then. Perhaps the most relevant for this article comes from the work of Bernstein and Zelevinsky in the late 1970s on the representation theory of the affine Hecke algebra Hn associated to the general linear group GLn .F / over a non-archimedean local field F (e.g. see [39]). This is even richer algebraically since, unlike CSn , its finite dimensional representations are no longer completely reducible. Bernstein and Zelevinsky showed that the direct sum M ŒRep.Hn / ŒRep.H / WD n0

of the Grothendieck groups of the categories of finite dimensional Hn -modules for all n 0 again has a natural structure of graded Hopf algebra. Moreover this graded Hopf algebra is isomorphic to the coordinate algebra ZŒN of a certain direct limit N of groups of upper unitriangular matrices. In [40], Zelevinsky went on to formulate a padic analogue of the Kazhdan–Lusztig conjecture, which was proved by Ginzburg [7]. Zelevinsky’s conjecture implies that the basis for ZŒN arising from the isomorphism

Supported in part by NSF grant no. DMS-1161094.

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classes of irreducible Hn -modules for all n 0 coincides with the Lusztig–Kashiwara dual canonical basis. The dual version of this theorem was proved by Ariki in [1], who also investigated certain finite dimensional quotients of Hn called cyclotomic Hecke algebras, which categorify the integrable highest weight modules of the corresponding Lie algebra g D sl1 . Ariki’s work includes the case that the defining parameter of Hn is a primitive pth root of unity, when sl1 is replaced by the affine Kac–Moody p . algebra sl Quiver Hecke algebras were discovered independently in 2008 by Khovanov and Lauda [18], [19] and Rouquier [30]. They are certain Hecke algebras attached to symmetrizable Cartan matrices. It appears that Khovanov and Lauda came upon these algebras from an investigation of endomorphisms of Soergel bimodules (and related bimodules which arise from cohomology of partial flag varieties), while Rouquier’s motivation was a close analysis of Lusztig’s construction of canonical bases in terms of perverse sheaves on certain quiver varieties. In a perfect analogy with the BernsteinZelevinsky theory just described, these algebras categorify the coordinate algebra of the unipotent group N associated to a maximal nilpotent subalgebra of the Kac–Moody algebra g arising from the given Cartan matrix. In fact the picture is even better: the quiver Hecke algebras are naturally Z-graded, so that the Grothendieck groups of their categories of finite dimensional graded representations are ZŒq; q 1 -modules, with q acting by degree shift. The resulting “graded” Grothendieck groups categorify a ZŒq; q 1 -form for the quantum group f that is half of the quantized enveloping algebra Uq .g/. Moreover there is an analogue of Ariki’s theorem, conjectured originally by Khovanov–Lauda and proved by Varagnolo–Vasserot [32] and Rouquier [31, Corollary 5.8] using geometric methods in the spirit of [7]. There are even cyclotomic quotients of the quiver Hecke algebras which Kang–Kashiwara [13], Rouquier [31, Theorem 4.25] and Webster [35] have used to categorify integrable highest weight modules. Rouquier also observed in (finite or affine) type A that the quiver Hecke algebras become isomorphic to the affine Hecke algebras discussed earlier (at a generic parameter or a root of unity) when suitably localized (see [30, Proposition 3.15] and also [4] in the cyclotomic setting). Thus Ariki’s theorem is a special case of the Rouquier–Varagnolo–Vasserot categorificiation theorem just mentioned (see [5]). Even more variations on the quiver Hecke algebras have subsequently emerged, including a twisted version related to affine Hecke algebras of type B introduced by Varagnolo and Vasserot [33], and the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka [16]. The latter superalgebras generalize Wang’s spin Hecke algebras [34] and the odd nil Hecke algebra of Ellis, Khovanov and Lauda [10], and give a completely new “supercategorification” of the same quantum groups/highest weight modules as above (see [15]). We also mention the work [14] which connects quiver Hecke algebras to quantum affine algebras, potentially providing a direct algebraic link between the categorifications of ZŒN arising via quiver Hecke algebras and the ones introduced by Hernandez and Leclerc in [12].

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For the future perhaps the most exciting development arising from these new algebras is the introduction again by Khovanov–Lauda [20] and Rouquier [31] of certain 2-categories called 2-Kac–Moody algebras. These categorify Lusztig’s idempotented version UP q .g/ of the quantized enveloping algebra of g (see [20], [35], [37]). In the case g D sl2 this goes back to work of Chuang–Rouquier [8] and Lauda [24]. In the introduction of [30], Rouquier promises to define a tensor product on the 2-category of dg 2-representations of the 2-Kac–Moody algebra, the ultimate goal being to construct 4-dimensional TQFTs in fulfillment of predictions made long ago by Crane and Frenkel [9]. Webster has also suggested a more down-to-earth diagrammatic approach to constructing categorifications of tensor products of integrable highest weight modules in finite types in [35], [36]. In this article we will not discuss at all any of these higher themes, aiming instead to give a gentle and self-contained introduction to the quiver Hecke algebras and their connection to Lusztig’s algebra f, focussing just on the case of symmetric Cartan matrices for simplicity. In the last section of the article we specialize further to finite type and explain some of the interesting homological properties of quiver Hecke algebras in that setting, similar in spirit to those of a quasi-hereditary algebra, despite being infinite-dimensional. As we go we have included proofs or sketch proofs of many of the foundational results, before switching into full survey mode later on. To improve readability, references to the literature are deferred to the end of each section.

2 Quiver Hecke algebras In this opening section, we give a general introduction to the definition and structure of quiver Hecke algebras. Gradings. Fix once and for all an algebraically closed ground field K. Everything (vectorLspaces, algebras, modules, …) will be Z-graded. vector space P For a graded n V , its graded dimension is Dim V WD .dim V /q , where q is a V D n n n2Z n2Z formal variable. Of course this only makes sense if V is locally finite dimensional, i.e. the graded pieces of V are finite dimensional. Typically V will also be bounded below, i.e. Vn D 0 for n 0, in which case Dim V is a formal Laurent series in q. We write q m V for the upward degree shift by m steps, so q m V is the graded vector m space with .qP V /n WD Vnm , and then Dim q mL V D q m Dim V . More generally, n n ˚n . Finally we write for f .q/ D n2Z fn q , we write f .q/V n2Z .q V / L for Hom.V; W / for the graded vector space n2Z Hom.V; W /n , where Hom.V; W /n denotes the linear maps f W V ! W that are homogeneous of degree n, i.e. they map Vm into WmCn . Note then that End.V / WD Hom.V; V / is a graded algebra. Demazure operators. Recall that the symmetric group Sn is generated by the basic transpositions t1 ; : : : ; tn1 subject only to the braid relations ti tiC1 ti D tiC1 ti tiC1 and ti tj D tj ti for ji j j > 1, plus the quadratic relations ti2 D 1. The length `.w/ of

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w 2 Sn is #f1 i < j n j w.i / > w.j /g. We will denote the longest element of Sn by wŒ1;n . This is the permutation 1 7! n, 2 7! n 1, 3 7! n 2, …, n 7! 1, n q n which is of length 12 n.n 1/. Letting Œn WD qqq 1 and ŒnŠ WD ŒnŒn 1 : : : Œ2Œ1, we have the well-known factorization of the Poincaré polynomial of Sn : X

1

q 2`.w/ D q 2 n.n1/ ŒnŠ :

(2.1)

w2Sn

Let Sn act on the polynomial algebra Poln WD KŒx1 ; : : : ; xn by permuting the variables. Viewing Poln as a graded algebra with each xi in degree 2, this is an action by graded algebra automorphisms. So the invariants form a graded subalgebra Symn WD PolSn n , namely, the algebra of symmetric polynomials. This is again a free P polynomial algebra generated by the elementary symmetric polynomials er WD 1i1 0; see Definition 2.10 in [4]; for two simple-minded collections fX1 ; : : : ; Xn g and fX10 ; : : : ; Xn0 g in Dfd .A/, we have fX1 ; : : : ; Xn g fX10 ; : : : ; Xn0 g () HomC .Xi ; †m Xj0 / D 0 for all i , j and m < 0; see [75, Section 7]. Let us define the bijections in Theorem 4.1 as in [68], [75]. silt.A/ ! smc.A/: Let M D M1 ˚ ˚ Mn be a basic silting object in per.A/,

where M1 ; : : : ; Mn are indecomposable. The corresponding simple minded collection fX1 ; : : : ; Xn g is the unique collection (up to isomorphism) in Dfd .A/ satisfying for any i; j D 1; : : : ; n ´ k if i D j and m D 0; Hom.Mi ; †m Xj / D (4.1) 0 otherwise: More precisely, it follows from Keller’s Morita theorem for triangulated categories ([59, Theorem 4.3]) that there is a non-positive dg algebra B together with a triangle equivalence D.B/ ! D.A/ taking B to M . The collection fX1 ; : : : ; Xn g is the image of a complete collection of pairwise non-isomorphic simple H 0 .B/-modules under this equivalence. silt.A/ ! t-str.A/: Let M be a silting object in per.A/. The corresponding t -structure .D 0 ; D 0 / on Dfd .A/ is defined as D 0 D fX 2 Dfd .A/ j Hom.M; †m X / D 0 for m > 0g; D 0 D fX 2 Dfd .A/ j Hom.M; †m X / D 0 for m < 0g: The object M can be characterised as an additive generator of the category fN 2 per.A/ j Hom.N; †X / D 0 for any X 2 D 0 g: A nice consequence of the bijectivity of this map is that the heart of any element in t-str.A/ is equivalent to the category of finite-dimensional modules over a finitedimensional algebra, namely, the endomorphism algebra of the corresponding silting object.

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silt.A/ ! co-t-str.A/: Let M be a silting object in per.A/. The corresponding co-t -structure .P0 ; P0 / on per.A/ is defined as

P0 D the smallest full subcategory of per.A/ which contains f†m M j m 0g and which is closed under taking extensions and direct summands; P0 D the smallest full subcategory of per.A/ which contains f†m M j m 0g and which is closed under taking extensions and direct summands: We point out that this map is well defined and bijective in a much more general setting, see [12], [4], [68], [80]. t-str.A/ ! smc.A/: Let .D 0 ; D 0 / be a bounded t -structure on Dfd .A/ with length heart. The corresponding simple-minded collection is a complete collection of pairwise non-isomorphic simple objects of the heart D 0 \ D 0 . t-str.A/ ! co-t-str.A/: Let .D 0 ; D 0 / be a bounded t -structure on Dfd .A/ with length heart. The corresponding co-t-structure .P0 ; P0 / is defined as P0 D fN 2 per.A/ j Hom.N; †X / D 0 for any X 2 D 0 g; P0 D fN 2 per.A/ j Hom.N; †X / D 0 for any X 2 D 0 g: smc.A/ ! t-str.A/: Let fX1 ; : : : ; Xn g be a simple-minded collection of Dfd .A/. The corresponding t-structure .D 0 ; D 0 / on Dfd .A/ is defined as

D 0 D the smallest full subcategory of Dfd .A/ which contains f†m Xi j 1 i n; m 0g and which is closed under taking extensions and direct summands; D 0 D the smallest full subcategory of Dfd .A/ which contains f†m Xi j 1 i n; m 0g and which is closed under taking extensions and direct summands: smc.A/ ! co-t-str.A/: Let fX1 ; : : : ; Xn g be a simple-minded collection of Dfd .A/. The corresponding co-t-structure .P0 ; P0 / on per.A/ is defined as L P0 D fN 2 per.A/ j Hom.N; †m niD1 Xi / D 0 for m > 0g; L P0 D fN 2 per.A/ j Hom.N; †m niD1 Xi / D 0 for m < 0g:

This map was not directly defined in [75]. We take it as the composition of smc.A/ ! t-str.A/ and t-str.A/ ! co-t-str.A/. co-t-str.A/ ! silt.A/: Let .P0 ; P0 / be a bounded co-t -structure on per.A/. The corresponding silting object M of per.A/ is an additive generator of the coheart, i.e. add.M / D P0 \ P0 . co-t-str.A/ ! t-str.A/: Let .P0 ; P0 / be a bounded co-t -structure on per.A/. The corresponding t-structure .D 0 ; D 0 / is defined as D 0 D fX 2 Dfd .A/ j Hom.N; †X / D 0 for any N 2 P0 g; D 0 D fX 2 Dfd .A/ j Hom.N; †X / D 0 for any N 2 P0 g:

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Next, consider the subsets 2-silt.A/, 2-smc.A/, int-t-str.A/ and int-co-t-str.A/ of silt.A/, smc.A/, t-str.A/ and co-t-str.A/. Set FA D fcone.f / j f is a morphism in addper.A/ .A/g per.A/: By definition a 2-term silting object of per.A/ is exactly a silting object belonging to FA . Corollary 4.3. There is a commutative diagram of bijections which commute with mutations and which preserve partial orders. 2-silt.A/ o

/ int-co-t-str.A/ TTTT 5 O TTTT jjjjjj j T jjjj TTTTTT ) ju jjj / 2-smc.A/ int-t-str.A/ o

Proof. It suffices to check that the following six maps restrict. co-t-str.A/ ! t-str.A/: These two maps are defined by taking perpendicular subcategories. It is easy to check that they restrict to intermediate objects. std std silt.A/ ! co-t-str.A/: Let .P0 ; P0 / denote the standard co-t -structure on std std per.A/. One checks that FA D †P0 \ P0 holds. By Lemma 3.10, a co-t -structure is intermediate if and only if its co-heart is contained in FA , namely, the corresponding silting object is 2-term. Thus the two maps silt.A/ $ co-t-str.A/ restrict to bijections 2-silt.A/ $ int-co-t-str.A/. smc.A/ ! t-str.A/: Dual to silt.A/ $ co-t-str.A/. The proof of Corollary 4.3 works for A being a homologically smooth non-positive dg algebra or a finite-dimensional algebra. Theorem 4.4. There is a commutative diagram of bijections which commute with mutations and preserve partial orders. 2-silt./ oYYYY Y I

/ int-co-t-str./ fMMM YYYYYY II Y M YYYYYY II YYYYYY MMMM II YYYY, $ / 2-smc./ int-t-str./ o O O _ o _ _ _ _ _ _ / 2-silt.J / Y Y int-co-t-str.J / Y Y Y fM II II M Y Y Y II Y Y Y Y Y MM I$ Y/, int-t-str.J / o 2-smc.J /

The following proof of this theorem still works if the pair .; J / is replaced by .A; H 0 .A//, where A is a homologically smooth non-positive dg algebra such that H 0 .A/ is finite-dimensional.

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Proof. The top square (respectively, the bottom square) is obtained by applying Corollary 4.3 to (respectively, J ). Next, we will define the vertical maps. We state the bijectivity and compatibility with mutations for some of them: that for others follow from the commutativity of the diagram. Consider the canonical projection p W ! J . Let p W per./ ! per.J / (respectively, p W Dfd .J / ! Dfd ./) be the induction (respectively, restriction) along the projection p. The functor p is extensively studied by King and Qiu in [74] for the case of a Dynkin quiver with trivial potential. 2-silt./ ! 2-silt.J /: The assignment M 7! p .M / defines a bijection from 2-silt./ to 2-silt.J / which commutes with mutations, by applying Propositions A.3 and A.6 to the dg algebra A D . More explicitly, observe that p induces an additive equivalence addper./ ./ ! addper.J / .J / D proj J . Since M belongs to 2-silt./, there is a triangle P

f

/Q

/M

/ †P

with P; Q 2 addper./ ./. Then p .M / is the cone of the morphism p .f / W p .P / ! p .Q/, which is a morphism proj J . int-co-t-str./ ! int-co-t-str.J /: Consider the assignment induced by the projection: .P0 ; P0 / 7! .H0 ; H0 /, where H0 D the smallest full subcategory of per.J / which contains p .P0 / and which is closed under extensions and direct summands; H0 D the smallest full subcategory of per.J / which contains p .P0 / and which is closed under extensions and direct summands; which clearly preserves partial orders. Let .P0 ; P0 / be an intermediate co-t-structure on per./ and let M be the 0 0 corresponding silting object. Let .H0 ; H0 / be defined as above. Let .H0 ; H0 / be the co-t-structure on per.J / corresponding to the silting object p .M /. Then 0 0 and H0 H0 . It follows that both equalities hold, .H0 ; H0 / is a H0 H0 bounded co-t -structure on per.J / and it corresponds to p .M /. This shows the welldefinedness of the map int-co-t-str./ ! int-co-t-str.J / as well as the commutativity of the inner square of the diagram. 2-smc.J / ! 2-smc./: Consider the assignment induced by the projection: fX1 ; : : : ; Xn g 7! fp .X1 /; : : : ; p .Xn /g. Let fX1 ; : : : ; Xn g be a 2-term simple-minded collection in D b .mod J /. We want to show that fp .X1 /; : : : ; p .Xn /g is a 2-term simple-minded collection in Dfd ./. Let M D M1 ˚ ˚ Mn be the basic 2-term silting object corresponding to fX1 ; : : : ; Xn g, where M1 ; : : : ; Mn are indecomposable. Then as we have shown, there is a 2-term basic silting object N D N1 ˚ ˚ Nn such that p .Ni / D Mi for all i .

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Then Hom.Ni ; †m p .Xj // D Hom.p .Ni /; †m Xj /

D Hom.Mi ; †m Xj / ´ k if i D j and m D 0; D 0 otherwise: It follows by (4.1) that fp .X1 /; : : : ; p .Xn /g is the simple-minded collection corresponding to the silting object N . This shows the well-definedness of the map 2-smc.J / ! 2-smc./ as well as the commutativity of the diagonal square of the diagram. int-t-str.J / ! int-t-str./: The assignment .C 0 ; C 0 / 7! .D 0 ; D 0 /, where D 0 D the smallest full subcategory of Dfd .J / which contains p .C 0 / and which is closed under extensions and direct summands; D 0 D the smallest full subcategory of Dfd .J / which contains p .C 0 / and which is closed under extensions and direct summands; defines a bijection int-t-str.J / ! int-t-str./. The proof for the well-definedness of this map and the commutativity of the right square of the diagram is similar to the case for int-co-t-str./ ! int-co-t-str.J /. 4.2 The Amiot cluster category. Let .Q; W / be a Jacobi-finite quiver with potential, y D .Q; W / and J D Jy.Q; W /. Recall from Section 2.5 that per./ contains Dfd ./ as a triangulated subcategory. The Amiot cluster category of .Q; W / is defined as the triangle quotient C.Q;W / WD per./=Dfd ./: Let W per./ ! C.Q;W / be the canonical projection functor. Theorem 4.5. The following statements hold. (a) ([7], Theorem 3.5) The Amiot cluster category C.Q;W / is Hom-finite and 2Calabi–Yau. Moreover, the object ./ is a cluster-tilting object in C.Q;W / . Its endomorphism algebra is isomorphic to J . (b) ([7], Proposition 2.9) The functor W per./ ! C.Q;W / induces an additive equivalence F ! C.Q;W / : (c) ([7], Proposition 2.12) For X; Y 2 F , there is a short exact sequence 0 ! Homper./ .X; †Y / ! HomC.Q;W / ..X /; †.Y // ! D Homper./ .Y; †X / ! 0:

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These results were proved by Amiot [7] for non-complete Ginzburg dg algebras and non-complete Jacobian algebras, but her proof works also in the complete setting. When Q is acyclic, the Amiot cluster category C.Q;0/ is triangle equivalent to the cluster category CQ , thanks to the theorem of Keller and Reiten [71, Theorem 2.1]. 4.3 The correspondences between silting objects, support -tilting objects and cluster-tilting objects. Let .Q; W / be a Jacobi-finite quiver with potential, D y .Q; W /, J D Jy.Q; W / and p W ! J be the canonical projection of dg algebras. Let C.Q;W / be the Amiot cluster category of .Q; W / and W per./ ! C.Q;W / be the canonical projection functor. Recall that s -tilt.J / denotes the set of isomorphism classes of basic support -tilting modules over J (Section 3.3) and c-tilt.C.Q;W / / denotes the set of isomorphism classes of basic cluster-tilting objects in C.Q;W / (Section 3.8). Theorem 4.6. There is a commutative diagram of bijections which commute with mutations. / 2-silt.J / 2-silt./ >> P >> >> >> >> >> s -tilt.J / O

#

c-tilt.C.Q;W / /

The map 2-silt./ ! 2-silt.J / was defined in Theorem 4.4. 2-silt./ ! s-tilt.J /: This map is defined as taking the 0-th cohomology: M 7! H 0 .M / D Homper./ .; M /, see [53]. 2-silt.J / ! s-tilt.J /: This map is defined as taking the 0-th cohomology: M 7! H 0 .M / D HomH b .proj J / .J; M /. For the well-definedness, bijectivity and compatibility of mutations of this map, see [2, Theorem 3.2 and Corollary 3.9]. The commutativity of the top triangle follows from the fact that the equivalence p W addper./ ./ ! proj J is just H 0 . c-tilt.C.Q;W / / ! s-tilt.J /: Recall from Theorem 4.5 (a) that ./ is a clustertilting object of C.Q;W / with endomorphism algebra J . So there is a functor HomC.Q;W / ../; ‹/ W C.Q;W / ! mod J:

The desired map takes a cluster-tilting object M to HomC.Q;W / ../; M /, see Theorem 4.1 of [2]. 2-silt./ ! c-tilt.C.Q;W / /: The assignment M 7! .M / defines a bijection 2-silt./ ! c-tilt.C.Q;W / /, which is known to the experts. We give a proof for the

173

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convenience of the reader. Recall that a silting object of per./ is 2-term if and only if it belongs to F . Proposition 4.7. The equivalence W F ! C.Q;W / induces a bijection from 2-silt./ to c-tilt.C.Q;W / /. Moreover, this bijection commutes with mutations, that is, if M is a basic 2-term silting object in per./ and N is an indecomposable direct summand of M such that N .M / is 2-term, then .N / ..M // D .N .M //. Proof. Let X be an object of F . Looking at the short exact sequence in Theorem 4.5 (c) with Y D X , we obtain that HomC.Q;W / ..X /; †.X // D 0 if and only if Homper./ .X; †X / D 0. Thus we have .X/ is a cluster tilting object in CA () HomC.Q;W / ..X /; †.X // D 0 and j.X /j D j./j

by Proposition 3.16

() Homper./ .X; †X / D 0 and jXj D jj () X is a presilting object and jXj D jj by Lemma A.2 () X is a silting object by Proposition 3.14: This proves the first statement. The second statement holds because f is a minimal approximation in F if and only if .f / is a minimal approximation in C.Q;W / . The commutativity of the left triangle of the diagram follows from the definitions of the three maps and the fact that W F ! C.Q;W / is an equivalence. c-tilt.C.Q;W / / ! 2-silt.J /: The functor HomC.Q;W / ../; ‹/ W C.Q;W / ! mod J restricts to an additive equivalence addC.Q;W / ..// ! proj J . Let M be a clustertilting object of C.Q;W / . Then there is a triangle Q1

f

/ Q0

/M

/ †Q1

(4.2)

with Q1 and Q0 in addC.Q;W / ..//. The desired map takes M to the cone of the morphism HomC.Q;W / ../; f / in H b .proj J /, see [2, Theorem 4.7]. By applying HomC.Q;W / ../; ‹/ to the triangle (4.2), we see that HomC.Q;W / ../; M / is the cokernel of HomC.Q;W / ../; f /, i.e. the 0-th cohomology of its cone. The commutativity of the right triangle of the diagram follows. Remark 4.8. Reduction techniques analogous to 2-Calabi–Yau reduction (Section 3.8), silting reduction and -tilting reduction, are respectively introduced by Aihara and Iyama in [4] and by Jasso in [56]. The compatibility of these reductions are studied in [64], [54], [56]. 4.4 The correspondences between silting objects, t-structures, torsion pairs and support -tilting modules. Let .Q; W / be a Jacobi-finite quiver with potential, D y .Q; W / and J D Jy.Q; W /.

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Theorem 4.9. There is a commutative diagram of bijections which commute with mutations and preserve partial orders. 2-silt./ o

/ int-t-str./ O bFF FF B FF B FF B FF B FF B FF B! _ o _ _ _ _ _ _ _ _ _ / 2-silt.J / int-t-str.J / < | xx x | x | xx | xx x | x | xx }| xx / f-tors.J / s -tilt.J / B

The maps in the upper square were defined in Theorem 4.4 and the maps in the left triangle were defined in Theorem 4.6. s -tilt.J / ! f-tors.J /: The assignment T 7! Fac.T / defines a bijection from s -tilt.J / to f-tors.J /, see [2, Theorem 2.6]. The module T can be characterised as an additive generator of the subcategory fN 2 mod J j Ext1J .N; M / D 0 for all M 2 Fac.T /g: f-tors.J / ! int-t-str.J /: This map is defined as the Happel–Reiten–Smalø tilt (see Section 3.4). There is the following general result.

Theorem 4.10 ([11], Theorem 3.1, and [103], Proposition 2.1). Let .C 0 ; C 0 / be a bounded t-structure on C with heart A. The Happel–Reiten–Smalø tilt induces a bijective map from the set of torsion pairs of A to intermediate t -structures with respect to .C 0 ; C 0 /. Its inverse takes .C 00 ; C 00 / to .C 00 \ A; .†1 C 00 / \ A/. However, it is difficult to directly prove that this map is well defined, namely, a functorially finite torsion class is taken to a bounded t -structure with length heart. We use the commutativity of the lower square of the diagram. To show the commutativity, let M be a 2-term silting object and let .D 0 ; D 0 / denote the corresponding t structure on D b .mod J /. We need to prove the equality Fac.H 0 .M // D D 0 \ mod J . Let X 2 Fac.H 0 .M //. Then there is a positive integer m and a surjection 0 H .M /˚m ! X . Since M is 2-term, there is a morphism M ! H 0 .M / such that taking H 0 we obtain the identity map. Composing the above two maps, we obtain a map f W M ˚m ! X . Form the triangle M ˚m

f

/X

/N

/ †M ˚m .

(4.3)

0 By taking cohomologies we see that H i .N / D 0 for i 0, and hence N 2 †Dstd , 0 0 b 0 where .Dstd ; Dstd / denotes the standard t-structure on D .mod J/. Since .D ; D 0 /

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Ordered exchange graphs

is intermediate, it follows that N belongs to D 0 . Applying Hom.M; ‹/ to the above triangle (4.3), we obtain a long exact sequence :::

/ Hom.M; †i M ˚n /

/ Hom.M; †i X /

/ Hom.M; †i N /

/ :::.

The two outer terms vanish for i > 0, so the middle term also vanishes for i > 0, i.e. X 2 D 0 . This shows the inclusion Fac.H 0 .M // D 0 \ mod J . Let X be a non-zero object in D 0 \ mod J . We claim that Hom.M; X / ¤ 0. Indeed, if Hom.M; X / D 0, then Hom.M; †i X / D 0 for i 0, so †1 X 2 D 0 . 0 since .D 0 ; D 0 / is intermediate. This implies that Thus X 2 †D 0 †Dstd 0 X D 0 because †Dstd \ mod J D 0. Now take a basis f1 ; : : : ; fm of Hom.M; X / and form the following triangle: N

/ M ˚m

.f1 ;:::;fm /

/X

/ †N .

(4.4)

Applying Hom.M; ‹/ to this triangle we obtain a long exact sequence / Hom.M; †i X /

Hom.M; †i M ˚m /

/ Hom.M; †iC1 N /

/ Hom.M; †iC1 M ˚m /.

If i > 0, then Hom.M; †iC1 N / vanishes because its two neighbours vanish. If i D 0, then the leftmost map is clearly surjective, and hence Hom.M; †N / D 0. Therefore 0 , in particular, H 1 .N / D 0. Now taking cohomologies of the N 2 D 0 Dstd triangle (4.4) gives us an exact sequence H 0 .M /˚n

/X

/ H 1 .N / D 0.

So X 2 Fac.H 0 .M //. This shows the inclusion D 0 \ mod J Fac.H 0 .M //. f-tors.J / ! int-t-str./: This map is defined as the Happel–Reiten–Smalø tilt, similar to the map f-tors.J / ! int-t-str.J /. Remark 4.11. Let fX1 ; : : : ; Xn g be a 2-term simple-minded collection of Dfd ./ (respectively, D b .mod J /). In Section 4.1 we constructed a bounded t -structure .D 0 ; D 0 / whose heart A is a length category with simple objects X1 ; : : : ; Xn . By Theorem 4.10, there is a torsion pair .T ; F / of mod J such that .D 0 ; D 0 / is the Happel–Reiten–Smalø tilt of the standard t -structure at .T ; F /. In particular, A has a torsion pair .†F ; T /. Therefore, for any i D 1; : : : ; n, the object Xi belongs to either †F or T . 4.5 Ordered exchange graphs and reachable objects. Let .Q; W / be a Jacobi-finite y quiver with potential, D .Q; W / and J D Jy.Q; W /. Gluing the three diagrams in Theorems 4.4, 4.6 and 4.9 we obtain the diagram in Figure 2.

176

T. Brüstle and D. Yang 2-silt./ oYYYYY F YYY

/ int-co-t-str./ cFF YYYYYYY FF YYYYYYY FF YYYYYYY Y YYYY, F / 2-smc./ int-t-str./ o O \9 O 9 9 9 9 9 _ _ _ _/ _ o _ _ _ _ _ _ 2-silt.J / Y Y int-co-t-str.J / 9 Y Y Y FF cF Y Y Y 9 F FF Y Y Y Y F FF 9 Y Y Y F# 9 Y Y, F / 2-smc.J / int-t-str.J / o :: hQQQ :: QQQ :: QQQ : QQQ QQQ ::: QQQ : & / f-tors.J / s -tilt.J / dII II II II II II , FF FF FF #

c-tilt.C.Q;W / /

Figure 2

Via the bijections each of the sets in the diagram is equipped with a mutation operation and a partial order. Thanks to Theorem 3.6, they all have the structure of ordered exchange graphs and in this way the diagram becomes a commutative diagram of isomorphisms of ordered exchange graphs. It is known that these graphs are connected when they have a finite connected component (Proposition 3.7) and when Q is acyclic and W D 0 ([17, Proposition 3.5]). In general they are not connected, for example, when Q is the quiver @ 2 = @ =====

a1 a2

o 1o

c2 c1

b b2=1====

=

3

and W D c1 b1 a1 C c2 b2 a2 c1 b2 a1 c2 b1 a2 , see [89, Example 4.3]. Demonet and Iyama informed us that in this example the graph has precisely two connected components. The distinguished objects 2 2-silt./; fp .S1 /; : : : ; p .Sn /g 2 2-smc./; 0 0 .Dstd; ; Dstd; / 2 int-t-str./;

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Ordered exchange graphs std; std; .P0 ; P0 / 2 int-co-t-str./;

JJ 2 2-silt.J /; fS1 ; : : : ; Sn g 2 2-smc.J /; 0 0 .Dstd;J ; Dstd;J / 2 int-t-str.J /; std;J std;J ; P0 / 2 int-co-t-str.J /; .P0

JJ 2 s-tilt.J /; .mod J; 0/ f-tors.J /; ./ 2 c-tilt.C.Q;W / / correspond to each other under the above bijections and they are the unique sources, where fS1 ; : : : ; Sn g is a complete collection of non-isomorphic simple J -modules, p W ! J is the canonical projection homomorphism and W per./ ! C.Q;W / is the canonical projection functor. The objects in the same connected component as these distinguished objects in the ordered exchange graphs are called reachable objects. Adding the condition ‘reachable’ to the above sets we obtain subsets r: 2-silt./, r: 2-smc./, r: int-t-str./, r: int-co-t-str./, r: 2-silt.J /, r: 2-smc.J /, r: int-t-str.J /, r: int-co-t-str.J /, r: s-tilt.J /, r: f-tors.J / and r: c-tilt.C.Q;W / / and a commutative diagram of isomorphisms of ordered exchange graphs (Figure 3), which, by definition, are all connected. For an acyclic quiver with trivial potential, it follows from [77, Theorem 1.1] that Bernstein–Gelfand–Ponomarev reflections (i.e. mutations at sinks/sources) induce flipflops of these ordered exchange graphs. 4.6 From cluster-tilting objects to clusters. Let .Q; W / be a Jacobi-finite quiver y with potential such that Q is a cluster quiver and W is non-degenerate, D .Q; W /, J D Jy.Q; W / and let C.Q;W / denote the Amiot cluster category of .Q; W / and W per./ ! C.Q;W / denote the canonical projection functor. Assume Q0 D f1; : : : ; ng. Let M be an object of C.Q;W / . Recall that there is a triangle P 1

/ P0

/M

/ †P 1

with P 1 and P 0 in add..//. Define the index of M as ind.M / WD ŒP 0 ŒP 1 2 K0

split

.add..///: Ln The object ./ has a canonical decomposition ./ D iD1 .i /, where i D split ei . Thus Œ.1 /; : : : ; Œ.n / form a (standard) basis for K0 .add..///. Via split this basis we identity K0 .add../// with Zn and identify the index ind.M / of M with the corresponding n-tuple of integers .ind1 .M /; : : : ; indn .M //.

178

T. Brüstle and D. Yang r: 2-silt./ oYYYY YYYY F

/ r: int-co-t-str./ cFF YYYYYY FF YYYYYY F YYYYYY YY YYYY FF , / r: 2-smc./ r: int-t-str./ o O \9 O 9 9 9 9 9 _ _ _ _/ _ o _ _ _ _ _ _ r: 2-silt.J / Y Y r: int-co-t-str.J / 9 FF Y Y Y cF Y Y Y9 F FF Y9 Y Y F FF Y Y Y F# 9 Y Y, F / r: 2-smc.J / r: int-t-str.J / o :: hQQQ :: QQQ :: QQQ : QQQ QQQ ::: QQQ : & / r: f-tors.J / r: s -tilt.J / dII II II II II II , FF FF FF #

r: c-tilt.C.Q;W / /

Figure 3

Theorem 4.12 ([25], Theorem 2.3, and [87], Proposition 3.1). Let M and N be two rigid objects of C.Q;W / . Then M is isomorphic to N if and only if ind.M / D ind.N /. Define the F-polynomial of M as X FM .y1 ; : : : ; yn / WD .Gr d .Hom../; †M ///y1d1 : : : yndn ; d

where d runs over all n-tuples of non-negative integers, Gr d denotes the variety of submodules with dimension vector d and is the Euler–Poincaré characteristic. Now we define a map, the Caldero–Chapoton map: CC W

/ ZŒx ˙1 ; : : : ; x ˙1 ; xnC1 ; : : : ; x2n ; n 1

C.Q;W / M

/ x ind1 .M / : : : x indn .M / F .yO ; : : : ; yO /; M 1 n n 1

Q b where yOi D xnCi jnD1 xj j i for i D 1; : : : ; n, and B.Q/ D .bj i /1j;in is the matrix of Q. This map plays a central role in the theory of additive categorification of cluster algebras, which has proved powerful in understanding cluster algebras, for example in proving a number of Fomin and Zelevinsky’s conjectures. It was originally defined by Caldero and Chapoton [19] to use quiver representations to categorify cluster algebras

Ordered exchange graphs

179

(without coefficients) with defining quiver being of Dynkin type. This work was generalised to all acyclic quivers by Caldero and Keller [21], [20] (see also Hubery [50]) and further to 2-Calabi–Yau triangulated categories by Fu and Keller [36] and by Palu [85], and to Amiot cluster categories of (not necessarily Jacobi-finite) quivers with potential by Plamondon [88], [87]. In parallel, instead of objects in C.Q;W / , Derksen, Weyman and Zelevinsky constructed in [28] the Caldero–Chapoton map for decorated representations over the Jacobian algebra J and gave the first complete proof of some of Fomin and Zelevinsky’s conjectures in the skew-symmetric case; Nagao constructed in [82] the Caldero–Chapoton map for certain objects of per./ and related it to Donaldson– Thomas invariants. Geiß, Leclerc and Schröer took a different approach for stably 2-Calabi–Yau Frobenius categories arising from preprojective algebras in [38], [39], [40] and later they proved in [41] that the two approaches are closely related. Let Cl.Q/ denote the set of clusters of the cluster algebra AQ with principal coefficients (Section 3.10). An important feature of the Caldero–Chapoton map is Theorem 4.13 ([87]). The map CC induces a bijection / Cl.Q/;

r: c-tilt.C.Q;W / /

M

/ fCC.M1 /; : : : ; CC.Mn /; xnC1 ; : : : ; x2n g;

which commutes with mutations. Here M D M1 ˚ ˚ Mn is a decomposition of M into the direct sum of indecomposable objects. Remark 4.14. (a) With C.Q;W / being replaced by a suitable subcategory, Theorem 4.13 holds for any cluster quiver, see [87]. (b) Thanks to [23, Corollary 5.5], with suitable modification of the CC map we can replace in the above theorem the cluster algebra AQ by a cluster algebra with arbitrary coefficients with defining quiver Q. Theorem 4.13 is a consequence of the following Theorem 4.15, isomorphism (4.5) and Proposition 4.16. y is A priori one has to work with the quiver with potential .Q0 ; W /, where Q0 D Q the framed quiver associated to Q with frozen vertices fn C 1; : : : ; 2ng (Section 3.9). y 0 ; W / and J 0 D Jy.Q0 ; W /. They are related to and J by D Let 0 D .Q 0 =.enC1 ˚ ˚e2n / and J D J 0 =.enC1 ˚ ˚e2n /. Let C.Q0 ;W / be the Amiot cluster y W / and 0 W per. 0 / ! C.Q0 ;W / be the canonical projection functor. category of .Q; L 2n 0 0 0 Then 0 D subcategory U of C.Q0 ;W / iD1 i with i D ei . Consider L the full 0 which consists of objects M such that Hom. niD1 0 .nCi /; †M / vanishes. Denote by r: c-tiltLniD1 0 .nCi / .C.Q0 ;W / / the set of isomorphism classes of reachable basic L 0 cluster-tilting objects of C.Q0 ;W / which have niD1 0 .nCi / as a direct summand.

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Theorem 4.15 ([87], Theorem 3.7 and its proof). The map CC induces a bijection r: c-tiltLn

i D1

0 . 0 nCi /

/ Cl.Q/;

.C.Q0 ;W / / M

/ fCC.M1 /; : : : ; CC.Mn /; xnC1 ; : : : ; x2n g;

0 0 which commutes with mutations. Here M D M1 ˚ ˚Mn ˚ 0 .nC1 /˚ ˚ 0 .2n / is a decomposition of M into the direct sum of indecomposable objects. L 0 It follows from Theorem 3.17 that U= add. niD1 0 .nCi // is naturally a 2-Calabi– Yau triangulated category and the additive quotient functor L / U= add. n 0 . 0 // U iD1 nCi

induces an isomorphism of exchange graphs c-tiltLn

i D1

0 . 0 nCi /

Ln

/ c-tilt.U= add.

.C.Q0 ;W / /

iD1

0 0 .nCi // :

L 0 object of U= add. niD1 0 .nCi //. In fact, In particular, 0 . 0 / is a cluster-tilting Ln 0 0 Keller shows that U= add. iD1 .nCi // is triangle equivalent to C.Q;W / and the image of 0 . 0 / is , see [64, Theorem 7.4]. So we obtain an additive quotient functor ˆW U

/ C.Q;W /

which induces an isomorphism of exchange graphs r: c-tiltLn

i D1

0 . 0 nCi /

.C.Q0 ;W / /

/ r: c-tilt.Q;W / :

(4.5)

Recall that for an object in C.Q0 ;W / , the index is an element in K0 .add. 0 . 0 /// which is a free abelian group of rank 2n and the F-polynomial is a polynomial in 2n variables y1 ; : : : ; y2n . Part (a) of the following result is a combination of the proof of [87, Theorem 3.13] and [89, Proposition 3.14]. Proposition 4.16. Let M be an indecomposable object of U which is not isomorphic 0 to 0 .nCi / for any i D 1; : : : ; n. 0 (a) For any i D 1; : : : ; n, the coefficient of ŒnCi in ind.M / is trivial. Moreover, if 0 we identify Œi with Œi for 1 i n, then ind.M / D ind.ˆM /.

(b) The polynomial FM .y1 ; : : : ; yn ; ynC1 ; : : : ; y2n / is constant in ynC1 ; : : : ; y2n . Moreover, FM .y1 ; : : : ; yn ; ynC1 ; : : : ; y2n / D FˆM .y1 ; : : : ; yn /. Proof. (b) This proof is due to Plamondon. It suffices to show: (b1) As a J 0 -module, HomC.Q0 ;W / . 0 . 0 /; †M / is supported on J ; (b2) HomC.Q0 ;W / . 0 . 0 /; †M /, as a J -module, is isomorphic to HomC.Q;W / ../; †ˆM /. 0 (b1) is clear because by the definition of U we have HomC.Q0 ;W / . 0 .nCi /; †M / D 0 for any i D 1; : : : ; n. (b2) follows from [55, Lemma 4.8] since the functor ˆ is a Calabi–Yau reduction and we have ˆ. 0 . 0 // D ./.

Ordered exchange graphs

181

4.7 The correspondences between silting objects, simple-minded collections, cluster-tilting objects, clusters, ice quivers, g-matrices and c-matrices. Let .Q; W / be a Jacobi-finite quiver with potential such that Q is a cluster quiver and W is nony degenerate. Let D .Q; W /, J D Jy.Q; W / and let C.Q;W / denote the Amiot cluster category and Cl.Q/ denote the clusters of the cluster algebra AQ with principal coefficients (Section 3.10). Theorem 4.17. There is a commutative diagram of bijections which commute with mutations / r: 2-smc./ r: 2-silt./ o PPP PPP PPP PP' r: c-tilt.C.Q;W / /

~~ ~~ ~ ~ ~~ ~ ~ Cl.Q/ ~ nn ~~ ~~ nnnnn ~ ~ nn ~~~wnnn g-mat.Q/ o

/ mut.Q/ MMM MMM MMM M& / c-mat.Q/

The four maps r: 2-silt./ $ r: 2-smc./, r: 2-silt./ ! r: c-tilt.C.Q;W / / and r: c-tilt.C.Q;W / / ! Cl.Q/ were defined in Sections 4.1, 4.3 and 4.6, respectively. We will define the other maps and check the commutativity of the diagram. Again we will state the bijectivity only for some maps. Cl.Q/ ! g-mat.Q/: This map is part of the definition of g-matrices, see Section 3.12. By definition it commutes with mutations. r: 2-silt./ ! g-mat.Q/: Assume that Q0 D f1; : : : ; ng. Put i D ei . Then each i is indecomposable in per./ and D 1 ˚ ˚ n is a silting object in per./. Hence, by Theorem 3.12, the Grothendieck group K0 .per.// is free of rank n and the isomorphism classes Œ1 ; : : : ; Œn form a basis of K0 .per.//. Let M D M1 ˚ ˚ Mn be a basic 2-term silting object of per./ with Mi indecomposable for each i. Then the isomorphism classes ŒM1 ; : : : ; ŒMn form a basis of K0 .per.// for the same reason as above. Let TM; be the invertible matrix representing a change of basis from fŒ1 ; : : : ; Œn g to fŒM1 ; : : : ; ŒMn g, i.e. .ŒM1 ; : : : ; ŒMn / D .Œ1 ; : : : ; Œn /TM; : Theorem 4.18 ([82], Theorem 6.18). The assignment M 7! TM; defines a map r: 2-silt./ ! g-mat.Q/, which commutes with mutations. r: c-tilt.C.Q;W / / ! g-mat.Q/: Let M D M1 ˚ ˚ Mn be a basic cluster-tilting object of C.Q;W / with Mi indecomposable for each i . Applying the map ind of taking indices to each Mi , we obtain a matrix TM such that

.ind.M1 /; : : : ; ind.Mn // D .Œ.1 /; : : : ; Œ.n //TM :

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By [87, Proposition 3.6 and the proof of Theorem 3.7] and Proposition 4.16, the assignment M 7! TM defines a map r: c-tilt.C.Q;W / / ! g-mat.Q/, which is the composition r: c-tilt.C.Q;W / / ! Cl.Q/ ! g-mat.Q/:

This map is bijective by Theorem 4.12. That the composition r: 2-silt./ ! r: c-tilt.C.Q;W / / ! g-mat.Q/

is r: 2-silt./ ! g-mat.Q/ follows immediately from the definitions of these three maps. Cl.Q/ ! mut.Q/: Let u be a cluster of AQ . According to Theorem 3.21, there is a unique seed .u; R/. Then the assignment u 7! R defines a map from Cl.Q/ to mut.Q/ which commutes with mutations. mut.Q/ ! c-mat.Q/: This map is part of the definition of c-matrices, see Section 3.11. By definition it commutes with mutations. g-mat.Q/ ! c-mat.Q/: According to [84, Theorem 1.2], taking inverse transpose T 7! T tr defines a bijection between the set of g-matrices and the set of cmatrices which commutes with mutations. One checks the commutativity of the lower square of the desired diagram by using the fact that all the four maps commute with mutations. The bijectivity of Cl.Q/ ! mut.Q/ and mut.Q/ ! c-mat.Q/ is a consequence of the fact that these two maps are surjective and the other two maps in the square are bijective. r: 2-smc./ ! c-mat.Q/: Let X be an object of Dfd ./. Then the cohomologies of X are finite-dimensional modules over J D H 0 ./. Define the dimension vector dim.X/ of X as X .1/i dim.H i .X //: dim.X / WD i2Z

This yields a map dim W r: 2-smc./ fX1 ; : : : ; Xn g

/ Mn .Z/; / .dim.X1 /; : : : ; dim.Xn //;

where Mn .Z/ is the set of n n matrices with integer entries. This map has range

c-mat.Q/ and commutes with mutations, see [66, Theorem 7.9 and Section 7.10]. In

conjunction with Remark 4.11 this implies the sign-coherence of c-vectors, as observed by Keller in [66, Theorem 7.9]. The commutativity of the outer square of the desired diagram follows from the fact that all four maps commute with mutations. But next we give an explanation from a different point of view. Let fS1 ; : : : ; Sn g be a complete set of pairwise non-isomorphic simple J -modules, viewed as a collection of objects in Dfd ./ via the restriction functor p . Recall that both fŒS1 ; : : : ; ŒSn g and fŒX1 ; : : : ; ŒXn g are bases of the Grothendieck group K0 .Dfd .//. In fact, the matrix dim.X1 ; : : : ; Xn / is precisely the matrix representing a change of basis from fŒS1 ; : : : ; ŒSn g to fŒX1 ; : : : ; ŒXn g.

183

Ordered exchange graphs

Observe that there is the Euler form K0 .per.// K0 .Dfd .// .M; X /

/ Z; /P

i i2Z .1/

dim Hom.M; †i X /;

which is non-degenerate. Let M D M1 ˚ ˚Mn be a basic 2-term silting object in per./ with Mi indecomposable for each i and let fX1 ; : : : ; Xn g be the corresponding 2-term simple-minded collection of Dfd ./. Thanks to (4.1), the sets fŒM1 ; : : : ; ŒMn g and fŒX1 ; : : : ; ŒXn g form dual bases of K0 .per.// and K0 .Dfd .// with respect to the Euler form. As a consequence, the matrices representing changes of bases from fŒ1 ; : : : ; Œn g to fŒM1 ; : : : ; ŒMn g and from fŒS1 ; : : : ; ŒSn g to fŒX1 ; : : : ; ŒXn g are related by taking tr the inverse of the transpose. Namely, dim.ŒX1 ; : : : ; ŒXn / D TM; , so the outer square of the diagram is commutative. Remark 4.19. (a) Recall from Section 4.1 that the bijection r: 2-smc.J / ! r: 2-smc./ is induced by the restriction p along the morphism p W ! J , and hence preserves dimension vectors. Thus, combining it with the bijection r: 2-smc./ ! c-mat.Q/, we obtain a bijection dim W r: 2-smc.J / ! c-mat.Q/: This map is expected to help to understand c-mat.Q/ using representation theory over the finite-dimensional algebra J . (b) Let C D .cij /1i;j n be a c-matrix and M D M1 ˚ ˚ Mn be the corresponding reachable basic cluster-tilting object of C.Q;W / . Then there is a triangle p.i /

/L

j Wcij >0

˚cij

Mj

/L

j Wcij 0.

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Let A be a non-positive dg algebra. Then A is a silting object of per.A/ because Homper.A/ .A; †i A/ D H i .A/

vanishes for i > 0. Conversely, let C be an idempotent complete algebraic triangulated category which has a silting object P . Here a triangulated category is said to be algebraic if it is triangle equivalent to the stable category of a Frobenius category. Then by Keller’s Morita theorem for triangulated categories ([61, Theorem 3.8 (b)]), there is a non-positive dg algebra A such that there is a triangle equivalence C ! per.A/ which takes P to A (see for example [75, Lemma 4.1] for a detailed proof). Part (a) of the following theorem is obtained by combining Proposition 6.2.1 and Proposition 5.2.2 of [12], part (b) implicitly appears in [7, Section 2.1] (see for example [58, Section 2.4] for a detailed proof) and part (c) is easy to prove by using (b). Theorem A.1. Let A be a non-positive dg algebra. std std (a) The pair .P0 ; P0 / is a bounded co-t-structure on per.A/ with co-heart the std std category addper.A/ .A/, where P0 (respectively, P0 ) is the smallest full subcati egory of per.A/ which contains † A for i 0 (respectively, i 0) and which is closed under extensions and direct summands. We will refer to this co-t -structure as the standard co-t -structure on per.A/. 0 0 (b) The pair .Dstd ; Dstd / is a bounded t -structure on Dfd .A/ with heart being 0 0 equivalent to mod H 0 .A/, where Dstd (respectively, Dstd ) is the full subcategory of Dfd .A/ containing those dg A-modules whose cohomologies are concentrated in non-positive degrees (respectively, non-negative degrees). We will refer to this t-structure as the standard t -structure on Dfd .A/.

(c) Suppose that H 0 .A/ is finite-dimensional. Then a complete collection of pairwise non-isomorphic simple H 0 .A/ modules, considered as a collection of objects in Dfd .A/, is simple-minded. Let A be a non-positive dg algebra. Set FA D fcone.f / j f is a morphism in addper.A/ .A/g per.A/: Objects of FA will be called 2-term objects of per.A/ for obvious reasons. The following is an easy observation. Lemma A.2. Let X be an object of FA . Then Hom.X; †p X / D 0 for p 2. In particular X is a presilting object if and only if Hom.X; †X / D 0. A.2 The induction functor. Let A be a non-positive dg algebra. Let AN D H 0 .A/. N Then the induction functor Denote by p the canonical projection A ! A. N p W per.A/ ! per.A/ N ! addper.A/ restricts to an additive equivalence addper.A/ .A/ N .A/ and induces a canonN of Grothendieck groups. ical isomorphism K0 .per.A// ! K0 .per.A//

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Ordered exchange graphs

A.2.1 The bijection N induces a bijection Proposition A.3. The induction functor p W per.A/ ! per.A/ between the sets of isomorphism classes of 2-term silting objects. We need the following two 4-lemmas. Lemma A.4. Consider the following commutative diagram of abelian groups with exact rows: / M0 / M1 / M2 M 1

g

f

/ N0

N 1

i

h

/ N1

/ N 2.

(a) If f is surjective, g is injective and i is injective, then h is injective. (b) If f is surjective, h is surjective and i is injective, then g is surjective. Proof of Proposition A.3. We first show that if X is an object of FA then X is a silting object if and only if p .X / is a silting object. There is a triangle /X

X0

/ X 00

/ †X 0

with X 0 2 addper.A/ .A/ and X 00 2 † addper.A/ .A/. Applying the two functors 00 0 Homper.A/ .‹; X 00 / (respectively, Homper.A/ N .‹; p .X //) and Homper.A/ .‹; X / (respec0 tively, Homper.A/ N .‹; p .X //) to this triangle (respectively, its image under p ), we obtain two commutative diagrams .X 0 ; †1 X 00 /

/ .X 00 ; X 00 /

f1

/ .X; X 00 /

f2

/ .X 0 ; X 00 / D 0

f3

.p .X 0 /; †1 p .X 00 // / .p .X 00 /; p .X 00 // / .p .X /; p .X 00 // / .p .X 0 /; p .X 00 // D 0,

.X 0 ; X 0 /

g1

/ .X 00 ; †X 0 /

/ .X; †X 0 /

g2

/ .X 0 ; †X 0 / D 0

g3

.p .X 0 /; p .X 0 // / .p .X 00 /; †p .X 0 // / .p .X /; †p .X 0 // / .p .X 0 /; †p .X 0 // D 0.

N Since p W addper.A/ .A/ ! addper.A/ N .A/ is an equivalence, it follows that f1 , f2 , g1 and g2 are bijective. Therefore by Lemma A.4, f3 and g3 are bijective. Applying Homper.A/ .X; ‹/ (respectively, Homper.A/ N .p .X /; ‹/) to the above triangle (respectively, its image under p ), we obtain the following commutative diagram: .X; X 00 /

f3

/ .X; †X 0 /

g3

/ .X; †X /

h

/ .X; †X 00 / D 0

.p .X /; p .X 00 // / .p .X /; †p .X 0 // / .p .X /; †p .X // / .p .X /; †p .X 00 // D 0.

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By Lemma A.4, h is bijective. Therefore X is a silting object if and only if p .X / is a silting object. The bijectivity is a consequence of the following proposition. Proposition A.5. Let be the ideal of FA consisting of morphisms factoring through morphisms †X ! Y with X; Y 2 addper.A/ .A/. Then 2 D 0 and p induces an equivalence FA = ! FAN . In particular, p is full, detects isomorphisms, preserves indecomposability and induces a bijection between isomorphism classes of objects of FA and those of FAN . Proof. That 2 D 0 holds is because Hom.A; †A/ vanishes. Next we show that p induces an equivalence FA = ! FAN . Then the last statement follows immediately. N Let Y 2 FAN . Then Y Š cone.g/ for a morphism g in addper.A/ N .A/. Since N p W addper.A/ .A/ ! addper.A/ N .A/ is an equivalence, it follows that there is a morphism f in addper.A/ .A/ such that g D p .f /. Take X D cone.f /. Then p .X / Š cone.p .f // D cone.g/ D Y . This shows that p W FA ! FAN is dense. N it follows that the Since there is no non-trivial morphism from †AN to AN in per.A/, image of a morphism in under p is zero. Let X; Y 2 FA . There are triangles in per.A/: PX1

u

/ P0 X

v

/X

w

/ †P 1 ; X

PY1

u0

/ P0 Y

v0

/Y

w0

/ †P 1 Y

with PX1 ; PX0 ; PY1 ; PY0 2 addper.A/ .A/. Applying Homper.A/ .PX0 ; ‹/ respectively 0 Homper.A/ N .p .PX /; ‹/ to the triangle containing Y respectively its image under p , we obtain the following commutative diagram with exact rows / .P 0 ; P 0 / X Y

.PX0 ; PY1 / f1

/ .P 0 ; Y / X

f2

f3

/ .P 0 ; †Y 1 / D 0 X

.p .PX0 /; p .PY1 // / .p .PX0 /; p .PY0 // / .p .PX0 /; p .Y // / .p .PX0 /; †p .PY1 // D 0:

The maps f1 and f2 are bijective, implying that f3 is bijective too. 1 Applying Homper.A/ .†PX1 ; ‹/ respectively Homper.A/ N .p .†PX /; ‹/ to the triangle containing Y respectively its image under p , we obtain the following commutative diagram with exact rows: .†PX1 ; PY0 /

.†p .PX1 /; p .PY0 //

/

g4

/

/

.†PX1 ; Y /

g1

.†p .PX1 /; p .Y //

/

.†PX1 ; †PY1 /

g2

.†p .PX1 /; †p .PY1 //

/ .†PX0 ; †PY1 /

g3

/ .†p .PX0 /; †p .PY1 //:

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Ordered exchange graphs

The vector space .†p .PX1 /; p .PY0 // is trivial and the maps g2 and g3 are bijective. By Lemma A.4 (b), g1 is surjective. A straightforward diagram chasing shows that Ker.g1 / D Im.g4 /. Applying Homper.A/ .‹; Y / respectively Homper.A/ N .‹; p .Y // to this triangle respectively its image under p , we obtain the following commutative diagram with exact rows: .†PX0 ; Y /

/

˛

h1

.†p .PX0 /; p .Y //

˛0

/

/ .X; Y /

ˇ

.†PX1 ; Y / g1

.†p .PX1 /; p .Y //

ˇ0

/

/

h3

.p .X /; p .Y //

0

/

.PX0 ; Y /

/

ı

f3

.p .PX0 /; p .Y //

ı0

/

.PX1 ; Y /

h5

.p .PX1 /; p .Y //:

Recall that g1 is surjective and f3 is bijective, and similarly one shows that h1 is surjective and h5 is bijective. It follows from Lemma A.4 (b) that h3 is surjective. We claim that Ker.h3 / D Im.ˇ B g4 /. Notice that for 2 Homper.A/ .†PX1 ; PY0 / we have that ˇ B g4 . / D v 0 B B w belongs to . So the proof is complete. To prove the claim, take ' 2 Ker.h3 /. Then h4 B .'/ D 0 B h3 .'/ D 0, implying that .'/ D 0. So there is '1 2 Homper.A/ .PX1 ; Y / such that ' D ˇ.'1 /. Then 0 ˇ B h2 .'1 / D h3 B ˇ.'1 / D 0. So there is '2 2 Homper.A/ N .†p .PX /; p .Y // such that g1 .'1 / D ˛ 0 .'2 /. Since h1 is surjective, there is '3 2 Homper.A/ .†PX0 ; Y / such that '2 D h1 .'3 /. Let '10 D '1 ˛.'3 /. Then g1 .'10 / D 0 and ˇ.'10 / D '. Because Ker.g1 / D Im.g4 /, this finishes the proof of the claim. A.2.2 Compatibility with mutations. Assume further that per.A/ is Hom-finite. Proposition A.6. Let M be a 2-term silting object of per.A/ and N be an indecompos able direct summand of M such that N .M / is again 2-term. Then p .N / .p .M // D p .N .M //. Proof. By definition, N .M / D N ˚L, where N is given by the triangle in per .A/ f

N

/E

/ N

/ †N ;

where the morphism f is a minimal left add.L/-approximation. Applying the triangle N functor p we obtain a triangle in per.A/ p .N /

p .f /

/ p .E/

/ p .N /

/ †p .N / ;

where p .f / is a left add.p .L//-approximation. On the other hand, p .N / is indecomposable and p .M / D p .N / ˚ p .L/. By definition, p .N / .p .M // D N p .N / ˚ p .L/, where p .N / is given by the triangle in per.A/ p .N /

g

/ E0

/ p .N /

/ †p .N /;

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where the morphism g is a minimal left add.p .L//-approximation. Therefore p .f / factors through g and we have the following octahedron:

p .N / pp8 bEEEE p p EE pp EE ppp p EE p p p EE p p EE p pp EE o o EE 00 p .N / E h EE bEEQQQ EE EE QQQ E EE Q( ( EE p .N / / p .f / p .E/ EE 7 EE pp8 p EE 777 p pp EE 7 ppp EE g7 h p p EE 77 EE7 ppp E7 pppp p 0 E

The morphism h splits, and hence E 00 belongs to add.p .L//. Consider the triangle p .N /

/ p .N /

/ E 00

/ †p .N / .

Since p .L/ ˚ p .N / is a silting object, the last morphism in the above triangle vanishes. Therefore the triangle splits. Because both p .N / and p .N / are indecomposable, they must be isomorphic and the desired result follows.

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[90] I. Reiten, Cluster categories. In Proceedings of the International Congress of Mathematicians. Volume I, Hindustan Book Agency, New Delhi 2010, 558–594. [91] J. Rickard, Morita theory for derived categories. J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. [92] J. Rickard, Equivalences of derived categories for symmetric algebras. J. Algebra 257 (2002), no. 2, 460–481. [93] J. Rickard and R. Rouquier, Stable categories and reconstruction. Preprint, arXiv:1008.1976 [math.RT]. [94] J. Rickard and A. Schofield, Cocovers and tilting modules. Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 1, 1–5. [95] C. Riedtmann and A. Schofield, On a simplicial complex associated with tilting modules. Comment. Math. Helv. 66 (1991), no. 1, 70–78. [96] C. M. Ringel, Appendix: Some remarks concerning tilting modules and tilted algebras. Origin. Relevance. Future. In Handbook of tilting theory, London Math. Soc. Lecture Note Ser. 332, Cambridge University Press, Cambridge 2007, 413–472. [97] P. Seidel and R. Thomas, Braid group actions on derived categories of coherent sheaves. Duke Math. J. 108 (2001), no. 1, 37–108. [98] D. Speyer and H. Thomas, Acyclic cluster algebras revisited. In Algebras, quivers and representations, Abel Symposia 8, Springer, Berlin 2013, 275–298. [99] L. Unger, The partial order of tilting modules for three-point-quiver algebras. In Representation theory of algebras (Cocoyoc, 1994), CMS Conf. Proc. 18, Amer. Math. Soc., Providence, RI, 1996, 671–679. [100] L. Unger, The simplicial complex of tilting modules over quiver algebras. Proc. London Math. Soc. (3) 73 (1996), no. 1, 27–46. [101] L. Unger, Combinatorial aspects of the set of tilting modules. In Handbook of tilting theory, London Math. Soc. Lecture Note Ser. 332, Cambridge University Press, Cambridge 2007, 259–277. [102] J. Wei, Semi-tilting complexes. Israel J. Math. 194 (2013), no. 2, 871–893. [103] J. Woolf, Stability conditions, torsion theories and tilting. J. London Math. Soc. (2) 82 (2010), no. 3, 663–682. [104] D. Xie, BPS spectrum, wall crossing and quantum dilogarithm identity. Preprint, arXiv:1211.7071 [hep-th]. [105] A. Zelevinsky, What is …a cluster algebra? Notices Amer. Math. Soc. 54 (2007), no. 11, 1494–1495. [106] Y. Zhou and B. Zhu, Mutation of torsion pairs in triangulated categories and its geometric realization. Preprint, arXiv:1105.3521 [math.RT]. [107] Y. Zhou and B. Zhu, Maximal rigid subcategories in 2-Calabi–Yau triangulated categories. J. Algebra 348 (2011), 49–60.

Introduction to Donaldson–Thomas invariants Sergey Mozgovoy

1 Lecture 1 Donaldson–Thomas invariants are numbers that virtually count sheaves on 3-Calabi– Yau manifolds. One of the motivation for their study is the MNOP conjecture [11] (proved for example for toric 3-Calabi–Yau varieties) that asserts a close relationship between Donaldson–Thomas invariants and Gromov–Witten invariants of the same 3-Calabi–Yau manifold (the latter invariants virtually count curves on the manifold). In [9] Kontsevich and Soibelman developed a framework for the refined Donaldson– Thomas invariants of non-commutative 3-Calabi–Yau varieties, that is, of triangulated 3-Calabi–Yau A1 -categories. This framework allows us to shift the study of Donaldson–Thomas invariants from 3-Calabi–Yau varieties to other sources of 3Calabi–Yau categories. One of such sources, closely related to representation theory, consists of quivers with potentials. The goal of these lectures is to define and study refined (or quantized) Donaldson–Thomas invariants of the categories associated to quivers with potentials. According to Kontsevich and Soibelman [9], the Donaldson– Thomas invariants correspond to counting functions of BPS states in physics. Therefore it is natural to expect the integrality and positivity properties for the Donaldson–Thomas invariants. In the first lecture we will quickly introduce numerical Donaldson–Thomas invariants for one simple example and we will see how it can be quantized. In the second lecture we will introduce refined Donaldson–Thomas invariants for arbitrary quivers and we will see that the example from the first lecture corresponds to the generalized Kronecker quiver. We will also discuss various properties of refined Donaldson– Thomas invariants. In the third lecture we will introduce refined Donaldson–Thomas invariants for quivers with potentials and compute them in some situations. x D QŒŒx1 ; x2 be 1.1 Numerical DT invariants. Let m > 0 be an integer and let A a commutative algebra with multiplication x1a x2b x1c x2d D .1/m.ad bc/ x1aCc x2bCd : x by For any .a; b/ 2 N 2 nf0g, define an automorphism Ta;b of the algebra A x1 7! x1 .1 x1a x2b /mb ;

x2 7! x2 .1 x1a x2b /ma :

The following result was proved by Reineke [17] and Kontsevich and Soibelman [10].

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x a;b Theorem 1.1 (Integrality conjecture). There exist uniquely determined integers 2 for .a; b/ 2 N nf0g such that Y x T1;0 T0;1 D Ta;ba;b ; (1) a b"

where the product over .a; b/ 2 N 2 nf0g is taken in increasing order of ab . The integers x a;b are called numerical Donaldson–Thomas invariants. Example 1.2. For m D 1 one can directly check that T1;0 T0;1 D T0;1 T1;1 T1;0 : For m D 2 one has 2 T1;0 T0;1 D T0;1 T1;2 T2;3 : : : T1;1 : : : T3;2 : : : T2;1 : : : T1;0 :

1.2 Poisson algebra approach. Let us give an alternative definition of the automorphisms Ta;b . Define a skew-symmetric form h ; i on Z2 by h.a; b/; .c; d /i D m.ad bc/: Then for any ˛ D .a; b/ 2 N 2 nf0g the automorphism T˛ D Ta;b is given by x ˇ 7! x ˇ .1 x ˛ /h˛;ˇ i D x ˇ .1 .1/h˛;ˇ i x ˛ /h˛;ˇ i ;

ˇ 2 N 2:

x D QŒŒx1 ; x2 given by It preserves the Poisson algebra structure on A fx ˛ ; x ˇ g D h˛; ˇi x ˛ x ˇ D .1/h˛;ˇ i h˛; ˇi x ˛Cˇ : Remark 1.3. A Poisson algebra is a commutative algebra together with a Lie bracket f ; g (called in this situation a Poisson bracket) satisfying the Leibnitz rule ff; g hg D ff; gg h C g ff; hg: To see that T˛ preserves the Poisson bracket we write T˛ as an adjoint automorphism of the corresponding Lie algebra: Lemma 1.4. We have

T˛ D Ad exp

X x n˛ n1

n2

:

Proof. We can write

Ad exp

X 1 x n˛ x n˛ .x ˇ / D ad 2 n kŠ n2 k0

D

k

.x ˇ /

X 1 h˛; ˇi x n˛ k h˛; ˇi x n˛ x ˇ D exp xˇ : kŠ n n

k0

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Therefore

Ad exp

X x n˛ n2

n1

ˇ

.x / D exp

X x n˛ h˛;ˇ i n

n1 ˛ h˛;ˇ i

x ˇ D T˛ .x ˇ /:

D .1 x / Remark 1.5. The series Li2 .x/ D

xˇ

X xn n1

n2

that occurred in the previous lemma is called a dilogarithm. Given a lattice D Zr with a skew-symmetric form h ; i we can, similarly to the x D QŒŒx1 ; : : : ; xr : above discussion, define a Poisson algebra structure on A x ˛ x ˇ D .1/h˛;ˇ i x ˛Cˇ ;

fx ˛ ; x ˇ g D .1/h˛;ˇ i h˛; ˇi x ˛Cˇ :

x by Similarly, for any ˛ 2 N 2 nf0g, define a Poisson algebra automorphism T˛ of A x ˇ 7! x ˇ .1 x ˛ /h˛;ˇ i : As before, we can prove that

T˛ D Ad exp

X x n˛ n1

n2

:

x can be quantized [16]. This means that 1.3 Quantization. The Poisson algebra A 1 we can define an associative algebra structure on Aı D QŒq ˙ 2 ŒŒx1 ; : : : ; xr (or 1 A D Q.q 2 /ŒŒx1 ; : : : ; xr ) 1

x ˛ B x ˇ D .q 2 /h˛;ˇ i x ˛Cˇ x D Aı j such that A

1

q 2 D1

, that is, x ' Aı =.q 12 1/ A

as Poisson algebras, with the Poisson bracket on the right given by ff; gg D

fg gf q1

1

.mod .q 2 1//:

Indeed 1

fx ˛ ; x ˇ g D

1

.q 2 /h˛;ˇ i .q 2 /h˛;ˇ i ˛Cˇ x q1

.1/h˛;ˇ i h˛; ˇi x ˛Cˇ The automorphism T˛ can also be quantized.

1

.mod .q 2 1//:

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Lemma 1.6. T˛ is the specialization at q 2 D 1 of the adjoint operator Ad.E.x ˛ //, where X 1 1 .q 2 x/n 1 E.x/ D exp 2 Q.q 2 /ŒŒx: n n 1q n1 Proof. For any ˇ 2 , define an algebra automorphism Sˇ of A by Sˇ .x ˛ / D q h˛;ˇ i x ˛ . Then x ˛ B x ˇ D q h˛;ˇ i x ˇ B x ˛ D xˇ B Sˇ .x ˛ / and more generally

f B x ˇ D x ˇ B Sˇ .f /;

f 2 A :

We can write E.x ˛ / B x ˇ B E.x ˛ /1 D x ˇ B Sˇ E.x ˛ / B E.x ˛ /1 D x ˇ B exp x ˇ exp

X

1

1 .q nh˛;ˇ i 1/.q 2 x ˛ /n n 1 qn n1

X

x n˛ n n1

h˛;ˇ i

1

T˛ .x ˇ / .mod .q 2 1//:

Remark 1.7 (Quantum dilogaritm). One can show, using the q-binomial theorem, that X

1

1 .q 2 x/n E.x/ D exp n 1 qn n1

D

X .q 12 x/n n0

.q/n

X .q 12 /n2 xn; D jGL .F /j n q n0

(2)

where .q/n D .1 q/ : : : .1 q n / and for the last equality we assume q to be the number of elements in a finite field Fq . The series E.x/ is called the (exponential of the) quantum dilogarithm. Remark 1.8. Define the plethystic exponential Exp W m ! 1 C m on the maximal ideal 1 m Q.q 2 /ŒŒx1 ; : : : ; xr by the rules k

Exp.q 2 x ˛ / D

Exp.f C g/ D Exp.f / Exp.g/;

1 k

1 q 2 x˛

:

One can show that X

1 Exp.f / D exp n n1

nf

;

1

n f .q 2 ; x1 ; : : : ; xr /

n

D f .q 2 ; x1n ; : : : ; xrn /:

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In particular

1

q2 E.x/ D Exp x D Exp 1q

x

: (3) 1 1 q 2 q 2 If we substitute operators T˛ by the elements E.x ˛ / 2 A in the integrality conjecture (1), it is natural to expect an identity of the form (the lattice is D Z2 ) Y E.x1 / B E.x2 / D E.x1a x2b /a;b a b"

for some a;b 2 Q.q/. This is indeed the case if we use the plethystic power map f g WD Exp.g Log.f //;

(4)

where Log W 1 C m ! m is inverse to the plethystic exponential Exp. More precisely, the following result is true. Theorem 1.9 (Kontsevich–Soibelman [10]). There exist uniquely determined polyno1 mials ˛ 2 ZŒq ˙ 2 for ˛ 2 N 2 nf0g such that Y Y ˛ x ˛ ˛ ˛ E.x1 / B E.x2 / D ; E.x / D Exp 1 1 q 2 q 2 ˛" ˛" where the products over ˛ D .a; b/ 2 N 2 nf0g are taken in increasing order of ab . The polynomials ˛ are called quantized (or refined) Donaldson–Thomas invariants. Remark 1.10. The specialization of Ad E.x ˛ /˛ at q 2 D 1 is T˛˛ .1/ . Therefore x ˛ D ˛ .1/. Indeed, given an element P 2 1 C m such that the limit 1

.q 1/ log.P /j

1

q 2 D1

1

is well-defined, the specialization of Ad P at q 2 D 1 exists and equals : Ad exp .q 1/ log.P /j 1 q 2 D1

In our case we obtain

˛ ˛

.q 1/ log E.x /

1

q2 1 ˛ .q 2 /x ˛ D .q 1/ log Exp 1q X 1 q1 n 1 D ˛ .q 2 /.q 2 x ˛ /n n n1q n1 ˛ .1/

X x n˛

n1

n2

1

.mod .q 2 1//:

Therefore the specialization of Ad E.x ˛ /˛ is given by X x n˛ Ad exp ˛ .1/ D T˛˛ .1/ : 2 n n1

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2 Lecture 2 2.1 Quantum affine space of a quiver. Let Q D .Q0 ; Q1 / be a quiver and let D ZQ0 ;

C D N Q0 :

The category Rep.Q/ of finite-dimensional representations of Q over a field k is hereditary, that is, Exti .M; N / D 0 for any i 2 and M; N 2 Rep.Q/. Define the Euler–Ringel form on by X X .˛; ˇ/ D ˛i ˇi ˛i ˇj ; ˛; ˇ 2 : a W i!j

i2Q0

Its basic property is X .1/i dim Exti .M; N / D .dimM; dimN /;

M; N 2 Rep.Q/;

i0

where dimM D .dim Mi /i2Q0 2 N Q0 is the dimension vector of the representation M . Define a skew-symmetric form h ; i on by h˛; ˇi D .˛; ˇ/ .ˇ; ˛/: Example 2.1. Let Q be the m-Kronecker quiver. This is the quiver with two vertices 1, 2 and m arrows from 2 to 1. 1

1

o o

:: :

2

m

Then h˛; ˇi D m.ad bc/;

˛ D .a; b/; ˇ D .c; d /:

This is the same skew-symmetric form that we seen in the first lecture. 1

As before, we define an associative algebra A D Q.q 2 /ŒŒx1 ; : : : ; xr (where r is the number of vertices of Q) with a product 1

x ˛ B x ˇ D .q 2 /h˛;ˇ i x ˛Cˇ : This algebra is called the quantum affine space of the quiver Q. 2.2 Stability conditions. A stability function (see [3, §2]) on the quiver Q is a group homomorphism Z W ! C such that Z.dimM / 2 HC D fre i' j r > 0; ' 2 .0; 1g;

0 ¤ M 2 Rep.Q/:

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Define the phase of ˛ 2 C nf0g to be the number '.˛/ 2 .0; 1 such that Z.˛/ D re i'.˛/ . Define '.M / D '.dimM /. A representation M is called Z-semistable if for any subrepresentation 0 ¤ N ¨ M '.N / '.M /: Remark 2.2. Let Z D d C i r for d; r W ! R. Define the slope .˛/ D ˛ 2 C nf0g. Then '.˛/ '.ˇ/ iff .˛/ .ˇ/

d.˛/ r.˛/

for

and we can use the slope function to verify the semistability. Theorem 2.3 (Harder–Narasimhan filtration, see e.g. [3], §2). For any representation M there exists a unique filtration 0 D M0 ¨ M1 ¨ ¨ Mn D M with semistable Mi =Mi1 and '.M1 =M0 / > > '.Mn =Mn1 /: 2.3 Harder–Narasimhan formula. For any ˛ 2 C nf0g, let M Homk .k˛i ; k˛j / R.Q; ˛/ D e W i!j sst .Q; ˛/ R.Q; ˛/ and let RZ Q be the subset of semistable representations. They admit an action of GL˛ .k/ D i2Q0 GL˛i .k/ by conjugation and R.Q; ˛/= GL˛ .k/ parametrizes the set of isomorphism classes of Q-representations having dimension vector ˛. Now let k D Fq be a finite field. For any ray l HC (subset in HC of the form R>0 for some 2 HC ), define

X

AZ l D1C

1

.q 2 /.˛;˛/

Z.˛/2l

sst jRZ .Q; ˛/j ˛ x 2 A : jGL˛ .Fq /j

(5)

Theorem 2.4 (Wall-crossing formula/Harder–Narasimhan formula, see e.g. [15], Proposition 4.12). We have Õ Y lHC

AZ l D

X ˛2C

1

.q 2 /.˛;˛/

jR.Q; ˛/j ˛ x DW AQ ; jGL˛ .Fq /j

(6)

where the ordered product over rays is taken in clockwise order. In particular, the product on the left is independent of the stability function Z.

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We will sketch the proof of the above theorem. Let A D Rep.Q; k/ be the category of Q-representations over k D Fq (or any other exact k-linear category with finite Hom and Ext1 ). The Hall algebra HQ D H.A/ of the category A (see e.g. [18], [19]) is spanned over Q by all isomorphism classes of objects in A. Multiplication is given by ŒM B ŒN D

X

X gMN ŒX ;

X gMN D jfU X j X=U ' M; U ' N gj:

ŒX

yQ D Similarly, we define the completed Hall algebra H

Q ŒM 2A

Q ŒM .

yQ ) is associative. Theorem 2.5 (Ringel [18]). The algebra HQ (and H Proposition 2.6 (Reineke [15]). The map y op ! A ; IW H Q 1

ŒM 7! .q 2 /.dimM;dimM /

x dimM ; jAut M j

is an algebra homomorphism. Remark 2.7. Theorem 2.5 can be proved for any exact category. Proposition 2.6 can be proved only for hereditary categories (i.e. abelian categories with vanishing Exti for i 2). The heredity assumption is an important limitation and is the reason why one has to develop more involved techniques to prove the wall-crossing formulas for 3-Calabi–Yau categories, which we will discuss in the third lecture. The previous result implies that relations in the Hall algebra can be translated into relations in the quantum torus. In particular, for any ray l 2 HC define AZ l D1C

X

ŒM ;

ŒM Z-sst dimM 2Z 1 .l/

where the sum runs over all isomorphism classes of Z-semistable representations M of Q with dimM 2 Z 1 .l/. Then, by the Harder–Narasimhan filtration 2.3, we obtain Ô Y lHC

ŒAZ l D

X ŒM ; ŒM

where the sum on the right runs over all isomorphism classes of representations of Q. y op ! A , we obtain the Harder–Narasimhan forApplying the integration map I W H Q mula in the quantum torus (6).

Introduction to Donaldson–Thomas invariants

203

2.4 Donaldson–Thomas invariants Definition 2.8 (see [10], Definition 21). Assume that l HC is a ray such that h˛; ˇi D 0

whenever Z.˛/; Z.ˇ/ 2 l:

(7)

1

1 2 Define the Donaldson–Thomas invariants Z .l/ by the formula ˛ 2 Q.q / for ˛ 2 Z P Z ˛ Y Z.˛/2l ˛ x Z ˛ Z ˛ E.x / D Exp Al D ; (8) 1 1 q 2 q 2 Z.˛/2l

where E.x/ D Exp

x

q

1 2

1 q 2

is the quantum dilogarithm.

Remark 2.9. Condition (7) is automatically satisfied if the quiver Q is symmetric, that is, if the number of arrows from i to j equals the number of arrows from j to i for any vertices i; j in Q. In this case the bilinear form h ; i is identically zero. Remark 2.10. Condition (7) means that the subalgebra of A generated by x ˛ with Z.˛/ 2 l is commutative, with multiplication coinciding with the usual multiplication 1 in Q.q 2 /ŒŒx1 ; : : : ; xr . That is the reason why we can talk about the plethystic exponent on this subalgebra. Although we can use equation (8) without the assumption (7), this is unjustified for two reasons: we apply the plethystic exponential which is defined 1 on the commutative algebra Q.q 2 /ŒŒx1 ; : : : ; xr , while AZ lives in the quantum affine l space A , but more importantly, as direct computations show, the invariants Z ˛ do not have any nice properties in contrast to the situation where condition (7) is satisfied (more on this later). Example 2.11. Let Q be a quiver with one vertex and no loops. Then Rep Q is equivalent to the category of finite-dimensional vector spaces and (see Remark 1.7) AQ D

X

1

.q 2 /n

n0

2

xn D E.x/: # GLn .Fq /

There exists just one stability function up to equivalence (we say that two stability functions are equivalent if the corresponding phase functions induce the same partial preorder on C nf0g) and we have 1 D 1 and n D 0 for n > 1. Example 2.12. Let Q be the m-Kronecker quiver as in Example 2.1. There are essentially just two non-trivial stability conditions on Q: the one with '.e1 / > '.e2 / and the other with '.e1 / < '.e2 /. (1) Let Z be a stability function such that '.e1 / > '.e2 /. Let M D .M1 ; M2 ; f / be a representation (here M1 , M2 are vector spaces and f D .fi W M2 ! M1 /iD1;:::;m is an m-tuple of linear maps). Then M 0 D .M1 ; 0; 0/ is a subrepresentation and we have '.M 0 / > '.M /. This implies that if M is semistable

204

S. Mozgovoy

then either M1 or M2 is zero. Semistable objects concentrated in the first vertex form a category equivalent to the category of vector spaces. The corresponding generating function AZ is equal, by the previous example, to E.x1 /. Similarly, l1 of representations concentrated in the second vertex the generating function AZ l2 is equal to E.x2 /. We obtain

Õ Y lHC

AZ D E.x1 / B E.x2 /: l

(2) Let ZC be a stability function such that '.e1 / < '.e2 /. Then '.a; b/ > '.c; d / if and only if ab < dc . This means that the product over rays in clockwise order corresponds to the product over pairs .a; b/ taken in increasing order of ab . By the Harder–Narasimhan formula we obtain ZC Y E.x1 / B E.x2 / D E.x1a x2b /a;b : a b"

This is precisely the equation discussed in the first lecture. In view of the last example we can interpret Theorem 1.9 as a statement about integrality of Donaldson–Thomas invariants for the m-Kronecker quiver. A similar statement is true for an arbitrary quiver: Theorem 2.13 (Kontsevich–Soibelman [10]). Let Z be a stability function on the quiver Q and let l HC be a ray satisfying condition (7). Then, for any ˛ 2 Z 1 .l/, ˙1 2 . we have Z ˛ 2 ZŒq Direct computations show that the Donaldson–Thomas invariants should be positive in the following sense: Conjecture 2.14. Under the conditions of Theorem 2.13, for any ˛ 2 Z 1 .l/, the 1 2 polynomial Z ˛ .q / has non-negative integer coefficients. This conjecture was proved in the case of symmetric quivers (see Remark 2.9). In this case the quantum affine space is commutative and therefore the order in the product of the Harder–Narasimhan formula (6) is irrelevant. We can rewrite (6) as P Z ˛ Y ˛ x Z AQ D : Al D Exp 1 1 q 2 q2 lHC This implies that the Donaldson–Thomas invariants Z ˛ are independent of Z (we denote them just by ˛ ). Theorem 2.15 (Efimov [4]). Let Q be a symmetric quiver. Then, for any ˛ 2 C nf0g, 1 the polynomial ˛ .q 2 / has non-negative integer coefficients.

Introduction to Donaldson–Thomas invariants

205

3 Lecture 3 In this lecture we will define refined Donaldson–Thomas invariants for quivers with potential. The reason we are interested in quivers with potential is that they provide simple examples of 3-Calabi–Yau categories while Donaldson–Thomas invariants are specially designed as invariants of such categories [9]. Moreover, in some cases, 3-Calabi–Yau varieties are derived equivalent to the categories associated to quivers with potential. In these cases we can reduce the computation of Donaldson–Thomas invariants on the 3-Calabi–Yau variety to a representation-theoretic problem, which is usually easier to deal with. 3.1 Jacobian algebra. Let P k be a field, Q D .Q0 ; Q1 / be a quiver, and W be a potential on Q, that is, W D u cu u is a linear combination of cycles in Q. Given a cycle u D a1 : : : an and an arrow a 2 Q1 , we define the cyclic derivative X @u aiC1 : : : an a1 ; : : : ai1 D @a iWai Da

as an element of the path algebra kQ. We can extend cyclic derivatives to potentials by linearity. We define the Jacobian algebra JQ;W as the quotient algebra

JQ;W D kQ

ı @W j a 2 Q1 : @a

Example 3.1. Let Q be the quiver with one vertex and three loops x, y, z. Let W D xyz zyx: Then we have

@W D Œy; z; @x

@W D Œz; x; @y

@W D Œx; y: @Z

Therefore JQ;W ' kŒx; y; z, which is the structure ring of k3 , and is a 3-Calabi–Yau algebra. Example 3.2. Let G SL3 .C/ be a finite abelian subgroup. We can decompose the induced representation V D C 3 D 1 ˚ 2 ˚ 3 ; where i are irreducible one-dimensional representations satisfying 1 2 3 D 1. Define the McKay quiver Q of .G; V /: y of irreducible representations of G. (1) The set of vertices is the set G

206

S. Mozgovoy

y (2) The set of arrows ! is a basis of HomG . ; ˝ V / for any ; 2 G. We can parametrize arrows in Q as y i D 1; 2; 3; 2 G;

ai W i ! ;

and we can parametrize 3-cycles in Q (up to cyclic shift) as u W ! 1 2 ! 1 ! ;

y 2 S3 : 2 G;

Finally, define a potential as the linear combination of 3-cycles X W D sgn u :

u =

Theorem 3.3. We have JQ;W ' CŒx; y; zÌG. In particular, JQ;W is a 3-Calabi–Yau algebra. In the case of a non-abelian subgroup G SL3 .C/ one can still endow the McKay quiver with a potential in a canonical way and the Jacobian algebra is Morita-equivalent to CŒx; y; z Ì G (see [6]). 3.2 Ginzburg dg algebra. In general, the Jacobian algebra JQ;W is not a 3-Calabi– Yau algebra. Nevertheless, we can always associate with .Q; W / a differential graded 3-Calabi–Yau algebra, called a Ginzburg dg algebra [6], closely related to JQ;W . y to have the same set of vertices as Q and the following Define a new graded quiver Q arrows: (1) an arrow a W i ! j of degree 0 for any arrow a W i ! j in Q, (2) an arrow a W j ! i of degree 1 for any arrow a W i ! j in Q, (3) a loop ti W i ! i of degree 2 for any vertex i in Q. Define the Ginzburg differential graded algebra Q;W as the path algebra kQ with the following differential: X @W da D 0; da D Œa; a ei : ; dti D ei @a a2Q1

It follows from our definitions that Q;W is concentrated in non-negative degrees. Moreover H 0 .Q;W / ' JQ;W : Theorem 3.4 (Keller–Van den Bergh [8]). The algebra Q;W is a 3-Calabi–Yau algebra. Proposition 3.5 (Amiot [1]). The derived category D.Q;W / has a natural t-structure with heart naturally equivalent to Mod JQ;W .

Introduction to Donaldson–Thomas invariants

207

3.3 Donaldson–Thomas invariants. We have seen that the category of modules over the Jacobian algebra JQ;W can be considered as the heart of a triangulated 3-Calabi– Yau category. In this situation Kontsevich and Soibelman [9] managed to construct the Donaldson–Thomas invariants. Here is a brief description of their approach: (1) Construct an algebra homomorphism I W H.JQ;W / ! A (this is the most difficult step). 2 A counting Z-semistable modules over (2) Define the generating series AZ l JQ;W . Q AZ is independent of Z. (3) Prove the wall-crossing formula: Õ l2HC l (4) Define the DT invariants of the Jacobian algebra by the formula Y E.x ˛ /˛ : AZ l D Z.˛/2l

We will define the Donaldson–Thomas invariants for quivers with potential admitting a so-called cut. A cut of .Q; W / is a subset C Q1 such that W is homogeneous of degree 1 with respect to the grading defined on arrows by ´ 1; a 2 C; deg a D 0; a … C: The significance of a cut is that it defines an action of k on R.Q; ˛/, t .Ma /a2Q1 D .t deg a Ma /a2Q1 ; such that the map w˛ W R.Q; ˛/ ! k;

M 7! tr.W jM /;

is equivariant with respect to this action. Lemma 3.6. The space of representations of the Jacobian algebra R.JQ;W ; ˛/ R.Q; ˛/ coincides with the degeneracy locus of the map w˛ : fM 2 R.Q; ˛/ j dw˛ .M / D 0g: In order to define an analogue of the series AZ from the previous lecture (see l formula (5)), we would like to substitute the set of Z-semistable Q-representations sst sst RZ .Q; ˛/ by the set of Z-semistable JQ;W -representations RZ .JQ;W ; ˛/. Unfortusst nately, the space RZ .JQ;W ; ˛/ has, in general, singularities and is not well-suited for our purposes. As we have seen, this space can be identified with the degeneracy locus

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sst .Q; ˛/ ! k. Therefore one uses certain invariant of this function, called of w˛ W RZ a motivic vanishing cycle [2], in order to define an analogue of the series AZ . In the l presence of a cut, this invariant can be written as a difference of motivic classes of w˛1 .0/ and w˛1 .1/ (see [2, Proposition 1.10]). Therefore, for a finite field k D Fq , we define [13], [14]

X

AZ l;Q;W D

1

.q 2 /.˛;˛/

Z.˛/2l

sst sst jw˛1 .0/ \ RZ .Q; ˛/j jw˛1 .1/ \ RZ .Q; ˛/j ˛ x : jGL˛ .Fq /j

Using this definition it is not difficult to prove the wall-crossing formula by the same methods as we applied in the previous lecture. More precisely, we have [13] Õ Y l2HC

AZ l;Q;W D

X

jw˛1 .0/j jw˛1 .1/j ˛ x DW AQ;W : jGL˛ .Fq /j

1

.q 2 /.˛;˛/

˛2C

(9)

In particular, the product on the left is independent of the stability function Z. We define the Donaldson–Thomas invariants Z ˛;Q;W in the same way as in the previous lecture (under the assumption (7)) P Z ˛ Y Z.˛/2l ˛;Q;W x Z ˛ Z Al;Q;W D E.x / ˛;Q;W D Exp : (10) 1 1 q 2 q 2 Z.˛/2l Remark 3.7. If W is zero, then w˛1 .0/ D R.Q; ˛/ and w˛1 .1/ is empty. Therefore X

AZ l;Q;W D

1

.q 2 /.˛;˛/

Z.˛/2l

sst jRZ .Q; ˛/j ˛ x : jGL˛ .Fq /j

This formula coincides with the formula (5) for quivers without potential. Conjecture 3.8. In the same way as for quivers without potentials (see Conjecture 2.14) 1 ˙1 2 and Z 2 we conjecture that Z ˛;Q;W 2 ZŒq ˛;Q;W .q / has non-negative coefficients. y be the same quiver that we constructed from Example 3.9. Let Q be a quiver and let Q y by the formula Q in the definition of the Ginzburg algebra. Define a potential on Q X X X W D .tj aa ti a a/ D ti Œa; a : i2Q0

.a W i!j /2Q1

a2Q1

y W / admits a cut: The pair .Q; C D fti j i 2 Q0 g: Therefore we can define the series AZ y

l;Q;W

, AQ;W and the Donaldson–Thomas invariy

y is symmetric and therefore the ants using the above formulas. Note that the quiver Q

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Donaldson–Thomas invariants Z y are independent of Z. We denote them just by ˛;Q;W ˛ . One can show that AQ;W D y

X

q Q .˛;˛/

˛2C

jR.…Q ; ˛/j ˛ x ; jGL˛ .Fq /j

P x x where …Q D kQ=. a2Q1 Œa; a / is the preprojective algebra of Q (Q is a subquiver y containing arrows a, a for a 2 Q1 ). of Q For example, let Q consist of one vertex and one loop. Then R.…Q ; n/ D f.A; B/ 2 Mat 2nn j AB D BAg: The number of points in this set was computed half a century ago by Feit and Fine [5] X 1 qx n D Exp : D jGLn .Fq /j 1 q 1 1 q 1k x n n1 n1

X jR.…Q ; n/j n0

X

k0

We can generalize this result to arbitrary quivers [12]: X ˛2C

q Q .˛;˛/

jR.…Q ; ˛/j ˛ x D Exp jGL˛ .Fq /j

P

a˛ .q/x ˛ ; 1 q 1 ˛

where a˛ .q/ is the polynomial counting absolutely indecomposable representations of Q having dimension vector ˛ over finite fields. Comparing this formula with the definition of Donaldson–Thomas invariants (10) we obtain 1

1

˛ .q 2 / D q 2 a˛ .q/: According to the Kac conjecture, proved recently in [7], the polynomials a˛ .q/ have 1 1 non-negative integer coefficients. This implies that ˛ 2 ZŒq ˙ 2 and ˛ .q 2 / have non-negative coefficients, justifying the positivity conjecture 3.8.

References [1] C. Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential. Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2525–2590. [2] K. Behrend, J. Bryan, and B. Szendr˝oi, Motivic degree zero Donaldson-Thomas invariants. Invent. Math. 192 (2013), 111–160. [3] T. Bridgeland, Stability conditions on triangulated categories. Ann. of Math. (2) 166 (2007), no. 2, 317–345, [4] A. I. Efimov, Cohomological Hall algebra of a symmetric quiver. Compos. Math. 148 (2012), no. 4, 1133–1146,

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[5] W. Feit and N. J. Fine, Pairs of commuting matrices over a finite field. Duke Math. J. 27 (1960), 91–94. [6] V. Ginzburg, Calabi-Yau algebras. Preprint, arXiv:math/0612139 [math.AG]. [7] T. Hausel, E. Letellier, and F. Rodriguez-Villegas, Positivity of Kac polynomials and DTinvariants for quivers. Ann. of Math. 177 (2013), no. 3, 1147–1168. [8] B. Keller, Deformed Calabi-Yau completions. With an appendix by Michel Van den Bergh, J. Reine Angew. Math. 654 (2011), 125–180, [9] M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. Preprint, arXiv:0811.2435 [math.AG] . [10] M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants. Commun. Number Theory Phys. 5 (2011), 231–352. [11] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. I. Compos. Math. 142 (2006), no. 5, 1263–1285. [12] S. Mozgovoy, Motivic Donaldson-Thomas invariants and McKay correspondence. Preprint, arXiv:1107.6044 [math.AG]. [13] S. Mozgovoy, On the motivic Donaldson-Thomas invariants of quivers with potentials. Math. Res Lett. 20 (2013), 121–132. [14] K. Nagao, Wall-crossing of the motivic Donaldson-Thomas invariants. Preprint, arXiv:1103.2922 [math.AG]. [15] M. Reineke, The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli. Invent. Math. 152 (2003), no. 2, 349–368. [16] M. Reineke, Poisson automorphisms and quiver moduli. J. Inst. Math. Jussieu 9 (2010), no. 3, 653–667, [17] M. Reineke, Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants. Compos. Math. 147 (2011), no. 3, 943–964, [18] C. M. Ringel, Hall algebras and quantum groups. Invent. Math. 101 (1990), no. 3, 583–591. [19] O. Schiffmann, Lectures on Hall algebras. Preprint, arXiv:math/0611617 [math.RT].

Cluster algebras and singular supports of perverse sheaves Hiraku Nakajima

Introduction In [27], the author found an approach to the theory of cluster algebras, based on perverse sheaves on graded quiver varieties. This approach gave a link between two categorical frameworks for cluster algebras, the additive one via the cluster category by Buan et al. [5] and the multiplicative one via the category of representations of a quantum affine algebra by Hernandez–Leclerc [12]. See also the survey article [20]. In [27, §1.5], the author asked four problems in the to-do list, to which he thought that the same approach can be applied. Except the problem (3), they have been subsequently solved in works by Qin [28], [29] and Kimura–Qin [18]. Let us explain other related problems, for which the approach does not work. We need a new idea to attack these. A problem, discussed in this paper, is a natural generalization of the problem (4) in the to-do list. The author asked to find a relation between the work of Geiß–Leclerc– Schröer [8] and [27] there. But the work [8] dealt with more general cases than those corresponding to [27]. A main conjecture says that every cluster monomial in the coordinate ring of a unipotent subgroup is a Lusztig’s dual canonical base element. Later the theory is generalized to the q-analog, where the canonical base naturally lives [9]. Therefore it is desirable to find a relation between the cluster algebra structure and perverse sheaves on the space of quiver representations, which give the canonical base of the quantum enveloping algebra. Another problem is not discussed here, but possibly related to the current one via [13]. In [12], Hernandez–Leclerc conjectured that the Grothendieck ring of representations of the quantum affine algebra has a structure of a cluster algebra so that every cluster monomial is a class of an irreducible representation. Those irreducible representations are simple perverse sheaves on graded quiver varieties. But what was proved in [27] is the special case of the conjecture only for a certain subalgebra of the Grothendieck ring. The first part of the conjecture has been subsequently proved in [13]. But the latter part, every cluster monomial is an irreducible representation, is still open. In this paper, we propose an approach to the first problem. It is not fully developed yet. We will give a few results, which indicate that we are going in the right direction. The new idea is to use the singular support of a perverse sheaf, which is a lagrangian

Supported by the Grant-in-aid for Scientific Research (No.23340005), JSPS, Japan.

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subvariety in the cotangent bundle. The latter is related to the representation theory of the preprojective algebra, which underlies the work [8]. The paper is organized as follows. In the first section, we briefly recall the canonical and semicanonical bases. In the second section, we review works of Geiß–Leclerc– Schröer [8], [9], where a quantum cluster algebra structure on a quantum unipotent subgroup is introduced. A subcategory, denoted by Cw , of the category of nilpotent representations of the preprojective algebra plays a crucial role. In the third section, we study how the singular support behaves under the restriction functor for perverse sheaves. The restriction functor gives a multiplication in the dual of the quantum enveloping algebra. Our main result is the estimate in Theorem 3.2. In the final section, we give two conjectures, which give links between the theory of [8], [9] and perverse sheaves via singular support. Acknowledgments. The main conjecture (Conjecture 4.2) was found in the spring of 2011, and has been mentioned to various people since then. The author thanks Masaki Kashiwara and Yoshihisa Saito for discussion on the conjecture. He also thanks the referee who points out a relation between the main conjecture and a conjecture in [7, §1.5].

1 Preliminaries 1.1 Quantum enveloping algebra. Let g be a symmetrizable Kac–Moody Lie algebra. We assume g is symmetric, as we use an approach to g via the Ringel-Hall algebra for a quiver. Let I be the index set of simple roots, P be the weight lattice, and P be its dual. Let ˛i denote the i th simple root. Let Uq be the corresponding quantum enveloping algebra, that is a Q.q/-algebra generated by ei , fi (i 2 I ), q h (h 2 P ) with certain relations. Let Uq be the subalgebra generated by fi . We set wt.ei / D ˛i , wt.fi / D ˛i , wt.q h / D 0. Then Uq is graded by P . The quantum enveloping algebra Uq is a Hopf algebra. We have a coproduct W Uq ! Uq ˝ Uq . It does not preserve Uq , but Lusztig introduced its modification r W Uq ! Uq ˝Uq such that r.fi / D fi ˝1C1˝fi and r is an algebra homomorphism with respect to the multiplication on Uq ˝ Uq given by .x1 ˝ y1 / .x2 ˝ y2 / D q .wt x2 ;wt y1 / x1 x2 ˝ y1 y2 ;

(1.1)

where xi , yi are homogeneous elements. We call r the twisted coproduct. Let A D ZŒq; q 1 . Then Uq has an A-subalgebra AUq generated by q-divided

powers fi.n/ D fin =ŒnŠ, where Œn D .q n q n /=.qq 1 / and ŒnŠ D ŒnŒn1 : : : Œ1. Then r induces AUq ! AUq ˝ AUq , which is denoted also by r.

1.2 Perverse sheaves on the space of quiver representations and the canonical base. Consider the Dynkin diagram G D .I; E/ for the Kac–Moody Lie algebra g,

Cluster algebras and singular supports of perverse sheaves

213

where I is the set of vertices, and E the set of edges. Note that G does not have an edge loop, i.e., an edge connecting a vertex to itself. Let H be the set of pairs consisting of an edge together with its orientation. So we have #H D 2#E. For h 2 H , we denote by i.h/ (resp. o.h/) the incoming (resp. outgoing) vertex of h. For h 2 H we denote by hN the same edge as h with the reverse orientation. Choose and fix an orientation of the graph, i.e., a subset H such x [ D H, \ x D ;. The pair .I; / is called a quiver. that Let V D .Vi /i2I be a finite dimensional I -graded vector space over C. The dimension of V is a vector dim V D .dim Vi /i2I 2 ZI0 : We define a vector space by M

def:

EV D

Hom.Vo.h/ ; Vi.h/ /:

h2

Let GV be an algebraic group defined by Y def: GL.Vi /: GV D Its Lie algebra is the direct sum

L i

i

gl.Vi /. The group GV acts on EV by

1 /h2 : B D .Bh /h2 7! g B D .gi.h/ Bh go.h/

The space EV parametrizes isomorphism classes of representations of the quiver with the dimension vector dim V together with a linear base of the underlying vector space compatible with the I -grading. The action of the group GV is induced by the change of bases. In [22], [24] Lusztig introduced a full subcategory PV of the abelian category of perverse sheaves on EV . Its definition is not recalled here. See [22, §2] or [24, Chapter 9]. Its objects are GV -equivariant. Let D .EV / be the bounded derived category of complexes of sheaves of C-vector spaces over EV . Let QV be the full subcategory of D .EV / consisting of complexes that are isomorphic to finite direct sums of complexes of the form LŒd for L 2 PV , d 2 Z. Let K .QV / be the Grothendieck group of QV , that is the abelian group with generators .L/ for isomorphism classes of objects L of QV with relations .L/C.L0 / D .L00 / whenever L00 is isomorphic to L ˚ L0 . It is a module over A D ZŒq; q 1 , where q corresponds to the shift of complexes in QV . Then K .QV / is a free A-module with a basis .L/ where L runs over PV . L Let us consider the direct sum V K .QV / over all isomorphism classes of finite dimensional I -graded vector spaces. Let Si be the I -graded vector space with dim Si D 1, dim Sj D 0 for j ¤ i . Then the corresponding space ESi is a single

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in K .QSi /. Then point. Let 1i be the constant sheaf on ESi , viewed as an element L Lusztig defined a multiplication and a twisted coproduct on V K .QV / such that the A-algebra homomorphism M K .QV / ˆ W AUq ! V

with ˆ.fi / D 1i is an isomorphism respecting twisted coproducts [22], [24]. The construction was motivated by an earlier work by Ringel [30]. L We here recall the definition of the twisted coproduct on V K .QV /. For the definition of the multiplication, see the original papers. Let W be an I -graded subspace of V . Let T D V =W . Let E.W / be the subspace of EV consisting of B 2 EV which preserves W . We consider the diagram

EV ; ET EW E.W / !

(1.2)

where is the inclusion and is the map given by assigning to B 2 E.W / its restriction to W and the induced map on T . Consider the functor Res D Š ./Œd W D .EW / ! D .ET EW /; def:

where d is a certain explicit integer, whose definition is omitted here as it is not relevant for the discussion in this paper. It is known that Res sends QW to the subcategory QT;W of D .ET EW / consisting of complexes that are isomorphic to finite direct sums of complexes of the form .L L0 /Œd for L 2 PT , L0 2 PW , d 2 Z. Therefore we have an induced A-linear homomorphism Res W K .QV / ! K .QT;W / Š K .QT / ˝A K .QW /: We take direct sum over V , T , W to get a homomorphism of an algebra with respect to the twisted multiplication (1.1). It corresponds to r on AUq under ˆ. Recall that K .QV / has an A-basis .L/, where L runs over PV . Taking the direct sum over V , and pulling back by the isomorphism ˆ, we get an A-basis of AUq . This is Lusztig’s canonical basis. Let us denote it by B .1/. Kashiwara gave an algebraic approach to B .1/. See [15] and references therein for details. He first introduced an A0 -form L .1/ of Uq , where A0 D ff 2 Q.q/ j f is regular at q D 0g. Then he also defined a basis, called the crystal base of L .1/=q L .1/. Then he introduced the global crystal base of Uq , which descends to the crystal base of L .1/=q L .1/. It turns out that the global crystal base and the canonical base are the same. See [11]. In this paper, we do not distinguish the crystal base and the global base, that is the canonical base. We denote both by B .1/. When we want to emphasize that a canonical base element b 2 B .1/ is a perverse sheaf, we denote it by Lb .

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215

1.3 Dual canonical base. There exists a unique symmetric bilinear form . ; / on Uq satisfying .1; 1/ D 1;

.fi ; fj / D ıij ;

.r.x/; y ˝ z/ D .x; yz/ for x, y, z 2 Uq : Our normalization is different from [24, Chapter 1] and follows Kashiwara’s as in [19], [9]. Under . ; /, we can identify the graded dual algebra of Uq , an algebra with the multiplication given by r, with Uq itself. Let B up .1/ denote the dual base of B .1/ with respect to . ; /. It is called the dual canonical base of Uq . For b1 ; b2 ; b3 2 B .1/, let us define rbb31 ;b2 2 A by r.b3 / D

X b1 ;b2 2B.1/

up

up

rbb31 ;b2 b1 ˝ b2 :

up

Let b1 , b2 , b3 2 B up .1/ be the dual elements corresponding to b1 , b2 , b3 respectively. Then we have X up up up rbb31 ;b2 b3 : b1 b2 D up

b3 2Bup .1/

Thus the structure constant is given by rbb31 ;b2 . 1.4 Lusztig’s lagrangian subvarieties and crystal. Let us introduce Lusztig’s lagrangian subvariety in the cotangent space of the space of quiver representations. The dual space to EV is M EV D Hom.Vo.h/ ; Vi.h/ /: x h2

The group GV acts on EV in the same way as on EV . The GV -action preserves the natural pairing between EV and EV . Considering E LV ˚ EV as a symplectic manifold, we have the moment map D .i / W EV ˚ EV ! i gl.Vi / given by X ".h/Bh BhN ; i .B/ D i.h/Di

x and ".h/ D 1 if h 2 and 1 where B has components Bh for both h 2 and , otherwise. Lusztig’s lagrangian ƒV [21], [22] is defined as ƒV D fB 2 EV ˚ EV j .B/ D 0; B is nilpotentg: def:

(1.3)

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This space parametrizes isomorphism classes of nilpotent representations of the preprojective algebra associated with the quiver .I; /, together with a linear base of the underlying vector space compatible with the I -grading. The action of the group GV is induced by the change of bases. The preprojective algebra is denoted by ƒ in this paper. This is a lagrangian subvariety in EV ˚ EV . (It was proved that ƒV is halfdimensional in EV ˚ EV in [22, 12.3]. And the same argument shows that it is also a lagrangian. Otherwise use [26, Theorem 5.8] and take the limit W ! 1.) Let Irr ƒV be the set of irreducible components of ƒV . Lusztig defined a structure of an abstract crystal (see [16, §3] for the definition) on Irr ƒV in [21], and Kashiwara– Saito proved that it is isomorphic to the underlying crystal of the canonical base B .1/ of Uq [16]. We denote by ƒb the irreducible component of ƒV corresponding to a canonical base element b 2 B .1/. 1.5 Dual semicanonical base. Let C.ƒV / be the Q-vector space of Q-valued constructible functions over ƒV , which is invariant under the GV -action. Lusztig defined an operator C.ƒT / C.ƒW / ! C.ƒV / for V D T ˚ W , under which the direct sum L V C.ƒV / is an associative algebra (see [22, §12]). If V D Si , then ƒSi is a single point. Let L 1i be the constant function on ƒSi with the value 1. Let C0 be the subalgebra of V C.ƒV / generated by the elements 1i (i 2 I ), and let C0 .ƒV / D C0 \ C.ƒV /. Then Lusztig (see [22, Theorem 12.13]) proved that C0 is isomorphic to the universal enveloping algebra U.n/ of the lower triangular subalgebra n of g by fi 7! 1i . Note that we have an embedding ƒT ƒW ! ƒV given by the direct sum, where V D T ˚ W as above. Then the restriction defines an operator C.ƒV / ! C.ƒT / ˝ C.ƒW /. Geiß–Leclerc–Schröer proved that it sends C0 .ƒV / to C0 .ƒT / ˝ C0 .ƒW /, and gives the natural cocommutative coproduct on U.n/ under the isomorphism C0 Š U.n/ (see [7, §4]). Let Y be an irreducible component of ƒV . Then we consider the functional Y W C.ƒV / ! Q given by taking the value on a dense open subset of Y . Then fY j Y 2 Irr ƒV g gives a base of C0 .ƒV /. This follows from [23, §3] together with the result of Kashiwara–Saito mentioned above. Under the isomorphism U.n/ Š C0 , the base Y is called the dual semicanonical base of U.n/gr , where U.n/gr denotes the graded dual of U.n/. We have a natural bijection b up 7! ƒb between the dual canonical base and the dual semicanonical base. However ƒb is different from the specialization of b up at q D 1 in general. See [7, §1.5] for a counter-example.

Cluster algebras and singular supports of perverse sheaves

217

2 Cluster algebras and quantum unipotent subgroups C C C We fix a Weyl group element w throughout this section. L Let w D \ w. /, C where is the set of positive roots. Then n.w/ D ˛2C g˛ is a Lie subalgebra w of g, where g˛ is the root subspace corresponding to the root ˛.

2.1 Quantum unipotent subgroup. Let us briefly recall the q-analog of the universal enveloping algebra U.n.w// of n.w/, denoted by Uq .w/. See [24, Chapter 40], [19, §4] and [9] for more details. (It is denoted by Aq .n.w// in [9].) 00 Let Ti be the braid group operator corresponding to i 2 I , where Ti D Ti;1 in the notation in [24]. Choose a reduced expression w D si1 si2 : : : si` . Then it gives ˇp D si1 si2 : : : sip1 .˛ip /, and we have C w D fˇp g1p` . We define a root vector Ti1 Ti2 : : : Tip1 .fip /: Let c D .c1 ; : : : ; c` / 2 Z`0 . We multiply q-divided powers of root vectors in the order given by ˇ1 , …, ˇ` : .c` /

L.c/ D fi1.c1 / Ti1 .fi2.c2 / / : : : .Ti1 : : : Ti`1 /.fi`

/:

Then the Q.q/-subspace spanned by L.c/ (c 2 Z`0 ) is independent of the choice of a reduced expression of w. This subspace is Uq .w/. Moreover L.c/ gives a basis of Uq .w/. It can be shown that Uq .w/ is a subalgebra of Uq . It is known that Uq .w/ is compatible with the dual canonical base, i.e., Uq .w/ \ up B .1/ is a base of Uq .w/. This is an interpretation of the main result due to Lusztig [25, Theorem 1.2], based on an earlier work by Saito [31]. (See [19, Theorem 4.25] for the current statement.) def:

Let B up .w/ D Uq .w/ \ B up .1/. Let B .w/ B .1/ be the corresponding subset in the canonical base. Then [25, Proposition 8.3] gives a parametrization of B .w/ as follows. Let L .1/ be the A0 -form used in the definition of the crystal base. Then one shows L.c/ 2 L .1/ and the set fL.c/ mod q L .1/g is equal to B .w/, where B .w/ is considered as a subset of L .1/=q L .1/. Let us denote by b.c/ 2 B .w/ the canonical base element corresponding to L.c/ mod q L .1/. Therefore b.c/ L.c/ mod q L .1/. The argument in the proof of [2, Theorem 3.13] shows that the transition matrix between the base fL.c/g and fb.c/g is upper triangular with respect to the lexicographic order on fcg. It is known that fL.c/g is orthogonal with respect to . ; / (see [24, Proposition 40.2.4]). Therefore we can deduce the corresponding relation between b up .c/ and Lup .c/ D L.c/=.L.c/; L.c//, where b up .c/ is the dual canonical base element corresponding to b.c/. (See [19, Theorem 4.29].)

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2.2 B .w/ and Kashiwara operators. Let us give a characterization of B .w/ in terms of Kashiwara operators on B .1/. Recall that B .1/ is an abstract crystal, and hence has maps wt W B .1/ ! P , "i W B .1/ ! Z, 'i W B .1/ ! Z (i 2 I ) together with Kashiwara operators eQi W B .1/ ! B .1/ t f0g, fQi W B .1/ ! B .1/ t f0g satisfying certain axioms. We denote by u1 the element in B .1/ corresponding to 1 in Uq . There is also an operator W B .1/ ! B .1/, which corresponds to the anti-involution W Uq ! Uq given by fi 7! fi . Therefore we have another set of maps and operators "i D "i , 'i D 'i , eQi D eQi , fQi D fQi . Let w D si1 si2 : : : si` as above. For i D i1 , we have "i .b/ D c1 ;

´ eQi b.c/ D

b.c0 / 0

if c1 ¤ 0; if c1 D 0;

fQi b.c/ D b.c00 /;

where c0 D .c1 1; c2 ; : : : ; c` /, c00 D .c1 C 1; c2 ; : : : ; c` /. In particular, B .w/ is invariant under eQi , fQi . Saito [31, Corollary 3.4.8] introduced a bijection ƒi W fb 2 B .1/ j "i .b/ D 0g ! fb 2 B .1/ j "i .b/ D 0g by

ƒi .b/ D .fQi /'i .b/ .eQi /"i .b/ b;

Q 'i .b/ .eQ /"i .b/ b: ƒ1 i .b/ D .fi / i

This is related to the braid group operator as follows. Let us consider w D si1 si2 : : : si` and w 0 D si2 : : : si` si1 D si1 wsi1 and corresponding PBW base elements .c` /

L D fi1.c1 / Ti1 .fi2.c2 / / : : : .Ti1 : : : Ti`1 /.fi` .c20 / i2

L0 D f

.c30 / i3

Ti2 .f

.c10 / i1

/ : : : .Ti2 : : : Ti` /.f

/;

/;

for .c1 ; : : : ; c` / 2 Z`0 , .c20 ; : : : ; c`0 ; c10 / 2 Z`0 . If c1 D 0, c2 D c20 , …, c` D c`0 , 0 D c10 , we have L D Ti1 L0 . Let b, b 0 be the canonical base elements corresponding to L and L0 respectively. Then Saito [31, Proposition 3.4.7] proved that the corresponding canonical base elements are related by b D ƒi1 b 0 . As a corollary of this result, we have a bijection ƒi

fb 2 B .w 0 / j "i .b/ D 0g ! fb 2 B .w/ j "i .b/ D 0g: This together with the invariance of B .w/ under eQi1 , fQi1 gives a characterization of B .w/ inductively in the length of w, starting from B .1/ D fu1 g. 2.3 A subcategory Cw . In view of §1.4 it is natural to look for a characterization of B .w/ in terms of Lusztig’s lagrangian subvarieties ƒV , or the representation theory

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of the preprojective algebra. It turns out to be related to the subcategory introduced by Buan–Iyama–Reiten–Scott [4], and further studied by Geiß–Leclerc–Schröer [8] and Baumann–Kamnitzer–Tingley [1]. We do not recall here the definition of the subcategory of the category of finitedimensional nilpotent representations of the preprojective algebra, denoted by Cw following [8]. This is because there are many equivalent definitions, and the author does not know what is the best for our purpose. See the above papers. Let ƒw V D fB 2 ƒV j B 2 Cw g, where we identify B with the corresponding representation of the preprojective algebra. This is an open subvariety in ƒV (see [10, Lemma 7.2]). We set C0w .ƒV / D ff 2 C0 .ƒV / j f .X / D 0 for X 2 Cw g: Therefore C0 .ƒV /=C0w .ƒV / consists of constructible functions on ƒw V. Let us consider C0w .ƒV /? D f 2 C0 .ƒV / j hf; i D 0 for f 2 C0w .ƒV /g: This space is spanned by an evaluation at a point X 2 ƒw V . As Cw is an additive category (in fact, it is closed under extensions), we have C0w .ƒT /? C0w .ƒW /? C0w .ƒV /? ; where the multiplication is given by the transpose of r, the natural cocommutative coproduct on U.n/. from the interpretation of r explained in §1.5. L wThis follows Therefore C0 .ƒV /? .C0 /gr Š U.n/gr is a subalgebra. By [8, Theorem 3.3] it is the CŒN.w/, which is the q D 1 limit of Uq .w/ ([19, Theorem 4.44]). By [8, Theorem 3.2] CŒN.w/ is compatible with the dual semicanonical basis, i.e., the intersection fƒb j b 2 B .1/g \ CŒN.w/ is a base of CŒN.w/. This base consists of ƒb such that ƒb intersects with the open subvariety ƒw V . The intersection is an open dense subset of ƒ . ƒb \ ƒw b V Finally we have fƒb j b 2 B .1/g \ CŒN.w/ D fƒb j b 2 B .w/g: This follows from [1, §5.5] and the characterization of B .w/ in §2.2. 2.4 Cluster algebras and Cw . Geiß–Leclerc–Schröer [8] have introduced a structure of the cluster algebra, in the sense of Fomin–Zelevinsky [6], on CŒN.w/. One of their main results says that dual semicanonical base of CŒN.w/ contains cluster monomials. We review their theory only briefly here. See the original paper for more details. The construction is based on Cw in §2.3. A ƒ-module T is rigid if Ext 1ƒ .T; T / D 0. It is easy to see from the formula dim Ext1ƒ .T; T / D 2 dim Homƒ .T; T / .dim V; dim V / (see e.g., [8, Lemma 2.1])

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that this is equivalent to saying that the orbit through T is open in ƒV , where V is the I -graded vector space underlying T , and . ; / is the Cartan matrix for the graph G . We say T is Cw -maximal rigid if Ext 1ƒ .T ˚ X; X / D 0 with X 2 Cw implies X is in add.T /, the subcategory of modules which are isomorphic to finite direct sums of direct summands of T . In Cw , there is a distinguished Cw -maximal module, denoted by Vi in [8], where i D .i1 ; : : : ; i` / is a reduced expression of w. It is conjectured that every Cw -maximal rigid module T is reachable, i.e., it is obtained from Vi using a sequence of operations, called mutations. This will be recalled below. Let T be a reachable Cw -maximal rigid module, and T D T1 ˚ ˚ T` be the decomposition into indecomposables. We assume that T is basic, which means Ti are pairwise non-isomorphic. In this case the number of summands ` is known to be equal to the length of w. If R 2 add.T /, it is a rigid ƒ-module, and hence the closure of the corresponding orbit is an irreducible component of ƒV for an appropriate choice of V . We denote it by R 2 CŒN.w/ the corresponding dual semicanonical base elements. From the identification of the coproduct r on U.n/gr , we see that c

R D Tc11 : : : T`` ;

(2.1)

c

where R D T1˚c1 ˚ ˚ T` ` with ci 2 Z0 . In the context of the cluster algebra theory, Ti (1 i `) are called cluster variables and R is a cluster monomial. Let T be a basic Cw -maximal rigid module, and Tk be a non-projective indecomposable direct summand of T . The mutation k .T / is a new basic Cw -maximal rigid module of the form .T =Tk / ˚ Tk , where Tk is another indecomposable module. Such a k .T / exists and is uniquely determined from T and Tk (see [8, Proposition 2.19] and the reference therein). We have dim Ext1ƒ .Tk ; Tk / D dim Ext1ƒ .Tk ; Tk / D 1 and Tk Tk D T 0 C T 00 ;

(2.2)

where T 0 , T 00 2 add.T =Tk / are given by short exact sequences 0 ! Tk ! T 0 ! Tk ! 0;

0 ! Tk ! T 00 ! Tk ! 0

respectively. Geiß–Leclerc–Schröer [9] have obtained a q-analog of the result explained above, namely they have introduced a structure of the quantum cluster algebra, in the sense of Berenstein–Zelevinsky [3], on the quantum unipotent subgroup Uq .w/. (This result was conjectured in [19].) The construction is again based on Cw . Let T be a reachable Cw -maximal rigid module. For R 2 add.T /, there is an element YR 2 Uq .w/, which satisfies the q-analog of (2.1, 2.2): c

YR D q ˛R YTc11 : : : YT`` ;

YTk YTk D q ŒTk ;Tk .q 1 YT 0 C YT 00 / for appropriate explicit ˛R , ŒTk ; Tk 2 Z. See [9, (10.17) and Proposition 10.5]. One of the main conjectures in this theory is

(2.3)

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Conjecture 2.4. All quantum cluster monomials YR are contained in B up .w/. This is closely related to their earlier open orbit conjecture [8, §18.3]: Conjecture 2.5. Suppose that an irreducible component ƒb of ƒV contains an open GV -orbit, then the corresponding dual semicanonical base element ƒb is equal to the specialization of the corresponding canonical base element b at q D 1. In fact, this is implied by Conjecture 2.4 for reachable rigid modules.

3 Singular supports under the restriction Let SS.L/ denote the singular support of a complex L 2 QV . See [17] for the definition. Lusztig proved that SS.L/ ƒV [22, 13.6]. In fact, we have finer estimates ƒb SS.Lb /

[

ƒb 0 :

(3.1)

b 0 2B.1/ 8i "i .b 0 /"i .b/

See [16, Theorem 6.2.2], but note that there is a misprint. See [16, Lemma 8.2.1] for the correct statement. F These estimates give us some relation between the canonical base and V Irr ƒV via singular supports. We study the behavior of singular supports under the functor Res in this section. 3.1 The statement. In order to state the result, we prepare notation. Let f W Y ! X be a morphism. Let TX , T Y (resp. T X , T Y ) be tangent (resp. cotangent) bundles of X , Y respectively. Let f 1 TX D Y X TX (resp. f 1 T X D Y X T X ) be the pull-back of TX (resp. T X ) by f . We have associated morphisms T Y

tf 0

f

f 1 T X ! T X;

where tf 0 is the transpose of the differential f 0 W T Y ! f 1 TX D Y X TX . We apply this construction for the morphisms , to get morphisms t 0

T E.W / 1 T EV ! T EV ; t 0

T E.W / 1 T .ET EW / ! T .ET EW /: Theorem 3.2. We have SS.Res.L// .t 01 .t 0 .1 .SS.L/////:

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Let us remark that the proof shows the following statement. Choose a complementary subspace of W in V , and identify V with W ˚ T . Then we have the induced embedding T .ET EW / T EV . Then we have SS.Res.L// T .ET EW / \ SS.L/:

(3.3)

The proof occupies the rest of this section. 3.2 Inverse image. If were smooth, we would have SS. L/ t 0 .1 .SS.L/// by [17, Proposition 5.4.5]. And if were proper, we would have SS.Š L/ Œ3 .t 01 .SS. L/// by [17, Proposition 5.4.4]. Therefore the assertion follows. However neither are true, so we need more refined versions of these estimates. In order to study the behavior of the singular support under the pull-back by a non-smooth morphism, we need several more notions related to cotangent manifolds from [17, Chapter VI]. We first recall the normal cone to S along M briefly. Suppose that M is a closed submanifold of a manifold X . Let TM X denote the normal bundle of M in X . Then one can define a new manifold XzM , which connects X and TM X in the following way: there are two maps p W XzM ! X; t W XzM ! R such that p 1 .X n M /, t 1 .R n f0g/ and t 1 .0/ are isomorphic to .X n M / .R n f0g/, X .R n f0g/ and TM X respectively. In our application, M is the zero section of a vector bundle X , and hence the normal bundle TM X is X itself. In this case, XzM is X R and p, t are the first and second projections. A general definition is in [17, §4.1] for an interested reader. Let be the inverse image of RC under t , and pQ the restriction of p to . Let S be a subset of X . The normal cone to S along M , denoted by CM .S / is defined by def: CM .S / D TM X \ pQ 1 .S /: If M is the zero section of a vector bundle X as before, we have CM .S / is identified with S itself under TM X Š X . Now we return back to a morphism f W Y ! X . We assume that f is a closed embedding for simplicity. We consider the conormal bundle TY X . We denote the t p0

p

projection TY X ! Y by p. We have morphisms T .TY X / p 1 T Y ! T Y as before. We consider T Y as a submanifold of T .TY X / via the composition t p0

T Y ,! p 1 T Y ! T .TY X /: See [17, (5.5.10)]. We consider TY X as a closed submanifold of T X . So we can define the normal cone to a subset of T X along TY X .

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If f W Y ! X is the embedding of the zero section to a vector bundle X , we can identify TY X with the dual vector bundle X . Then TY X ! T X is also the embedding of the zero section to a vector bundle. In fact, we have a natural identification T X Š T X , therefore TY X ! T X is identified with X ! T X . Therefore CTY X .A/ is identified with A itself under the isomorphism TTY X .T X / Š T X . Note also that T .TY X / is identified with T X Š T X . Under this identification, we have an isomorphism p 1 T Y Š f 1 T X which gives a commutative diagram f

tf 0

T X ? ? Šy

f 1 T X ! T Y ? ? Šy

T .TY X /

p 1 T Y ! T Y: t p0

(3.4)

p

Let A be a conic subset of T X , i.e., invariant under the RC -action, the multiplication on fibers. We define f # .A/ D T Y \ CTY X .A/: def:

Here we identify TTY X T X with T .TY X /. See [17, (6.2.3)]. Moreover we also have f # .A/ D p t p 01 .CTY X .A//: See [17, Lemma 6.2.1]. If f W Y ! X is the embedding of the zero section to a vector bundle X , we have f # .A/ D T Y \ A D t f 0 f1 .A/

(3.5)

by the commutative diagram (3.4). 3.3 Spaces. Let us describe relevant spaces explicitly. As E.W / is a linear subspace of EV , we have T E.W / D E.W / E.W / and E.W / is identified with EV =E.W /? , where ˚ E.W /? D B 0 2 EV j B 0 .W / D 0; Im B 0 W : Taking a complementary subspace of W in V , we identify T as an I -graded subspace of V . We then have the direct sum decomposition V Š W ˚ T and the induced projection V ! W . Then we have matrix notations of B and B 0 : 0 BT T 0 BT T BT0 W BD ; B0 D : 0 BW T BW W 0 BW W Similarly the space 1 .T EV / is nothing but E.W / EV , and identified with the space of linear maps B, B 0 of the forms 0 BT T 0 BT T BT0 W 0 BD ; B D : 0 0 BW T BW W BW BW T W

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The morphism t 0 W 1 .T EV / ! T E.W / is induced by the projection EV ! 0 E.W / . In the matrix notation, it is given by forgetting the component BW T. 1 The morphism W .T EV / ! T EV is the embedding E.W / EV ! EV EV . For .B; B 0 / 2 T .ET EW /, we have 0 0 0 BT T BT T 0 BD ; B D : 0 0 BW W 0 BW W For .B; B 0 / 2 1 .T .ET EW //, we have 0 BT T 0 BT T 0 BD ; B D BW T BW W 0

0

0 BW W

:

The morphisms t 0

T E.W / 1 .T .ET EW // ! T .ET EW / are given by taking appropriate matrix entries of B, B 0 . Since we have chosen an isomorphism V Š W ˚ T , we have the projection p W EV ! E.W / which gives a structure of a vector bundle so that is the embedding of the zero section. Therefore we have the commutative diagram (3.4) for f D , and hence (3.6) # .A/ D T E.W / \ A D t 0 1 .A/ by (3.5). 3.4 Proof. We first study the behavior of the singular support under the functor . Let L 2 QV . By [17, Corollary 6.4.4] we have SS. L/ # .SS.L//: In our situation, we have # .SS.L// D t 0 .1 .SS.L/// by (3.6). Next study the functor Š . Note that W E.W / ! ET EW is the projection of a vector bundle. Therefore the results in [17, §5.5] are applicable. A complex F in D .E.W // is conic if H j .F / is locally constant on the orbits of the RC -action for all j . In our situation, F D i L satisfies this condition. Then we have SS.Š .i L// T .ET EW / \ SS.i L/ D t 01 SS.i L/: See [17, Proposition 5.5.4] for the first inclusion and [17, (5.5.11)] for the second equality. Combining two estimates, we complete the proof of Theorem 3.2. The estimate (3.3) has been given during the proof.

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225

4 Conjectures 4.1 Quantum unipotent subgroup and singular supports. Let w be a Weyl group element as before. Motivated by §2.3, we introduce a subset B 0 .w/ in B .1/ by B 0 .w/ D fb 2 B .1/ j SS.Lb / \ ƒw V ¤ ;g; def:

where we suppose Lb 2 PV in the equation SS.Lb / \ ƒw V ¤ ;. Equivalently b … B 0 .w/ if and only if SS.Lb / is contained in the closed subvariety ƒV n ƒw V. 0 By (3.1), the condition ƒb \ ƒw ¤ ; implies b 2 B .w/. Therefore B .w/ V B 0 .w/ by §2.3. Let b … B 0 .w/. We have w w SS.Res.Lb // \ .ƒw T ƒW / SS.Lb / \ ƒV D ;

by (3.3) and the fact that Cw is an additive category. Writing Res.Lb / D we get

M b1 ;b2 Lb1 Lb2 Œn˚rbIn ;

w SS.Lb1 Lb2 / \ .ƒw T ƒW / D ;

b1 ;b2 if rbIn ¤ 0 for some n. This is because SS.L ˚ L0 / D SS.L/ [ SS.L0 / and

SS.LŒ1/ D SS.L/ (see [16, Chapter V]). In the notation in §1.3 we have rbb1 ;b2 D P b1 ;b2 n n rbIn q . We have an estimate SS.Lb1 Lb2 / SS.Lb1 / SS.Lb2 / [16, Proposition 5.4.1]. w However this does not imply SS.Lb1 / SS.Lb2 / \ .ƒw T ƒW / D ;, so we need a finer estimate. Since Lba (a D 1; 2) is a perverse sheaf, it corresponds to a regular holonomic D-module under the Riemann–Hilbert correspondence (see e.g., [14, Theorem 7.2.5]). Then the singular support of Lba is the same as the characteristic variety of the corresponding D-module [17, Theorem 11.3.3], [14, Theorem 4.4.5]. As the characteristic variety of the exterior product is the product of the characteristic varieties [17, (11.2.22)], we deduce SS.Lb1 Lb2 / D SS.Lb1 / SS.Lb2 /. Therefore we have w .SS.Lb1 / \ ƒw T / .SS.Lb2 / \ ƒW / D ;:

Therefore either b1 … B 0 .w/ or b2 … B 0 .w/. In other words, b1 ; b2 2 B 0 .w/ and L rbb1 ;b2 ¤ 0 implies b 2 B 0 .w/. Therefore b2B0 .w/ Q.q/b up is a subalgebra of Uq by §1.3. Our first conjecture is the following. Conjecture 4.1. B 0 .w/ D B .w/. In other words, if b … B .w/, then SS.Lb / ƒV n ƒw V. L This conjecture is also equivalent to saying that b2B0 .w/ Q.q/b up D Uq .w/.

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4.2 Cluster algebra and singular supports. Recall that Geiß–Leclerc–Schröer [9] have introduced the structure of a quantum cluster algebra on Uq .w/ and conjectured that quantum cluster monomials are contained in B up .w/. If this is true, we should have two formulas (2.3) for dual canonical base elements corresponding to YR , YTk , etc. Conversely (2.3) implies that YR , YTk are dual canonical base elements by induction on the number of mutations. Let us speculate why these formulas hold in terms of the corresponding perverse sheaves. The proposal here is the following conjecture: Conjecture 4.2. Let T be a reachable Cw -maximal rigid module and R 2 add.T /. Let ƒR be the closure of the orbit through R and bR the corresponding canonical base element. If another canonical base element b 2 B .w/ satisfies ƒR SS.Lb /; we should have b D bR . If Conjecture 4.1 is true, b 2 B .1/ with ƒbR SS.Lb / is contained in B .w/. Therefore the above conjecture holds for any b 2 B .1/. This conjecture is true for a special case when ƒR is the zero section EV of T EV and GV has an open orbit in EV . In fact, if SS.Lb / EV , we have supp.Lb / D EV . Then Lb is GV -equivariant and gives an irreducible GV -equivariant local system on the open orbit in EV . As the stabilizer of a point is connected from a general property from quiver representations, it must be the trivial rank 1 local system. Thus Lb is the constant sheaf on EV . In fact, the observation that supp.Lb / D EV implies Lb D the constant sheaf was used in a crucial way to prove the cluster character formula in [27]. If SS.Lb / is irreducible for all b, Conjecture 4.2 is obviously true. This condition is satisfied for g of type A4 , but not for A5 [16]. Let us remark a relation between the above conjecture and a conjecture in [7, §1.5]. This is pointed out by the referee to the author. Let us define the semicanonical base ffY g of U.n/ as the dual base of the dual semicanonical base fP Y g. In [7, §1.5] it is conjectured that the specialization of b is a linear combination mY fY (mY 2 Z), where the summation runs over irreducible components Y of SS.Lb /. (More precisely it is probably given by the characteristic cycle (see [14, 2.2.2] for the definition) of Lb .) Dually, an irreducible component Y D ƒb cannot be contained in other SS.Lb 0 / (b 0 ¤ b) if b up jqD1 D ƒb . This is nothing but our conjecture. Thus under the conjecture in [7, §1.5], our conjecture is equivalent to Conjecture 2.4 for reachable rigid modules. Let us explain how the first formula in (2.3) is related to Conjecture 4.2. We assume R D T1 ˚ T2 for brevity. From the assumption ƒR contains the product ƒıT1 ƒıT2 as an open dense subset. Here ƒıTi denotes the open orbit through Ti . Its closure is

227

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up

up

bT1 ;bT2

ƒTi . Suppose that b up appears in the product bT1 bT2 . Then rb

¤ 0. We have

SS.Lb / \ .ƒıT1 ƒıT2 / SS.Res.Lb // \ .ƒıT1 ƒıT2 /

SS.LbT1 LbT2 / \ .ƒıT1 ƒıT2 /

D .SS.LbT1 / \ ƒıT1 / .SS.LbT2 / \ ƒıT2 /;

where the first inclusion is by (3.3), the second as a shift of LbT1 LbT2 is a direct summand of Res Lb , and the third equality was observed above. The last expression is nonempty thanks to (3.1). Therefore we have SS.Lb / ƒR . Then Conjecture 4.2 up up up implies that b D bR . Therefore bT1 bT2 is a multiple of bR . A refinement of this up up up argument probably proves that bT1 bT2 is equal to bR up to a power of q. Let us turn to the second formula in (2.3). The same argument above implies that SS.Lb / \ .ƒıT ƒıTk / ¤ ; k

bT ;bTk

¤ 0. From what we have explained in §2.4, there are two irreducible if rb k components ƒT 0 , ƒT 00 , where T 0 and T 00 are ƒ-modules given by non-trivial extensions of Tk and Tk . As a non-trivial extension can degenerate to the trivial one, both ƒT 0 and ƒT 00 contain ƒıT ƒıTk . It is also easy to check that dim ƒıT ƒıTk D dim ƒV 1, k k where V is the underlying vector space of Tk ˚ Tk . Lemma 4.3. If an irreducible component Y of ƒV contains ƒıT ƒıTk , we have k either Y D ƒT 0 or D ƒT 00 . Proof. Take a sequence Zn of points of Y converging to the module Tk ˚ Tk , regarded as a point of Y . We may assume Zn 6Š Tk ˚ Tk . Then we have dim Hom.Zn ; Tk ˚ Tk / dim Hom.Tk ˚ Tk ; Tk ˚ Tk / for sufficiently large n by the upper semicontinuity of the dimension of cohomology groups. If the equality holds, we can take n 2 Hom.Zn ; Tk ˚ Tk / converging to the identity of Hom.Tk ˚ Tk ; Tk ˚ Tk / for n ! 1. In particular, n is invertible, hence Zn Š Tk ˚ Tk . This contradicts with our assumption. Therefore we have the strict inequality dim Hom.Zn ; Tk ˚Tk / < dim Hom.Tk ˚Tk ; Tk ˚Tk /. The same argument gives dim Hom.Tk ˚ Tk ; Zn / < dim Hom.Tk ˚ Tk ; Tk ˚ Tk /. Therefore dim Ext1 .Tk ˚ Tk ; Zn / D .dim V; dim V / C dim Hom.Tk ˚ Tk ; Zn / C dim Hom.Zn ; Tk ˚ Tk / .dim V; dim V / C 2 dim Hom.Tk ˚ Tk ; Tk ˚ Tk / 2 D Ext1 .Tk ˚ Tk ; Tk ˚ Tk / 2 D 0;

where we have used the formula in [8, Lemma 2.1]. The upper semicontinuity also shows Ext1 .T =Tk ; Zn / D 0, hence Zn 2 add.T =Tk /.

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As the inequality above must be an equality, we get dim Hom.Tk ˚ Tk ; Zn / D dim Hom.Zn ; Tk ˚ Tk / D dim Hom.Tk ˚ Tk ; Tk ˚ Tk / 1: Therefore dim Hom.Tk ; Zn / C dim Hom.Tk ; Zn / D dim Hom.Tk ; Tk ˚ Tk / C dim Hom.Tk ; Tk ˚ Tk / 1: Note that dim Hom.Tk ; Zn / dim Hom.Tk ; Tk ˚ Tk / and dim Hom.Tk ; Zn / dim Hom.Tk ; Tk ˚ Tk / by the semicontinuity. Hence the above implies that one of inequalities must be an equality. Suppose that the first one is an equality. Then we have dim Hom.Tk ; Zn / D dim Hom.Tk ; Tk ˚ Tk /; dim Hom.Tk ; Zn / D dim Hom.Tk ; Tk ˚ Tk / 1: The same argument shows that dim Hom.Zn ; Tk / D dim Hom.Tk ˚ Tk ; Tk / or dim Hom.Zn ; Tk / D dim Hom.Tk ˚ Tk ; Tk /. The first equality is impossible, as 0 D dim Ext1 .Tk ; Zn / ¤ dim Ext1 .Tk ; Tk ˚ Tk / D 1 and the above dimension formula. Therefore we have dim Hom.Zn ; Tk / D dim Hom.Tk ˚ Tk ; Tk / 1; dim Hom.Zn ; Tk / D dim Hom.Tk ˚ Tk ; Tk /: We take n 2 Hom.Tk ; Zn / converging to idTk ˚0 in Hom.Tk ; Tk ˚ Tk /. In particular, n is injective for sufficiently large n. We consider an exact sequence 0 ! Hom.Zn = Im n ; Tk / ! Hom.Zn ; Tk / ! Hom.Im n ; Tk /: The next term Ext1 .Zn = Im n ; Tk / vanishes, as we have dim Ext1 .Zn = Im n ; Tk / dim Ext1 .Tk ; Tk / D 0 by the upper semicontinuity. Therefore we have dim Hom.Zn = Im n ; Tk / D dim Hom.Tk ; Tk /: We take n 2 Hom.Zn = Im n ; Tk / converging to idTk . Then n is an isomorphism for sufficiently large n. Composing the projection p W Zn ! Zn = Im n with n , we have an exact sequence n Bp

n

0 ! Tk ! Zn ! Tk ! 0: This shows that Zn Š T 00 . When dim Hom.Tk ; Zn / D dim Hom.Tk ; Tk ˚ Tk /, the same argument shows that Zn Š T 0 . up

up

up

up

Now Conjecture 4.2 implies that bT bTk is a linear combination of bT 0 and bT 00 . k A refinement of the argument hopefully gives the second formula in (2.3).

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[19] Y. Kimura, Quantum unipotent subgroup and dual canonical basis. Kyoto J. Math. 52 (2012), 277–331. [20] B. Leclerc, Quantum loop algebras, quiver varieties, and cluster algebras. In Representations of algebras and related topics. EMS Ser. Congr. Rep., Eur. Math. Soc. Publ. House, Zürich 2011, 117–152. [21] G. Lusztig, Canonical bases arising from quantized enveloping algebras. II. In Common trends in mathematics and quantum field theories (Kyoto, 1990), Progr. Theoret. Phys. Suppl. 102, Kyoto 1991, 175–201. [22] G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras. J. Amer. Math. Soc. 4 (1991), 365–421. [23] G. Lusztig, Affine quivers and canonical bases. Inst. Hautes Études Sci. Publ. Math. 76 (1992), 111–163. [24] G. Lusztig, Introduction to quantum groups. Progr. Math. 110, Birkhäuser, Boston, Mass., 1993. [25] G. Lusztig, Braid group action and canonical bases. Adv. Math. 122 (1996), 237–261. [26] H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76 (1994), 365–416. [27] H. Nakajima, Quiver varieties and cluster algebras. Kyoto J. Math. 51 (2011), 71–126. [28] F. Qin, Quantum cluster variables via Serre polynomials. With an appendix by Bernhard Keller. J. Reine Angew. Math. 668 (2012), 149–190. [29] F. Qin, t-Analogue of q-characters and bases of quantum cluster algebras. Preprint, arXiv:1207.6604 [math.QA]. [30] C. M. Ringel, Hall algebras and quantum groups. Invent. Math. 101 (1990), 583–591. [31] Y. Saito, PBW basis of quantized universal enveloping algebras. Publ. Res. Inst. Math. Sci. 30 (1994), 209–232.

Representations and cohomology of finite group schemes Julia Pevtsova

0 Introduction This survey is based on the lecture given by the author at the International Conference on Representations of Algebras in Bielefeld in 2012. Jon Carlson gave a talk on this subject at ICRA XII in Toru´n in 2007 with the survey [13] published in the same series. In the current article we try to pick up where Carlson left off although some overlaps to set the stage were unavoidable. We also focus on the general case of a finite group scheme as opposed to a finite group highlighted in [13]. Even though this is an article about finite group schemes which is a much more general class than finite groups, we start by recalling the foundations of the classical theory of support varieties which motivated later developments featured in this survey. In his seminal papers [48], Quillen gave a general description of the maximal ideal spectrum of the commutative algebra Hev .GI k/ for a finite group G in terms of the elementary abelian p-groups of G. Following work of Alperin and Evens [1], Carlson extended Quillen’s work to introduce and study the support variety of a finite dimensional G-module, a closed conical subvariety of Spec Hev .GI k/ which became an important invariant of modular representations. An essential feature of the support variety of a module of an elementary abelian p-group E, conjectured by Carlson [12] and proved by Avrunin–Scott [3], is that it admits a description in terms of cyclic shifted subgroups of the group algebra kE with no recourse to cohomology. Such a description is referred to as “rank variety”. Representations of a finite group scheme are equivalent to modules of its group algebra (also known as the algebra of measures) which is a finite dimensional cocommutative Hopf algebra. Hence, from the representation theoretic point of view, finite group schemes fall between finite groups and finite dimensional Hopf algebras. Some of the formalism involving cohomology, such as the definition of support variety originally developed for finite groups, works equally well for any finite dimensional Hopf algebra. But any argument that uses reduction to Sylow subgroups or any deeper “local-to-global” methods from finite group theory usually fails miserably for other classes of finite group schemes. From the point of view of a Hopf algebraist this is not surprising: group algebras of finite groups are generated by the group-like elements whereas group algebras corresponding to other examples of finite group schemes, such as restricted enveloping algebras, can be primitively generated. Hence, they can exhibit behaviour very different from finite groups. A case in point is finite generation of the

Partially supported by the National Science Foundation award DMS-0953011.

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cohomology algebra of a finite group scheme with coefficients in a field k of positive characteristic p. For finite groups this was known since the late 50s and early 60s, the reduction to a Sylow subgroup being a quick and easy step in the algebraic proof due to L. Evens [22]. For restricted Lie algebras finite generation of cohomology was proven in the 80s ([27], see also [2]) with the maximal unipotent subalgebra playing a rather different role from a Sylow p-subgroup and the passage from the unipotent to the general case being highly non-trivial. The finite generation of cohomology of any finite group scheme was proven in a celebrated theorem of Friedlander and Suslin in 1995 ([34]). Whether the cohomology ring of any finite dimensional Hopf algebra with trivial coefficients is finitely generated is still an open problem. The Friedlander–Suslin theorem on finite generation of cohomology unlocked the way for the applications of powerful geometric machinery to the study of representations and cohomology of a finite group scheme as pioneered by Quillen for finite groups. Suslin–Friedlander–Bendel developed a theory of rank varieties for infinitesimal group schemes based on one-parameter subgroups. They identified the spectrum of cohomology with the variety of one-parameter subgroups, and proved an analogue of the Avrunin–Scott theorem. Friedlander and the author subsequently unified the existing theories for finite groups and infinitesimal group schemes by introducing the notion of a -point and …-space. One interpretation of the main theorem of the theory of points, Theorem 3.8, is that it generalizes Quillen stratification and the Avrunin–Scott theorem to any finite group scheme. A byproduct of the theory of -points was the discovery of modules of constant Jordan type ([14]) which was an emerging subject at the time of publication of Carlson’s article [13]. There is now extensive literature on modules of constant Jordan type some of which is mentioned in the references although it seemed impossible to list every contribution. In particular, D. Benson is currently writing a book on the subject which is available in electronic form. The development that followed modules of constant Jordan type was construction of vector bundles associated to these modules. Somewhat surprisingly, the roles of different classes of finite group schemes got reversed at this stage. The more geometric nature of infinitesimal group schemes allows for construction of vector bundles on projective varieties of one-parameter subgroups whereas it is unknown whether an analogous construction exists for finite groups. One may ask “what about elementary abelian p-groups?”, the subject of [10]. As we argue in Example 1.13, from the representation-theoretic perspective, an elementary abelian p-group E is a very special case of an infinitesimal group scheme. Hence, constructions of [32] apply. Focusing on the case of an elementary abelian p-group brings considerable advantages: for example, Proj Hev .E; k/ is a projective space which provides for many special geometric tools and leads to stronger results than in the general case of an infinitesimal group scheme. We refer the reader to D. Benson’s article for material specific to the elementary abelian case. The organization of the paper follows the evolution of the subject. In the first two sections we recall the definitions and basic properties of finite group schemes, their

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representations, and cohomology. We give an overview of the scheme of one-parameter subgroups V .G/ for an infinitesimal group scheme G which plays an important role in the construction of the global p-nilpotent operator ‚ in Section 5. Section 3 is a dense summary of the theory of -points with applications to classification of thick tensor ideal subcategories of stmod G and determination of the representation type of G. In Section 4 we introduce modules of constant rank and constant Jordan type and construct non-maximal rank varieties. This section is somewhat skimpy and short on examples due to the fact that the material is covered in Benson’s article in the same volume. One can also find more on the properties of non-maximal support varieties and modules of constant Jordan type in Carlson’s article [13] as well as the original papers. In Section 5 we sketch the construction of the global nilpotent operator ‚ that acts on M ˝ kŒV .G/ for any module M of an infinitesimal group scheme G. The operator ‚ allows us to associate coherent sheaves on Proj V .G/ (which is homeomorphic to Proj H .G; k/) to any representation of G. For representations of constant rank the associated sheaves turn out to be vector bundles. In the same section we list some properties of these bundles and calculations for the restricted Lie algebra sl2 . In the final section we summarize the latest results of Carlson, Friedlander and the author from [15], [16]. We focus on a restricted Lie algebra g and replace the variety Proj V .G/ with the variety E.r; g/ which is the moduli space of the “elementary Lie subalgebras” of g of dimension r. If V .G/ was the variety of one-parameter subgroups, then E.r; g/ can be viewed as the variety of “r-parameter” subalgebras. These varieties are interesting geometric invariants of g on their own right. The beginning of Section 6 provides some information on what is known about them; many questions remain unanswered. Using restrictions to elementary subalgebras instead of one-parameter subgroups, we define modules of constant .r; j /-radical or socle rank (or type) that generalize modules of constant rank (or constant Jordan type). Replacing the operator ‚ of Section 5 with a more sophisticated construction we can associate vector bundles on E.r; g/ to modules of constant .r; j /-radical or socle rank. We finish the article with examples to illustrate our constructions. Throughout, k is assumed to be an algebraically closed field of characteristic p. The assumption of algebraically closed can often be relaxed if one is willing to work with schemes instead of varieties but we keep it to streamline the exposition. All vector spaces, algebras, and schemes are k-spaces, k-algebras, and k-schemes unless explicitly specified otherwise. The dual vector space of M is denoted by M # . For a field extension K=k, and a vector space M , we denote by MK D M ˝k K the base change from k to K. For a scheme X, we set XK D X Spec k Spec K. The author would like to thank the organizers, particularly Henning Krause and Rolf Farnsteiner, for the invitation to the meeting and for letting her experience such a unique and invigorating event as ICRA. She would also like to thank Eric Friedlander for introducing her to the subject and for generously sharing his ideas over the years that they have worked on it together.

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1 Finite group schemes: definitions and examples To define finite group schemes we take the “functor of points” approach to the definition of an affine scheme such as the one taken in [57], [38], or [21, Appendix]. We follow the terminology in [57]. An affine scheme X over a field k is a representable k-functor from the category of commutative k-algebras to sets. If X is represented by the algebra kŒX , then as a functor it is given by the formula X.A/ D Homk-alg .kŒX ; A/ for any commutative k-algebra A. The algebra kŒX is the coordinate algebra or the algebra of regular functions of X . An affine scheme X is algebraic if the coordinate algebra kŒX is a finitely generated algebra. By the Yoneda lemma, we have an antiequivalence of categories: ² ³ ² ³ affine algebraic finitely generated op schemes commutative algebras An affine group scheme G is an affine scheme that takes values in the category of groups, that is, a representable functor from commutative k-algebras to groups. The coordinate algebra kŒG is then a Hopf algebra and the anti-equivalence of categories above restricts to the following: ³ ² ² ³ affine algebraic finitely generated op group schemes commutative Hopf algebras For the purposes of this paper, all group schemes are affine algebraic. Moreover, our main interest lies within the following class: Definition 1.1. A group scheme G is finite if the coordinate algebra kŒG is a finite dimensional k-algebra. We have yet another anti-equivalence: ² ³ ² ³ finite group finite dimensional op schemes commutative Hopf algebras

(1.1)

Motivated by Example 1.5 below, we use the notation kG WD kŒG# and refer to this algebra as the group algebra of G. Dualizing the anti-equivalence (1.1) we arrive at the equivalence of categories that underlines the entire paper: ² ³ ² ³ finite group finite dimensional (1.2) schemes cocommutative Hopf algebras

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One can define representations of a finite group scheme G using the functorial perspective (see, for example, [57, Chapter 3]). The equivalence (1.2) will then imply that there is an equivalence between the category of representations of G and the category of kG-modules. In what follows, we shall use “representation of G” and “kG-module” interchangeably. Since the category of kG-modules is abelian with enough injectives, we can consider ExtnG .M; N / WD Ext nkG .M; N /, the Ext-groups in the category of kG-modules for n a non-negative integer. As usual, we set M ExtnG .k; k/; H .G; M / WD ExtG .k; M / H .G; k/ WD ExtG .k; k/ D n0

where k is the trivial module for kG given by the augmentation (equivalently, counit) map, and M is any kG-module. Since kG is a finite dimensional Hopf algebra, it is self-injective ([40], see also [38, Chapter 5]). Therefore, the category of kG-modules is Frobenius, that is, injective kG-modules are projective and vice versa ([24]). Due to the presence of the Hopf algebra structure on kG, there are two ways of defining the product on the cohomology algebra H .G; k/: the cup product and the Yoneda product. The cup product is defined by tensoring projective resolutions and composing with the diagonal approximation map whereas the Yoneda product is given by splicing extensions. The existence and compatibility of these two products leads to the graded commutativity of H .G; k/ by using a modified version of the classical Eckmann–Hilton argument. One can find details, for example, in [42, §3]. Theorem 1.2. Let G be a finite group scheme. Then H .G; k/ is a graded commutative k-algebra. We shall use the following notation throughout the paper: 8L < H2n .G; k/; p > 2; H .G; k/ D n0 :H .G; k/; p D 2: Theorem 1.2 implies that H .G; k/ is a commutative algebra. We next recall the key result of Friedlander and Suslin that allows one to employ techniques from algebraic geometry to study cohomology and representations of G. Theorem 1.3 (Main theorem, [34]). Let G be a finite group scheme, and M be a finite dimensional kG-module. Then H .G; k/ is a finitely generated k-algebra, and H .G; M / is a finitely generated module over H .G; k/. We make some historical remarks about the finite generation of the cohomology ring. For G a finite group with characteristic k dividing the order of G, the finite generation of H .G; k/ was proven by Golod [36], Venkov [56], and Evens [22] independently with Evens using purely algebraic techniques and adding the result about

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the module H .G; M /. It took another 35 years to prove the result for any finite group scheme. It is conjectured that H .A; k/ is finitely generated for any finite dimensional Hopf algebra A (see, for example, [23] where it is formulated in the context of tensor categories), with partial progress made in [35], [8], and [42]. We finish this section with a definition of a (cohomological) support variety. Let M be a finite dimensional kG-module. Then H .G; k/ ' Ext G .k; k/ acts on Ext G .M; M / / ::: / M via Yoneda product: namely, for any extension k / : : : / k and M we tensor the first extension with M and then concatenate. This defines a pairing ExtnG .k; k/ Extm G .M; M /

/ Ext nCm .M; M / G

and then the (algebra) action ExtG .k; k/ ExtG .M; M /

/ Ext .M; M / G

(we refer the reader to [9, I.2.6] for details. See also [9, I.3.2] where the same action is expressed in terms of cup product). Definition 1.4. Let IM D AnnH .G;k/ ExtG .M; M / be the annihilator ideal of Ext G .M; M / as a module over H .G; k/. The (cohomological) support variety of M , denoted by jGjM , is a closed subset of Spec H .G; k/ defined by the ideal IM . It has a canonical structure of an affine variety corresponding to the reduced affine scheme Spec.H .G; k/=IM /red . The algebra H .G; k/ is graded and the ideal IM is homogeneous. Hence, we can consider the “projectivized” support variety of M . We denote by Proj jGjM the closed points of the projective scheme Proj .H .G; k/=IM /, that is, the subset of homogeneous prime ideals in H .G; k/ of dimension one that contain IM . Examples of finite group schemes Example 1.5 (Finite groups). Let G be a finite group and kG be the group algebra. We z as follows: for a commutative k-algebra define the associated constant group functor G R, def z G.R/ G j0 .R/j ; z ' k jGj , where 0 .R/ is the set of connected components of Spec R. We have kŒG z ' kG. and k G 7!p

/ k be the Frobenius map. For Example 1.6 (Frobenius kernels). Let f W k a commutative k-algebra A we define its Frobenius twist as a base change over the Frobenius map: A.1/ WD A ˝f k. There is a k-linear algebra map FA W A.1/ ! A given by FA .a˝/ D ap . If G is a group scheme, then the Frobenius twist kŒG.1/ D

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kŒG ˝f k is again a Hopf algebra over k and, therefore, defines another group scheme G .1/ which we call the Frobenius twist of G. The algebra map FkŒG W kŒG .1/ D kŒG.1/ ! kŒG induces the Frobenius map of group schemes F D FG W G ! G .1/ : Definition 1.7. The r th Frobenius kernel of a group scheme G is the group scheme theoretic kernel of r iterations of the Frobenius map: G.r/ D Ker F r W G ! G .r/ : Definition 1.8. A finite group scheme G is called infinitesimal if the coordinate algebra kŒG is local. An infinitesimal finite group scheme G is of height r if for any x 2 kŒG r not a unit, x p D 0. An infinitesimal group scheme of height r is an infinitesimal group scheme of height r but not of height r 1. Frobenius kernels are examples of infinitesimal group schemes. In particular, they are highly non-reduced: for any field extension K=k there is only one K-point in G.r/ . If the group scheme G is defined over the prime field Fp , there is an isomorphism G .1/ ' G. By postcomposing with this isomorphism, we can consider the Frobenius map as a self-map G ! G. We describe this explicitly for a particular example of GLn . Example 1.9 (GLn ). Define the algebraic group (scheme) GLn as GLn .R/ D finvertible n n matrices over Rg 1 for any commutative k-algebra R. Then kŒGLn ' kŒXij ; det , 1 i; j n, with the Hopf algebra structure defined as follows:

coproduct

r W kŒGLn ! kŒGLn ˝ kŒGLn ; r.Xij / D

n P `D1

counit

W kŒGLn ! k;

.Xij / D ıij ;

antipode

S W kŒGLn ! kŒGLn ;

S.Xij / D

Xi` ˝ X`j ;

Adj.Xij / ; det

where Adj.Xij / is the adjoint matrix to .Xij /. The Frobenius map is given explicitly by the formula: p

F W GLn

.aij /!.aij /

/ GLn .

Hence, as a group scheme, r

p GLn.r/ .R/ D f.aij /1i;j n j aij 2 R; aij D ıij g:

As an algebra,

r

kŒGLn.r/ ' kŒXij =.Xijp ıij /;

with the Hopf algebra structure inherited from kŒGLn .

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Example 1.10 (Ga ). The additive group Ga is defined as follows: Ga .R/ WD RC ; the additive group of the ring R: We have kŒGa ' kŒT with the Hopf algebra structure determined by the following formulas: coproduct counit antipode

r W kŒT ! kŒT ˝ kŒT ; W kŒT ! k; S W kŒT ! kŒT ;

r.T / D T ˝ 1 C 1 ˝ T; .T / D 0; S.T / D T:

a7!ap

/ RC , and Frobenius kernels have The Frobenius map is given by F .R/ W RC the following form: r Ga.r/ .R/ D fa 2 R j ap D 0g: r

We have kŒGa.r/ ' kŒT =T p with the Hopf algebra structure inherited from kŒT , and kGa.r/ ' kŒu0 ; : : : ; ur1 =.up0 ; : : : ; upr1 / where the generators u0 ; : : : ; ur1 are the linear duals to T; T p ; : : : ; T p

r1

.

Example 1.11 (Restricted Lie algebras). As we explain below, the example of restricted Lie algebras is a special case of Example 1.6. Definition 1.12. A Lie algebra g over the field k is called restricted if it is endowed with the p th power operation ./Œp W g ! g satisfying the following axioms: (1) ad.x Œp / D ad.x/p for all x 2 g, (2) .x/Œp D p x Œp for 2 k, x 2 g, (3) .x C y/Œp D x Œp C y Œp C

p1 P

si .x;y/ , i

iD1 p1

in the formal expression ad.tx C y/

where si .x; y/ is the coefficient of t i1

.x/.

Classical Lie algebras such as gln , sln , son , sp2n defined over a field k of characteristic p are restricted, with the Œpth power map being matrix multiplication. The Lie algebra Lie G of any group scheme G over k has a naturally defined Œpth power operation (see [57]) and, hence, is restricted. To any restricted Lie algebra g over k we associate its restricted enveloping algebra: u.g/ D U.g/=hx p x Œp ; x 2 gi where U.g/ is the universal enveloping algebra of g. This is a finite dimensional cocommutative Hopf algebra with the coproduct r W u.g/ ! u.g/ ˝ u.g/ primitively generated; that is, r.x/ D x ˝ 1 C 1 ˝ x for any x 2 g.

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A restricted representation of a restricted Lie algebra g is a representation M of g as a Lie algebra which respects the Œpth power structure. Namely, for x 2 g, m 2 M , .x.x : : : .x m/ : : :/ D x Œp m: „ ƒ‚ … p

The structure of a restricted g representation on a vector space M is then equivalent to the structure of a u.g/-module. All Lie algebra representations in this paper are assumed to be restricted and will be referred to as g-modules. Using the equivalence (1.2), we can associate an infinitesimal group scheme of height 1 to any restricted Lie algebra g. Indeed, since u.g/ is a finite dimensional cocommutative Hopf algebra over k, its dual u.g/# is a commutative finite dimensional Hopf algebra. Hence, there exists a finite group scheme which we denote by gQ such that kŒg Q D u.g/# . One can show that gQ is necessarily an infinitesimal finite group scheme of height 1. Moreover, for any group scheme G, we have an isomorphism of Hopf algebras kŒG.1/ ' u.Lie G/# . The associations g 7! gQ and G 7! Lie G define an equivalence of categories between the restricted Lie algebras and finite infinitesimal group schemes of height 1. For any particular group scheme G we have an equivalence of categories of representations of the Frobenius kernel G.1/ and the restricted Lie algebra Lie G: Representations of G.1/ o

/ u.Lie G/-mod :

(See [20, II, §7, 3.9–3.12] for details.) Example 1.13 (Elementary abelian p-groups). Towards the end of the paper we focus on the case of a restricted Lie algebra or, equivalently, finite infinitesimal group scheme of height 1. Here, we explain how from the point of view of representation theory elementary abelian p-groups can be considered in that framework. Let E D Z=p r be an elementary abelian p-group of rank r. Let fg1 ; : : : ; gr g be generators of E and let xi D gi 1 2 kE. This choice of generators determines an isomorphism of algebras kE ' kŒx1 ; : : : ; xr =.x1p ; : : : ; xrp /: Let ga D Lie Ga be the Lie algebra of the additive group scheme Ga . Then p p u.ga / ' kŒt=t p , and u.g˚r a / ' kŒx1 ; : : : ; xr =.x1 ; : : : ; xr / ' kE. Therefore, the categories of representations of g˚r a and of an elementary abelian p-group of rank r are equivalent as abelian categories. It should be noted that they do have different tensor products since the Hopf algebra structures on kE and u.g˚r a / are different.

2 The spectrum of cohomology of a finite group scheme It would seem impossible to write about cohomology of finite group schemes without mentioning Quillen’s fundamental stratification theorem for finite groups. We start by recording the cohomology of an elementary abelian p-group.

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Theorem 2.1. Let E ' Z=p r be an elementary abelian p-group of rank r. Then ² kŒx1 ; : : : ; xr ˝ ƒ .y1 ; : : : ; yr / where deg xi D 2; deg yi D 1; p > 2; H .E; k/ ' kŒy1 ; : : : ; yr where deg yi D 1; p D 2: Theorem 2.1 implies that for an elementary abelian p-group E of rank r , Spec H .E; k/ ' Ar ; the affine r-space over k. We give the Quillen stratification theorem here in the “weak form” referring the reader to any of the excellent sources in the literature for further details (such as [9, II.5]). Let G be a finite group. For any elementary abelian p-subgroup E of G we have a restriction map on cohomology resE;G W H .G; k/ ! H .E; k/ and the induced map on spectra: resE;G W Spec H .E; k/ ! Spec H .G; k/. Then the weak form of Quillen stratification states that [ Spec H .G; k/ D resE;G Spec H .E; k/: E G

Moreover,

resE;G Spec H .E; k/ ' .Spec H .E; k//=WG .E/;

where WG .E/ is the Weyl group of E in G, that is, the normalizer of E modulo the centralizer of E. Hence, geometrically Spec H .G; k/ is a union of finite quotients of affine spaces. The “strong form” of Quillen’s theorem prescribes how this union is taken, or, equivalently, how the finite quotients of affine spaces stratify Spec H .G; k/. We shall see next that for a restricted Lie algebra the situation is quite different. Let g be a restricted Lie algebra. We denote by N (or N.g/) the nullcone of g: the set of all nilpotent elements of g. It can be given a structure of a variety (or a reduced scheme) by considering it as a closed subset of the linear space g ' Spec S .g# /. We also define the restricted nullcone of g as the subset Np D fx 2 g j x Œp D 0g of g consisting of all Œp-nilpotent elements. Similarly to N, it has a structure of a variety considered as a closed subset of g. For a connected reductive group G we denote by h its Coxeter number. The close relationship between the nullcone and cohomology is revealed in the following theorem. Theorem 2.2 ([28], [2]). Let G be a connected reductive group and assume that p > h. Let g D Lie G. Then Hodd .g; k/ D 0 and H .g; k/ ' kŒN.1/ :

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Even though the isomorphism between algebras fails in general, the following theorem holds for any restricted Lie algebra: Theorem 2.3 ([54]). For any restricted Lie algebra g there is a homeomorphism of varieties: Spec H .g; k/ ' Np : In fact, the map in the last theorem is more than a homeomorphism: the inverse is induced by a map of algebras W H .g; k/ ! kŒNp which is an isogeny (or an F -isomorphism as defined by Quillen). Namely, it has a nilpotent kernel and the image contains the p-th power of any element. The theorem identifying the spectrum of cohomology of an arbitrary restricted Lie algebra is a special case of the theory constructed by Suslin–Friedlander–Bendel [53], [54] for infinitesimal group schemes that we now briefly recall. For any affine group scheme G over k we consider the functor R 7! Homgrp. sch .Ga.r/;R ; GR /

(2.1)

sending a commutative k-algebra to the set of homomorphisms of group schemes over R. A map of group schemes of the form Ga.r/;R ! GR is referred to as a one-parameter subgroup of GR . Theorem 2.4 ([53]). The functor (2.1) is representable by an affine scheme Vr .G/ with the coordinate algebra kŒVr .G/. Equivalently, there is a natural isomorphism Homgrp. sch .Ga.r/;R ; GR / ' Homk-alg .kŒVr .G/; R/: Examples 2.5. (1) Let r D 1. Then the k-points of the scheme V1 .G/ coincide with the restricted nullcone of Lie G: V1 .G/.k/ ' Np .Lie G/. (2) For GLn , Vr .GLn / is the scheme of r-tuples of p-nilpotent pairwise commuting matrices. It was pointed out in [53] that this description can be extended to any reductive group of exponential type. In more recent work [43], [50], the identification of Vr .G/ with the scheme of r-tuples of Œp-nilpotent commuting elements of Lie G was further extended to any reductive group G under the assumption that p is at least the Coxeter number for G. For an infinitesimal group scheme of height r, we drop the subscript r in Vr .G/ and use the notation V .G/. The main results of [53], [54] amount to the following theorem. Theorem 2.6. Let G be an infinitesimal group scheme of height r. There exists a natural homomorphism of k-algebras W H .G; k/ ! kŒV .G/

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with nilpotent kernel whose image contains the p r -th power of any element of kŒV .G/. Consequently, the induced map of schemes V .G/ ! Spec H .G; k/ is a p-isogeny and the corresponding map of varieties of closed points is a homeomorphism: ‰ W V .G/.k/ ' Specm H .G; k/;

(2.2)

where Specm H .G; k/ is the maximal ideal spectrum of H .G; k/. Theorem 2.3 is a special case of Theorem 2.6 for r D 1 thanks to the correspondence between restricted Lie algebras and infinitesimal group schemes of height 1.

3 -points In the previous section we described the spectrum of cohomology in terms of a “model space” which was built from maps from some “easy” objects to the group scheme: either elementary abelian p-subgroups for finite groups or one-parameter subgroups for infinitesimal groups. We now explain a unified approach that works for any finite group scheme. We say that a map of algebras W A ! B is left (respectively, right) flat if makes B into a flat left (respectively, right) A-module. If B is a finite dimensional Frobenius algebra then any map of the form W kŒx=x p ! B is left flat if and only if it is right flat ([50]). Hence, we do not distinguish between left and right flat in the definition below. A finite group scheme U over a field K is abelian unipotent if the group algebra KU is a local commutative K-algebra. Recall that for a field extension K=k, we denote by KG D kG ˝k K the scalar extension from k to K. The finite group scheme over K corresponding to the group algebra KG is denoted by GK . Definition 3.1. Let G be a finite group scheme, and K=k be a field extension. A -point ˛K of G is a flat map of algebras ˛K W KŒx=x p ! KG such that there exists a unipotent abelian subgroup scheme of U GK and a commutative diagram ˛K W KŒx=x p SSSS )

KU

/ KG : u u u

where the map KU ! KGK is induced by the embedding U GK . Note that in the definition a -point, ˛K is only a map of algebras whereas the map KU ! KG is a map of Hopf algebras. We do not require that U be defined over k. Notation 3.2. For a -point ˛K W KŒx=x p ! KG, and a G-module M , we denote by ˛K .MK / the KŒx=x p -module obtained by pulling back MK via ˛K . Definition 3.3 ([30]). Let K; L be field extensions of k. Two -points ˛K W KŒx=x p ! KG, ˇL W LŒx=x p ! LG are equivalent, written ˛K ˇL , if the following condi .MK / is free if and only if tion holds: for any finite dimensional kG-module M , ˛K p p ˇL .ML / is free (as KŒx=x (respectively, LŒx=x )-module).

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Example 3.4. Let E be an elementary abelian group of rank r so that kE ' kŒx1 ; : : : ; xr =.x1p ; : : : ; xrp /: Let I D Rad.kE/ be the augmentation ideal. In this case the equivalence relation of Definition 3.3 has a very explicit interpretation. For simplicity, we restrict to points defined over the ground field k. Let ˛; ˇ W kŒx=x p ! kE be two -points (defined over k). Then ˛ ˇ if and only if there exists a scalar c 2 k such that .˛ cˇ/.t/ 2 I 2 (see [29, 2.2, 2.9]). Definition 3.5. For a finite group scheme G, we denote by ….G/ the set of equivalence classes of -points of G: f pointsg ; ….G/ D and by Œ˛K the equivalence class of ˛K . For a kG-module M , we define the …-support of M , ….G/M ….G/; as the subset of equivalence classes of -points ˛K such that ˛K .M / is not free as p KŒx=x -module.

Using …-supports, we endow ….G/ with Zariski topology in the following way: Proposition 3.6 ([30], 3.4). Declaring f….G/M ….G/ j M is finite dimensional kG-moduleg to be the closed subsets gives ….G/ a structure of a Noetherian topological space. The …-supports of modules satisfy a number of nice properties which would be familiar to anyone who has studied support varieties. One of them is the “tensor product theorem”. This property turns out to be surprisingly non-trivial due to the fact that points are not Hopf algebra maps and, hence, the restriction along a -point does not commute with tensor product. Theorem 3.7 ([29], 3.9, [30], 5.2). Let G be a finite group scheme, and M , N be kG-modules (not necessarily finite dimensional). Then ….G/M ˝N D ….G/M \ ….G/N : We define a continuous map ‰ W ….G/ ! Proj H .G; k/ in the following natural way: Let ˛K W KŒx=x p ! KG be a -point. It induces a non-trivial (that is, not landing in degree 0) map on cohomology ˛K W H .G; K/ ! p H .KŒx=x ; K/. The kernel of ˛K is a homogeneous ideal of H .G; K/. Then ‰

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\H .G; k/ sends the equivalence class of Œ˛K to the homogeneous prime ideal Ker ˛K where H .G; k/ is embedded into H .G; K/ D H .G; k/˝K by sending 2 H .G; k/ to ˝ 1 2 H .G; K/: ‰.Œ˛K / D Ker ˛K \ H .G; k/:

Theorem 3.8 ([30], 3.6). For any finite group scheme G, ‰ W ….G/ ! Proj H .G; k/ is a homeomorphism. For any finite dimensional kG-module M , ‰ restricts to a homeomorphism ….G/M ' Proj jGjM . Remark 3.9. There exists a scheme structure on ….G/ defined solely in terms of representation theory which makes the homeomorphism ‰ an isomorphism of schemes (see [30, §7]). Example 3.10. Let E ' Z=p r be an elementary abelian p-group, and choose generators fg1 ; : : : ; gr g. Then kE ' kŒx1 ; : : : ; xr =.x1p ; : : : ; xrp / with xi D gi 1 for 1 i r being the generators of the augmentation ideal I . For any ˛ D .˛1 ; : : : ; ˛r / 2 Ar , a cyclic shifted subgroup of kE is a cyclic subgroup generated by X˛ C1 where X˛ D ˛1 x1 C: : : ˛r xr . The space of all cyclic shifted subgroups plus f0g is naturally identified with I =I 2 ' S ..I =I 2 /# / ' Ar and denoted by V .E/. Then for any kE-module M , we have Carlson’s rank variety V .E/M defined as follows: V .E/M WD f˛ 2 V .E/ j M#hX˛ C1i is not free g [ f0g: The variety V .E/M is conical; we can consider the associated “projectivized” rank variety Proj V .E/M as a closed subset of Proj V .E/ ' P r . Example 3.4 implies that we have a homeomorphism Proj V .E/ ' ….E/ which restricts to Proj V .E/M ' ….E/M : Hence, in the case of an elementary abelian p-group Theorem 3.8 reduces to the “projectivized” version of the Carlson’s conjecture proved by Avrunin and Scott ([3]): there is a homeomorphism V .E/ ' Spec H .E; k/ which restricts to a homeomorphism between rank and support varieties: V .E/M ' jEjM . Example 3.11. For G an infinitesimal group scheme, Theorem 3.8 specializes to a “projectivized” version of Theorem 2.6. We describe how to go from a one-parameter subgroup to a -point and refer the reader to [30] for further details. Notation 3.12. Recall that kGa.r/ ' kŒu0 ; : : : ; ur1 =.up0 ; : : : ; upr1 / where ui is the i linear dual to T p . We fix a map of algebras W kŒx=x p ! kGa.r/ which sends x to ur1 .

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Let G be an infinitesimal group scheme of height r and let W Ga.r/ ! G be a oneparameter subgroup defined over the ground field k. It induces a map on group algebras W kGa.r/ ! kG. Precomposing with we get a -point B W kŒx=x p ! kG. The association 7! B (3.1) determines a homeomorphism between the varieties of k-points of Proj Vr .G/ and ….G/. Applications. Since kG is a Frobenius algebra for any finite group scheme G, we can associate to G the triangulated category stmod kG, or, equivalently, DSing .G/, the category of singularities of kG (see, for example, [39]; this is also discussed in [13]). Objects of stmod kG are finite dimensional kG-modules, and Hom-sets are quotients defined as follows: Homstmod kG .M; N / D

HomG .M; N / ; PHomG .M; N /

where PHomG .M; N / are G maps from M to N that factor through a projective module. Let T be a triangulated category. A thick subcategory C of T is a full triangulated subcategory closed under direct summands. If T is a symmetric monoidal triangulated category, then a thick subcategory C is tensor ideal if for any C 2 C and X 2 T , X ˝ C 2 C. Using the properties of …-supports, particularly the “tensor product theorem”, one can classify thick tensor ideal subcategories of stmod kG in terms of subsets of ….G/. The following result, conjectured by Hovey–Palmiery–Strickland [37], is a generalization to all finite group schemes of a theorem of Benson–Carlson–Rickard [11]. Theorem 3.13 ([30], 6.3). There is one-to-one order preserving correspondence between lattices of tensor ideal subcategories of stmod kG and subsets of ….G/ closed under specialization. For a small symmetric monoidal triangulated category T , P. Balmer defined the spectrum of T , Spec T , a ringed space that “classifies” the thick tensor ideal subcategories of T . Using Balmer’s spectrum and the scheme structure on ….G/ ([30, 7.5]), Theorem 3.13 can be given the following slick – and stronger – reformulation. Theorem 3.14 ([4]). Let G be a finite group scheme. There is an isomorphism of schemes Spec.stmod kG/ ' Proj H .G; k/: An essential component of the theory of …-supports is that it retains good properties even for infinite-dimensional modules – whereas the cohomological supports do not (in

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particular, the homeomorphism in Theorem 3.8 breaks down for infinite dimensional modules). This feature plays an important role in the proofs of Theorems 3.13 and 3.14. For finite groups the recent paper [18] gives a different proof of the classification without a recourse to infinite dimensional modules whereas for finite groups schemes in general the theory of -points is so far the only tool available to prove Theorem 3.13. Another application of a good “support variety theory” for finite group schemes is a geometric criterion to determine the representation type. Namely, we have the following theorem: Theorem 3.15 ([25], [26]). Let G be a finite group scheme. If dim Spec H .G; k/ 3, then the representation theory of G is wild.

4 Local Jordan type Definition 4.1. Let M be a finite dimensional kG-module, and let ˛K W KŒx=x p ! KG be a -point. The Jordan type of M at the -point ˛K is the Jordan canonical form of ˛K .x/ considered as an operator on MK . Note that ˛K .x/ is a p-nilpotent operator, that is, ˛K .x/p D 0. Hence, the eigenvalues are all zero, and the only Jordan blocks possible are the ones of size from 1 to p. We use the exponential notation JType.˛K ; M / D Œpap : : : Œ1a1 for the Jordan type where ai is the number of blocks of size i in the Jordan form of ˛K .x/. An equivalent way of thinking about JType.˛K ; M / is that it is the isomorphism type of the KŒx=x p -module ˛ .MK /. We often refer to JType.˛K ; M / as a local Jordan type of M . We start with a well-known example of what the local Jordan type of a module, considered at all -points simultaneously, can determine. The following theorem can be viewed as a generalization of the famous “Dade’s lemma” for elementary abelian p-groups [19]. Theorem 4.2. Let G be a finite group scheme, and M be a finite dimensional kGdim M module. Then M is projective if and only if JType.˛; M / D Œp p for all -points ˛ W kŒx=x p ! kG defined over k. We refer the reader to D. Benson’s article in the same proceedings for many examples and properties of local Jordan types of modules. We point out (as Benson does too) that in general the Jordan type is NOT independent of a representative of an equivalence class of -points. It is independent, though, if the -point is generic. We say that ˛K W KŒx=x p ! KG is a generic -point if the image of ˛K under the map ‰ of Theorem 3.8 is a generic point of Proj H .G; k/. The independence of JType.˛K ; M / of a representative of an equivalence class of Œ˛K in this case is the most difficult part of the following theorem.

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Theorem 4.3 ([33]). Let G be a finite group scheme, and ˛K W KŒx=x p ! KG be a generic -point of G. (1) The functor

˛K W stmod G ! stmod KŒx=x p

defined by sending a kG-module M to ˛K .M / is an exact functor which commutes with tensor products.

(2) For two different choices of representatives ˛K , ˇL of the equivalence class Œ˛K , the corresponding functors are naturally isomorphic. For a finite dimensional kG-module M and a -point ˛K , we denote by rk j f˛K ; M g the rank of ˛K .x j / as an operator on MK . Definition 4.4. Let M be a finite dimensional kG-module. We say that M has constant j -rank if rk j f˛; M g is independent of the choice of a -point ˛ defined over k. Since k is assumed to be algebraically closed, this definition is equivalent to requiring that rk j f˛K ; M g is independent of ˛K for any -point ˛K and any field extension K=k. Let M be a kŒx=x p -module, and let JType.x; M / D Œpap : : : Œ1a1 be the Jordan canonical form of x considered as an operator on M . The relations between the exponents ai and the ranks of x j as operators on M can be expressed explicitly with the following formulas: rkfx j ; M g D aj C1 C C ap :

(4.1)

Therefore, we can give two equivalent definitions of modules of constant Jordan type: Definition 4.5. A finite dimensional kG-module M is a module of constant Jordan type if it has constant j -rank for all j , 1 j p 1. Equivalently, M has the same Jordan type JType.˛; M / for any -point ˛ defined over k. In this case we refer to JType.˛; M / as Jordan type of M . Let ˛ W kŒx=x p ! kG be a -point. Just as the Jordan type JType.˛; M /, the rank of ˛.x/ as an operator on M is not well defined on an equivalence class of -points in general. But it is well defined when the rank is maximal. Theorem 4.6 ([33]). Let M be a finite dimensional kG-module, and let ˛ W kŒx=x p ! kG be a -point such that rk j .˛; M / is maximal among rk j .ˇ; M / for all -points ˇ W kŒx=x p ! kG. Then for any -point ˛ 0 equivalent to ˛, rk j .˛ 0 ; M / D rk j .˛; M /. Remark 4.7. A careful reader may notice a discrepancy between the definition of modules of constant Jordan type given here and in D. Benson’s article. Benson chooses generators of the augmentation ideal of kE for an elementary abelian p-group E, and then considers Jordan type of linear combinations of those generators. Our definition

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in terms of -points is “generator-independent” and might appear to be a stronger condition. Thanks to Theorem 4.6 these two approaches are equivalent. Indeed, if we choose a different set of generators of the augmentation ideal of kE, we do not change the equivalence class of a -point as seen in Example 3.4. Hence, Theorem 4.6 implies that the property of constant rank, and, hence, constant Jordan type, does not depend upon the choice of generators. Using local Jordan type, we can define new geometric invariants of modules. Definition 4.8. Let G be a finite group scheme, and M be a finite dimensional kGmodule. The non-maximal j -rank variety of M is the following subset of ….G/: ˚ j .G/M D Œ˛K 2 ….G/ j rk j .˛K ; M / is not maximal : The set .G/M D

p1 [

j .G/M

j D1

is called the non-maximal rank variety of M . Proposition 4.9 ([31]). (1) The varieties j .G/M are closed proper subvarieties of ….G/. j (2) .G/M D ; if and only if M is a module of constant j -rank. (3) .G/M D ; if and only if M is a module of constant Jordan type.

We refer the reader to [14] and [31] for further properties and examples of nonmaximal rank varieties. We mention here one of their obvious virtues: the non-maximal variety is always properly contained in the ambient space ….G/. Hence, it carries nontrivial information for any module – in particular, for all those modules for which the support variety coincides with ….G/. Before moving on to “global invariants” in the next section we point out one important property that non-maximal rank varieties share with support varieties: they are invariants of the connected components of the stable Auslander–Reiten quiver of G. Theorem 4.10 ([31]). Let G be a finite group scheme. Let ‚ be a connected component of the stable Auslander–Reiten quiver of kG. Then for any two indecomposable modules M , N belonging to ‚ and any j , 1 j p 1, j .G/M D j .G/N .

5 Global p-nilpotent operator and vector bundles on P .G / We now specialize to the case of an infinitesimal group scheme G (with a side note that elementary abelian p-groups are included in consideration by virtue of Example 1.13). Let G be an infinitesimal group scheme of height r. Recall that the functor of “one-parameter subgroups” of G, V .G/, is representable by the coordinate algebra A D kŒV .G/, so that we have a natural isomorphism (2.4) Homgrp. sch .Ga.r/;R ; GR / ' Homk-alg .A; R/

(5.1)

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for any commutative k-algebra R. Taking R D A, we get an isomorphism Homgrp. sch .Ga.r/;A ; GA / ' Homk-alg .A; A/:

(5.2)

Definition 5.1. The universal one-parameter subgroup of G is the one-parameter subgroup U W Ga.r/;A ! GA which corresponds to the identity map idA W A ! A via the isomorphism (5.2). By naturality, any one-parameter subgroup W Ga.r/;R ! GR can be obtained from U W Ga.r/;A ! GA via base change D U ˝A R where R is given the structure of an A-module via the map f W A ! R corresponding to in (5.1). The universal one-parameter subgroup U induces an A-linear homomorphism of coordinate algebras: U W kŒG ˝ A ! kŒGa.r/ ˝ A Dualizing, we get an A-linear homomorphism U W kGa.r/ ˝ A ! kG ˝ A: For the following definition we utilize the map W kŒx=x p ! kGa.r/ ' kŒu0 ; : : : ; ur1 =.up0 ; : : : ; upr1 / of Notation 3.12. Definition 5.2. The global p-nilpotent operator ‚ 2 kG ˝ kŒV .G/ D kG ˝ A is defined as the image of the generator x under the composition kŒx=x p

/ kGa.r/

/ kGa.r/ ˝ A

U

/ kG ˝ A;

that is, ‚ D U ..x/ ˝ 1/. Notation 5.3. Let be a point of V .G/ with the residue field k. / (hence, is given by a map f W A ! k. /). We denote by D ‚ ˝A k. / the specialization of ‚ 2 kG ˝ A at the point 2 V .G/.

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Let k./ W Ga.r/;k./ ! Gk./ be the one-parameter subgroup that corresponds to the point 2 V .G/, and let ˛k./ D k./; B W k. /Œx=x p ! k. /G be the -point defined by the one-parameter subgroup k./ as in (3.1). Then ˛k./ .x/ D ‚ ˝A k. / D ; and for any finite dimensional kG-module M , we have JType.˛k./ ; M / D JType. ; M /: For a kG-module M , ‚ determines a p-nilpotent A-linear operator: ‚M W M ˝ A ! M ˝ A given by the formula m ˝ f 7! ‚ .m ˝ f /: We give some explicit examples of ‚. Example 5.4. Let g be a restricted Lie algebra, and G be the corresponding infinitesimal group scheme of height 1. Then kG ' u.g/, and the reduced scheme V .G/red corresponds to the affine variety Np . We therefore have a projection map kG ˝ kŒV .G/ ! kG ˝ kŒV .G/red ' kG ˝ kŒNp ' u.g/ ˝ kŒNp . Abusing notation slightly, we consider the global p-nilpotent element ‚ as an element in u.g/ ˝ kŒNp , effectively factoring out the nilpotents in kŒV .G/. Let x1 ; : : : ; xn be a basis of g; and let y1 ; : : : ; yn be the dual basis of g# . The embedding Np g induces a surjective map of algebras S .g # / D kŒy1 ; : : : ; yn kŒNp . We denote by fY1 ; : : : ; Yn g the images of fy1 ; : : : ; yn g in kŒNp . With this notation, we have the following formula for the global p-nilpotent operator: ‚ D x1 ˝ Y1 C C xn ˝ Yn 2 u.g/ ˝ kŒNp : Any nilpotent element x D 1 x1 C C n xn 2 Np is a specialization of ‚ for some values .1 ; : : : ; n / of .Y1 ; : : : ; Yn /. For a restricted g-module M , the action of the operator ‚M W M ˝ kŒNp ! M ˝ kŒNp is given by the following formula: ‚M .m ˝ f / D

n X

xi m ˝ Yi f:

iD1

Example 5.5. Take g D g˚r a so that u.g/ ' kE for E an elementary abelian p-group of rank r. Then Np ' Ar is the affine r-space, and kŒAr ' kŒY1 ; : : : ; Yr . The global operator ‚ is given by the same formula as in Example 5.4: ‚ D x1 ˝ Y1 C C xr ˝ Yr : Hence, in the case of an elementary abelian p-group our operator ‚ is given by the same formula as the operator in Benson’s article in this volume, even though for historical reasons slightly different notation is used.

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For infinitesimal group schemes of higher height the formulas for ‚ become much more complicated. We refer the reader to multiple examples worked out in [32, §2]. Theorem 5.6 ([53], 1.11, [32], 2.10, 2.11). Let G be an infinitesimal group scheme of height r. The coordinate algebra kŒV .G/ is a graded connected k-algebra generated by homogeneous elements in degrees f1; p; : : : ; p r1 g. The global p-nilpotent operator ‚ 2 kG˝kŒV .G/ is homogeneous of degree p r1 where the grading on kG˝kŒV .G/ is induced by the grading on kŒV .G/ with kG ˝ k being assigned degree 0. Since kŒV .G/ is graded, there is a corresponding projective scheme Proj V .G/. Let P .G/ D Proj kŒV .G/red be the associated reduced projective scheme, and let OP .G/ be the structure sheaf. Since kŒV .G/ is generated in degrees dividing p r1 , the Serre twist OP .G/ .p r1 / is a locally free sheaf on P .G/ (see [32, 4.5]). The homogeneity of ‚ implies that for any kG-module M , ‚M induces a sheaf homomorphism z M W M ˝ OP .G/ ‚

/ M ˝ OP .G/ .p r1 /:

We define several sheaves on P .G/ associated to a kG-module M via the operator z M . Let ‚ z D M ˝ OP .G/ M denote the trivial vector bundle defined by the module M (equivalently, free OP .G/ z M once we twist it appropriately. To avoid module of rank dim M ). We can iterate ‚ j z ! M z .jp r1 / to denote the composition z WM cumbersome notation, we use ‚ M z ‚

z

z ‚

.p r1 /

.2p r1 /

‚M M M z z .p r1 / z .p r1 / M ! M ! M ! z M ..j 1/p r1 / ‚

z .jp r1 /: ! M An easy way to follow our convention is to keep in mind that all the sheaves we define z (no twists). should be subs, quotients or subquotients of M Definition 5.7. Let j be an integer, 1 j p. zj W M z !M z .jp r1 /g; z j ´ Ker f‚ Ker ‚ M M zj W M z .jp r1 / ! M z g; z j ´ Im f‚ Im ‚ M M zj W M z .jp r1 / ! M z g; z j ´ Coker f‚ Coker ‚ M M Fj .M / ´

z M \ Im ‚ z j 1 Ker ‚ M : z M \ Im ‚ zj Ker ‚ M

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Theorem 5.8 ([32], 5.1). Let G be a finite group scheme, and let M be a finite dimenzj , z j , Im ‚ sional module of constant j -rank for some j , 1 j p. Then Ker ‚ M M j z are algebraic vector bundles (equivalently, locally free coherent sheaves) Coker ‚ M on P .G/. Remark 5.9. It was mistakenly stated in [32, 4.13, 5.1] that Theorem 5.8 held “if and only if”. This mistake was pointed out to the author by J. Stark, see [52] for counterexamples. The following statement, though, is true. Theorem 5.10. Let G be a finite group scheme, and M be a finite dimensional kGmodule. The following are equivalent: (1) M is a module of constant Jordan type. (2) For each i , 1 i p, Fi .M / is an algebraic vector bundle on P .G/. Proof. Even though formulated for elementary abelian p-groups, the proof in [10, 7.4.12] goes through without change for any finite group scheme. See also [52, 3.9]. z j , Im ‚ z j , Coker ‚ z j , Fi .M / provide global geometric inThe sheaves Ker ‚ M M M variants of representations of G which carry more information than support varieties or local Jordan type. For a simple example, we show that the sheaves associated to representations distinguish between dual modules whereas local Jordan type is always the same for M and M # . Example 5.11. Let G D Ga.1/ Ga.1/ so that kG ' kŒx; y=.x p ; y p /. Let Wn be the .2n C 1/–dimensional kG-module represented by the following diagram: ::: ~~ ~~ ~ ~ y y ~x ~x ~~ ~~ ::: In the diagram the dots represent the basis of Wn as a k-vector space; the solid arrows give the action of x and the dotted arrows give the action of y. The action of both generators on the bottom row is trivial. Both Wn and its dual Wn# have constant Jordan type nŒ2 C Œ1 (see [14, §2]). We have P .G/ D P 1 , and the vector bundles F1 are different for Wn and Wn# : F1 .Wn / ' OP 1 .n/;

F1 .Wn# / ' OP 1 .n/:

There is a general duality: Proposition 5.12 ([32], 5.5). Let G be an infinitesimal group scheme, and let M be a finite dimensional kG-module of constant j -rank. There is an isomorphism of OP .G/ -modules: z j /_ ' Coker ‚ zj # .Ker ‚ M

where ./_ denotes the dual vector bundle.

M

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Even though finer invariants than support varieties and local Jordan types, the z j and others are by no means faithful. In particular, they vanish functors M 7! Ker ‚ M on the interesting class of modules with constant kernel/image property studied, in particular, in [17]. Definition 5.13 ([32], 5.11). Let M be a finite dimensional kG-module. We say that M has constant j -image property if there exists a subspace I.j / M such that for any point 2 V .G/, 6D 0, Imfj W Mk./ ! Mk./ g D I.j /k./ : Similarly, M has constant j -kernel property if there exists a subspace K.j / M such that for any point 2 V .G/, 6D 0, Kerfj W Mk./ ! Mk./ g D K.j /k./ : Proposition 5.14 ([32], 5.12). Let M be a kG-module of constant j -rank. Then the z j is trivial (i.e., a free coherent sheaf ) if and only if algebraic vector bundle Im ‚ M z j is trivial if and only if M has M has constant j -image property. Similarly, Ker ‚ M constant j -kernel property. Let HŒ1 .M / D

z Ker ‚ M zM Im ‚

p1

. Recall that a kG-module M is called endotrivial if

Endk .M / ' k in stmod G. It was shown in [14, §5] that an endotrivial module is a module of constant Jordan type with possible types Œpap Œ1 and Œpap Œp 1. One can also characterize endotrivial modules in terms of the sheaf HŒ1 .M /. Proposition 5.15 ([32], 5.17). Let G be an infinitesimal group scheme, and assume that G has a subgroup scheme isomorphic to Ga.1/ Ga.1/ or Ga.2/ . Let M be a module of constant Jordan type. Then HŒ1 .M / is a line bundle (i.e., an algebraic vector bundle of rank one) if and only if M is endotrivial. In particular, HŒ1 ./ induces a map from the group of endotrivial modules to the Picard group of P .G/. In [5], Balmer constructs a map in the opposite direction (after inverting p) for G a finite group. The connection between these two maps is yet to be understood. We give examples of some explicit calculations of bundles corresponding to sl2 modules. The nullcone N.sl2 / is a quadric in A3 defined by the equation z 2 C xy D 0. Hence, there is an isomorphism i W P .sl2 / ' P 1 (see, for example, [32, 5.8.1]). In the calculations below we identify P .sl2 / with P 1 via this isomorphism which allows us to use the standard notation for line bundles on P 1 . Recall that the representation theory of u.sl2 / is tame. The complete list of indecomposable sl2 -modules consists of the following four families (see, for example, [47]), with some overlap:

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(1) Weyl modules V ./. (2) Induced modules H0 ./ ' V ./# . (3) Modules ˆ ./ for 2 Pk1 . These modules do not generally have rational SL2 structure and do not have constant Jordan type. The projectivized support variety of ˆ ./ contains one point: . (4) Projective indecomposable modules Q./ for 0 p 1, with Q.p 1/ D V .p 1/ being the irreducible Steinberg module. zM It is shown in [51] that for any finite dimensional restricted sl2 -module M , Ker ‚ is a vector bundle, whether M has constant rank or not. Proposition 5.16 ([51]). Let D rp C a, 0 a p 1, r 0. z V ./ ' OP 1 ./ ˚ OP 1 .a C 2 p/˚r ; (1) Ker ‚ z 0 ' OP 1 .a/˚rC1 ; (2) Ker ‚ H ./ z ˆ ./ ' OP 1 .a C 2 p/˚r ; (3) Ker ‚ z Q.a/ ' OP 1 ./ ˚ OP 1 .a C 2 2p/. (4) Ker ‚ z uses properties of infinitesimal Remark 5.17. The construction of the global operator ‚ group schemes in an essential way. We can transfer the resulting constructions of sheaves and algebraic vector bundles to the variety Proj H .G; k/ via the isogeny ‰ of Theorem 2.6 but the structure of the variety of one-parameter subgroups V .G/ is crucial for the very definition of ‚. It is an open question whether an analogous construction exists for finite groups (apart from elementary abelian which are really just restricted Lie algebras in disguise).

6 Elementary subalgebras of restricted Lie algebras In this section we consider only infinitesimal group schemes of height one or, equivalently, restricted Lie algebras. As usual, all considerations apply to an elementary abelian p-group E of rank r once we choose an isomorphism kE ' u.g˚r a /. Definition 6.1. A subalgebra of a restricted Lie algebra g is called elementary if is an abelian Lie algebra with trivial Œpth power. Equivalently, ' g˚r a for some integer r > 0. We define E.r; g/ D f g j elementary subalgebra of dimension rg: Recall that for ' g˚r a , we have u./ ' kŒx1 ; : : : ; xr =.x1p ; : : : ; xrp /: For a vector space V of dimension n r, we denote by Grass.r; V / the Grassmannian of r-dimensional linear subspaces of V .

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Proposition 6.2. Let g be a restricted Lie algebra. For any r dim g, there exists a closed embedding E.r; g/ ,! Grass.r; g/ which gives E.r; g/ a structure of a projective algebraic variety. Except for some special cases, the varieties E.r; g/ are quite mysterious. In the case r D 1 they are familiar objects though: we get projectivizations of restricted nullcones. Hence, E.r; g/ can be viewed as a natural generalization of the rank variety of g. Constructions of rank varieties and generalized rank varieties for modules as well as of the global p-nilpotent operator ‚ and the associated sheaves all carry through for this new ambient variety E.r; g/. Hence, we get a whole slew of new invariants of modular representations as well as new techniques for constructing algebraic vector bundles on interesting projective varieties. Example 6.3. Let r D 1. Then E.1; g/ ' Proj kŒNp . In particular, if G is a reductive connected algebraic group and g D Lie G, then E.1; g/ is irreducible ([44], [55]). Example 6.4. Let r D 2. Let G be a connected reductive algebraic group, let g D Lie G, and assume that p is good for G. A result of Premet ([47]) implies that E.2; g/ is equidimensional; in the special case g D gln , p n, E.2; gln / is irreducible of dimension n2 5. Things get murky once r 3. On the other hand, there is quite a bit we can say if we consider maximal elementary subalgebras of a given Lie algebra g. Notation 6.5. Let g be a restricted Lie algebra. Let rk el .g/ D maxfr j there exists an elementary subalgebra g with dim D rg: For complex simple Lie algebras the dimension of a maximal abelian subalgebra was determined by Malcev in 1945 [41]; the general linear case was first considered by Schur at the turn of the last century [49]. We give some examples of calculations of rk el .g/ and the corresponding varieties E.r; g/ for restricted Lie algebras; more can be found in [16]. We denote by Grass.r; n/ the Grassmannian of r-planes in n-space, and by LG.n; n/ the Lagrangian Grassmannian of isotropic subspaces of maximal dimension in a 2ndimensional symplectic space. Theorem 6.6. Let g D gln (or sln ). (1) If n D 2m, then rk el .g/ D m2 , and E.m2 ; g/ ' Grass.m; 2m/: (2) If n D 2m C 1, m 2, then rk el .g/ D m.m C 1/, and E.m.m C 1/; g/ ' Grass.m; 2m C 1/ t Grass.m; 2m C 1/: Theorem 6.7. For g D sp2n , rk el .sp2n / D

E

n.nC1/ , 2

and

n.n C 1/ ; sp2n ' LG.n; n/: 2

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Modules of constant .r; j / rank and vector bundles on E.r; g/. For an elementary subalgebra g, and a g-module M , we denote by M # the restriction of M to . Then Radj .M # / is the j th radical of M as an u./ ' kŒx1 ; : : : ; xr =.x1p ; : : : ; xrp /module, and Socj .M # / is the j th socle. Definition 6.8. A restricted g-module M is a module of constant .r; j /-radical rank if the dimension of Radj .M # / is independent of 2 E.r; g/. We say that M is a module of constant .r; j /-socle rank if the dimension of Socj .M # / is independent of 2 E.r; g/. For r D 1 the notions of constant j -radical and j -socle rank are equivalent. Moreover, they are both equivalent to the notion of constant j -rank. For r > 1, the properties of constant .r; j /-radical and socle rank are independent of each other; one can find examples of this phenomenon in [15]. An analogue of modules of constant Jordan type for r > 1 is given in the following definition: Definition 6.9. A restricted g-module M is a module of constant r-radical type if it is a module of constant .r; j /-radical rank for any integer j > 0. Similarly, M is a module of constant r-socle type if it is a module of constant .r; j /-socle rank for any integer j > 0. Once again, for r D 1 both notions are equivalent to constant Jordan type. For r > 1, properties of constant radical type and constant socle type are independent. There are several constructions of modules of constant .r; j /-radical or socle rank for various combinations of r and j in [15]. Here, we give one family of examples which involves Carlson modules. For a positive degree cohomology class 2 Hm .G; k/, the Carlson module L is the kernel of the map O W m k ! k that corresponds to under the isomorphism Hm .G; k/ ' Homstmod G . m k; k/. Theorem 6.10 ([15], 5.5). Let E be an elementary abelian p-group of rank n, and let 2 Hm .E; k/ be a non-nilpotent positive dimensional cohomology class. If the hypersurface Z. / Proj H .E; k/ ' P n1 defined by the equation D 0 does not contain a linear subspace of dimension r 1 then L has constant r-radical type. Whereas modules L are never of constant Jordan type, for r > 1 there are plenty of examples when the condition of the theorem is satisfied (see [15, §3]). To construct algebraic vector bundles on E.r; g/ corresponding to representations of constant .r; j /-radical or socle rank, we need to replace the operator ‚ for r D 1 with a vector operator .‚1 ; : : : ; ‚r /. The operators ‚i are not defined globally on E.r; g/, and there are two equivalent approaches to their construction that lead to associated kernel, image, and cokernel sheaves which are defined globally on E.r; g/: (1) Local construction: this approach involves patching together images (respectively, kernels or cokernels) of explicit linear maps on an affine covering of E.r; g/:

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(2) Equivariant descent: this construction employs homogeneous linear operators ‚i W M ˝ kŒNpr .g/ ! M ˝ kŒNpr .g/Œ1 for i D 1; : : : ; r where Npr .g/ is the variety of p-nilpotent commuting elements of g. The kernels and images of these operators are GLr -equivariant which allows to pass to sheaves on E.r; g/. Details can be found in [15], [16]. The outcome of these somewhat technically involved constructions is the following theorem. For a quasi-projective variety X we denote by Coh.X / the category of coherent OX -modules. Theorem 6.11 ([16], §5). Let g be a restricted Lie algebra. There exist functors Im j ; Ker j W u.g/-mod ! Coh.E.r; g// such that the fiber of Im j .M / (respectively Ker j .M /) for a restricted g-module M at a generic point 2 E.r; g/ is naturally identified with Radj .M # / (respectively, Socj .M# /). More generally, for any locally closed subset X 2 E.r; g/, there exist functors Im j;X ; Ker j;X W u.g/-mod ! Coh.X / such that the fiber of Im j;X .M / (respectively, Ker j;X .M /) at a generic point 2 X is naturally identified with Radj .M # / (respectively, Socj .M # /). For an algebraic group G, the variety E.r; g/ for g D Lie G comes equipped with an action of G. The second part of Theorem 6.11 allows one to construct sheaves on G-orbits of E.r; g/. Under this construction, modules of constant .r; j / rank lead to algebraic vector bundles, in analogy with modules of constant j -rank. Proposition 6.12 ([16], 5.20, 5.21). Let M be a restricted g-module. If M is a module of constant .r; j /-radical rank (respectively, constant .r; j /-socle rank) then Im j .M / (respectively, Ker j .M /) is an algebraic vector bundle on E.r; g/ with fiber at 2 E.r; g/ naturally isomorphic to Radj .M # / (respectively, Socj .M # /). More generally, let X E.r; g/ be a locally closed subset. If dim Radj .M # / (respectively, dim Socj .M # /) is constant for all 2 X , then Im j;X .M / (respectively, Ker j;X .M /) is an algebraic vector bundle on X with fiber at 2 X naturally isomorphic to Radj .M # / (respectively, Socj .M # /). We finish with some computational examples of the kernel, image, and cokernel sheaves on E.r; g/. Example 6.13. Let g D sl2n , and let V be the defining representation of sl2n . We have E.n2 ; sl2n / ' Grass.n; 2n/ by Theorem 6.6. Assume p > 2n 2. Let n be the canonical ( tautological) rank n vector subbundle on Grass.n; 2n/. Then

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(1) Im 1 .V / D n . For any j m, (2) Im j .V ˝j / ' n˝j , (3) Im j .S j .V // ' S j .n /, (4) Im n .ƒn .V // ' ƒn .n /. Similar calculations can be done for other Grassmannians Grass.r; n/ by realizing them as GLn -orbits in E.r.n r/; sln / ([16, §6]). Example 6.14 ([16], 6.9). Let g D sp2n , let V be the defining for representation sp2n , and assume that p > 4n 2. By Theorem 6.7, E nC1 ; sp D LG.n; n/, 2n 2 the Lagrangian Grassmannian. We denote by nsp the canonical ( tautological) vector subbundle of rank n on LG.n; n/. With this notation we have exactly the same results as in Example 6.13: (1) Im 1 .V / D nsp ; For any j n, (2) Im j .V ˝j / ' .nsp /˝j , (3) Im j .S j .V // ' S j .nsp /; (4) Im n .ƒn .V // ' ƒn .nsp /: In our last example we demonstrate that using representations of Lie algebras one can easily realize tangent and cotangent bundles on Grassmannians. As with the examples above, it works very similarly for the special linear and the symplectic case. Example 6.15. Let g D sl2n (respectively, g D sp2n ). Consider g acting on itself via the adjoint Let X D E.n2 ; sl2n / ' Grass.n; 2n/ (respectively, representation. nC1 X D E 2 ; sp2n ' LG.n; n/). Assume p > 2n 2 (respectively, p > 4n 2). (1) Coker .g/ ' TX , the tangent bundle on X , (2) Im 2 .g/ ' X , the cotangent bundle on X . For other simple algebraic groups G one can make analogous calculations for bundles on homogeneous spaces associated with cominuscule parabolics by considering G-orbits of E.r; Lie G/ (see [16]).

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Superdecomposable pure-injective modules Mike Prest

1 Introduction Every finite-dimensional module is pure-injective. A superdecomposable module is one with no indecomposable direct summand. So it might seem that, in the context of modules over finite-dimensional algebras, the two adjectives in the title have little connection, at least as far as the finite-dimensional representation theory is concerned. In fact, the existence of superdecomposable pure-injective modules is a strong reflection of some aspects of the structure of morphisms in the category of finite-dimensional modules. In particular, the evidence to date is consistent with existence of superdecomposable pure-injectives being equivalent to having non-domestic representation type. In this expository paper, we explain this, describe some methods, due to Ziegler and Puninski, for proving existence of these modules and outline the proof of a recent result of Harland that, if R is a tubular algebra and r is a positive irrational, then there is a superdecomposable pure-injective module of slope r. Pointed modules and morphisms between them, as well as (the essentially equivalent) pp formulas and associated methods from the model theory of modules, figure heavily in Harland’s proof. Puninski’s proofs also use these methods, which we will explain, though we won’t use much model theory per se. When we consider pointed modules, we are interested not just in morphisms between modules but also in what these do to specified individual elements. This fits well with model theory, which has the notion of the type of an element in a structure; types keep track of what changes, as regards that element, when the structure changes. In model theory, the usual changes to a structure are to (elementarily) embed it in a larger one and/or to restrict automorphisms to those which fix some specified substructure. In the model theory of modules, however, the allowed changes are unrestricted: any homomorphism is fine. This still fits, in the module context, with the model theory because every homomorphism preserves pp formulas and there is a result (pp-elimination of quantifiers) which says that the model theory in modules is determined by pp formulas. We will explain what pp formulas, and the corresponding pp-types, are but we don’t need to go into model theory as such; all we need is this convenient terminology and technology for handling pointed modules plus some useful theorems that have been proved in this context. Our default is that modules are right R-modules. The category of these is denoted Mod-R; mod-R denotes the category of finitely presented modules. We write .M; N / rather than Hom.M; N /. We will use the notation pp for the lattice of pp formulas, equivalently pointed finitely presented modules, possibly decorated with a subscript to denote the ring and/or a superscript to denote the number of elements we’re pointing to.

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2 Pointed modules A pointed module is a pair .M; a/ consisting of a module and an element in that module. We preorder pointed modules by setting .M; a/ .N; b/ if there is a morphism from M to N taking a to b. Since a pointed module is equivalently a morphism with domain R (that which takes 1 2 R to the element of the pair), this is just the ordering on such morphisms given by f f 0 if f 0 factors initially through f (that is, f 0 D gf for some g). Write .M; a/ .N; b/ if both .M; a/ .N; b/ and .N; b/ .M; a/. If .A; a/ is a pointed module with A finitely presented then we refer to this as a pointed finitely presented module. The partial order we obtain on pointed finitely presented modules is naturally isomorphic to the lattice of pp formulas in one free variable, which is one of the basic structures in the model theory of modules. We explain this so that we can move easily between pointed modules and pp formulas. Before doing that, notice that we can generalise to n-pointed modules – pairs .M; a/ N consisting of a module and a specified n-tuple of its elements, thus replacing the top point .R; 1/ in the preordering by .Rn ; e/ N where eN D .e1 ; : : : ; en / is a basis for Rn . Even more generally, if A is a finitely presented module then we have the notion of A-pointed modules, equivalently morphisms with domain A. If we choose a generating tuple, aN D .a1 ; : : : ; an /, of elements of A then the poset of A-pointed finitely presented modules is thereby identified with the interval between .A; a/ N and .0; 0/ in the lattice of n-pointed finitely presented modules. The most general notion is that of an .A; a/-pointed N module, where the tuple aN does not necessarily generate A; again, this is just a representative of a point in the interval Œ.A; a/; N .0; 0/. In all these cases it is easy to check that the resulting partial order on (appropriately-)pointed finitely presented modules is a modular lattice, with join given by direct sum and meet given by pushout; note that if A and B are finitely presented then so is C in the pushout diagram below. a

/A R@ @@ @@c b @@ ˇ B ˛ /C

We will write .A; a/ ^ .B; b/ for (any pointed module in the equivalence class of) .C; c/ and .A; a/ C .B; b/ for .A ˚ B; a C b/. We will use pointed modules where the module is not necessarily finitely presented but, when it comes to considering a lattice structure, we restrict to pointed finitely presented modules. There will be a few occasions when we need to consider pp-types and not just pp formulas; then we use pointed pure-injective modules. This restriction is because the structure that we actually need, and that can be exactly expressed in terms of pp formulas (or finitely presented functors from mod-R to Ab) is perfectly reflected in existence of morphisms between finitely presented modules, or between pure-injective modules, but not between arbitrary modules where there are obstructions to existence

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of the morphisms that would faithfully reflect the pp formula/finitely presented functor structure. 2.1 Pointed finitely presented modules and pp formulas. Now we explain what pp formulas are and how they arise from pointed finitely presented modules. Fix a pointed finitely presented module .A; a/. Choose a finite generating tuple cN D .c1 ; : : : ; ck / for P A. Since A is finitely presented there are finitely many linear equations k1 xi rij D 0 (j D 1; : : : ; m) which are satisfied by cN and generate all other linear equations satisfied by c. N Combine these into the single matrix equation cH N D 0 where H is the matrix with ij -entry rij . Since cN generates A, the element a can be expressed as an R-linear P combination of the ci , say a D c s . So the following i i i Passertion is true in A: there are elements x1 ; : : : ; xn such that xH N D 0 and a D i xi si . That assertion is expressible by a formula in the standard formal language which one sets up when dealing with N D P the model theory of R-modules, namely the formula 9x1 ; : : : ; xn .xH 0 ^ a D i xi si /, where the symbol ^ is read as “and”. One might write the notation .a/ for this formula, showing the element a because it is a parameter coming from a particular module (whereas the symbols for multiplication by individual elements of the ring R are fixed, being built into the formal language). The corresponding formula without parameters would replacePa by a variable, say v, to give the formula .v/ which is 9x1 ; : : : ; xn .xH N D 0 ^ v D i xi si /, and then we would regard .a/ as the result of substituting the free variable v in .v/ by a particular element of a particular module. This is a fairly representative pp formula1 . Because the free variable v in .v/ is just a place-holder and the choice of variable name is irrelevant2 we may write just . Any formula constructed from .A; a/ as above (note that there were choices made) is said to generate the pp-type of a in A; we will discuss pp-types later and explain this terminology. What we have done here for (1-)pointed modules works as well for n-pointed modules; it will give us pp formulas with n free variables. The extension of the notion of pp formula to allow for more than one free (D unquantified D substitutable-byparameters variable) should be clear and the similarly-constructed pp formula is said to generate the pp-type of aN in A; the notation typically used is ppA .a/ N D hi. If is a pp formula with one free variable, such as that above, if M is any module and if m 2 M , then either m satisfies the condition expressed by or it doesn’t; if it does, then the notation used in model theory is M ˆ .m/ but we will rather write m 2 .M / where we denote by .M / the solution set of in M – the set of elements of M which satisfy the condition expressed by . Note that whether or not m satisfies depends on the containing module M (at least, if there are existentially quantified variables in ). Of course, if has n free variables then the solution set .M / will be a 1 The abbreviation pp (for positive primitive) is used for formulas such as .v/ which are existentially quantified conjunctions (repeated “and”) of linear equations; using matrix equations is just a convenient way of arranging a conjunction of equations. 2 as long as we avoid using one already in use

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subset of M n . The assignment of .M / to M actually defines a functor from Mod-R to Ab, the point being the following (which his easily proved, or see, e.g. [37, 1.1.7, §1.1.1]). Lemma 2.1. If M is any module and D .v1 ; : : : ; vn / is a pp formula then the solution set, .M /, of in M is a subgroup of M n – a pp-definable subgroup of M .3 If, moreover, f W M ! N is a morphism and if aN 2 .M / then f aN 2 .N /. That is, M 7! .M / defines a subfunctor of the n-th power, .Rn ; /, of the forgetful functor. If is a pp formula defined from the pointed module .A; a/ as above then this functor picks out the trace of .A; a/ on a module M , that is, the set of images of a under morphisms from A to M . Let us write tr.A; a/.M / for this set which, by the next lemma, equals .M /. Lemma 2.2 (see [37], 1.2.7). If .A; a/ N is an .n-/pointed finitely presented module and if is a pp formula constructed from .A; a/ N as above then, for any module M and m x 2 M n, m x 2 .M / iff there is a morphism f W A ! M with f aN D m. x This also shows that the ordering on pointed finitely presented modules is equivalent to the ordering on pp formulas, the latter being defined by if .M / .M / for every M 2 mod-R. (This implies that .M / .M / for every M 2 mod-R because every module is a direct limit of finitely presented modules, and functors of the form M 7! .M / commute with direct limits4 .) We regard pp formulas with the same solution set in every (finitely presented) module as equivalent and don’t distinguish between them, nor between them and their equivalence classes. We write ppn (or ppnR ) for the lattice of pp formulas in n free variables (for right R-modules) and usually drop the superscript when n D 1. By the discussion above (also see [37, 10.2.19]) we have the following. Proposition 2.3. The lattice of n-pointed finitely presented modules is naturally isomorphic to the lattice ppn of pp formulas with n free variables. This lattice is modular. More generally, if .A; a/ N is a pointed finitely presented module and if generates the pp-type of aN in A then the lattice of .A; a/-pointed N modules is naturally isomorphic to the interval Œ; 0 in ppn (where 0 denotes the class of the pp formula v1 D 0^ ^vn D 0). The terminology used in the model theory of modules when making use of this bijection is as follows. As we have seen already, starting with a pointed finitely presented module .A; a/ N we can construct a pp formula which generates the pp-type of aN in A: ppA .a/ N D hi. In the other direction, starting with a pp formula we can construct a free realisation of , that is a pair .A ; aN / with A finitely presented and generating the pp-type of a in A (see, e.g., [37, 1.2.14]). Or, more carefully if n > 1, a subgroup of M n pp-definable in M . This relation between and is one that can be checked fairly effectively, just in terms of the systems of equations they involve, in particular, without having to test it at every finitely presented module, see [37, 1.1.13]. 3 4

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We mention that the functor associated to an n-pointed finitely presented module n ; / .A; a/ N (equivalently to the associated pp formula), that is, the subfunctor of .RR defined by taking B 2 mod-R to fg aN W g 2 .A; B/g, is finitely generated, indeed, finitely presented, as an object of the functor category .mod-R; Ab/. For we have the a A ! A=aR ! 0 in mod-R and this induces an exact sequence of exact sequence R ! functors 0 ! .A=aR; / ! .A; / ! .R; /, where the image of the last morphism is exactly this functor. Since .A=aR; / and .A; / are finitely generated projectives in .mod-R; Ab/ this proves the claim. Indeed (see, e.g., [37, 10.2.2]) every finitely generated subfunctor of .Rn ; / has this form. 2.2 Relativising to a definable subcategory. Fix any module M . We define a quotient of the lattice of pointed finitely presented modules relative to M . This will be naturally isomorphic to the lattice of pp-definable subgroups of M , for which the usual notation is pp.M /, which notation we may as well use for the lattice thought of in terms of pointed finitely presented modules. That is, for pointed finitely presented modules .A; a/ and .B; b/ we set .A; a/ M .B; b/ if for every morphism f W B ! M there is a morphism g W A ! M with ga D f b. Clearly this is equivalent to tr.A; a/.M / tr.B; b/.M /. Then we set .A; a/ M .B; b/ if both .A; a/ M .B; b/ and .B; b/ M .A; a/ hold; we can then take pp.M / to be the resulting quotient lattice. Of course there are corresponding definitions for n-pointed modules obtained by putting a bar over a and b and a superscript n to “pp”. Lemma 2.4. Let .A; a/ and .B; b/ be pointed finitely presented modules with pushout diagram as shown and let M be any module. Then .A; a/ M .B; b/ iff every f W B ! M factors through ˛ W B ! C . a

R

/A

b

B

/C

˛

ˇ

Proof. ()) Take f W B ! M . Then we have the commutative diagram shown where g exists by assumption and so, by the pushout property, there is h as shown. In particular f factorises through ˛ as required. R

a

ˇ

b

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/A

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/C A f

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(() Given f W B ! M there is, by assumption, h W C ! M with f D h˛, so then f b D h˛b D hˇa so set g D hˇ W A ! M . Corollary 2.5. Let .A; a/ and .B; b/ be pointed finitely presented modules with pushout diagram as above. Then .A; a/ M .B; b/ iff every morphism from A to M factors through ˇ and every morphism from B to M factors through ˛. We comment here that we can assume when necessary that in a pointed finitely presented module .C; c/ N the module C is pp-essential over c, N meaning that there is no proper direct summand of C containing every entry of cN (if there is such a proper direct summand then it lies in the same equivalence class as .C; c/ N in the sense of the ordering in 2.3 so we may use it to replace .C; c/). N The reason for the terminology (which is from n [31]) is that it is equivalent to the image of the functor .cN ˝ / W .RR ˝ / ! .C ˝ / being essential in the functor .C ˝ / (which is an injective object of .R-mod; Ab/fp , [19, 5.5]), so we are actually looking here at injective hulls – which exist if R is an Artin algebra – in the subcategory of finitely presented functors on finitely presented left modules (equivalently, see, e.g., [37, §10.2.3], projective covers in the dual category of finitely presented functors on finitely presented right modules). A definable subcategory of Mod-R is one closed under direct limits, products and pure submodules5 . Relativising the pp-lattice to modules is equivalent to relativising to definable subcategories because if two modules M , M 0 generate the same definable subcategory then the induced orderings M and M 0 on pointed finitely presented modules are identical (e.g. by [37, 3.4.11]), so pp.M / and pp.M 0 / are naturally isomorphic (or identical, depending on how we view these). Every definable subcategory D is generated as such by some module (the direct sum, or direct product, of one copy of each indecomposable pure-injective in D will do) so we extend the notation and terminology to definable subcategories writing, for example, pp.D/. If M 0 is in the definable subcategory generated by M , in particular if M 0 is a pure submodule of M , and .A; a/ M .B; b/ then .A; a/ M 0 .B; b/ (see [37, 3.4.11]). 2.3 Elementary duality. If M is any module, S ! End.M / is a homomorphism of rings and S E is an injective cogenerator for left S-modules then we set M D HomS .S E; S M / to be the corresponding dual of M . Theorem 2.6 (e.g. [37], 3.4.17). If M and N generate the same definable subcategory, D, of Mod-R then M and N generate the same definable subcategory of R-Mod, denoted D.D/ and termed the (elementary) dual of D. Furthermore, M and M generate the same definable subcategory of Mod-R; that is, D 2 .D/ D D. In particular D.Mod-R/ D R-Mod. A If .v1 ; : : : ; vn / is a pp formula, say it is 9y1 ; : : : ; ym .v; N y/ N B D 0, where A, B are matrices (matching v, N resp. y) N with entries in R, then the elementary dual [32] of 5 The definable subcategories that we consider will arise naturally in particular contexts; for the general theory of definable categories one may look at [37] or [39].

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A vN is the pp formula for left modules which is 9zN I0n B zN D 0 where In is the n n identity matrix and 0 denotes a zero matrix of the correct size (and where vN and zN now are column vectors). In a moment we will rephrase this in terms of pointed modules but first we state the relation between pp.D/ and pp.D.D//. Theorem 2.7 (e.g. [37], ). If D is a definable subcategory of Mod-R then elementary duality op of pp formulas induces an anti-isomorphism of lattices pp.D.D// ' pp.D/ . Duality can be described in terms of pointed modules, as follows. If .A; a/ is a pointed finitely presented module, then choose a morphism Rm ! Rn , given by left multiplication by the matrix H , with cokernel A but also with H chosen so that the image of the element .1; 0; : : : ; 0/ 2 Rn is a. Now use the transpose map, right multiplication by H , to define a morphism R Rn ! R Rm . Let W R Rn ! R be projection to the first coordinate. Then form the pushout shown. Rn

H

R

l

/ Rm /L

If .A; a/ is a free realisation of the pp formula then .L; l/ will be a free realisation of its dual D.

3 Pure-injective modules Over Artin algebras the simplest definition of pure-injectivity is that a module N is pure-injective if it is a direct summand of a direct product of finite length modules. As the name suggests, the general definition is that a module is pure-injective if it is injective over pure monomorphisms. A pure monomorphism is one that begins a f

pure-exact sequence, that is, a short exact sequence 0 ! A ! B ! C ! 0 (of right modules) which, on tensoring with any (finitely presented) left A-module, gives an exact sequence of abelian groups. There is a conceptually quite distinct notion, algebraic compactness, which coincides with pure-injectivity (for modules, e.g. [66, Theorem 2], and for many other types of structure). A module N is algebraically compact if every collection of cosets of pp-definable subgroups of N with the finite intersection property (that is, the intersection of any finitely many of these cosets is non-empty) has non-empty intersection. Since every pp-definable subgroup of a module M is (by 2.1) closed under the action of End.M /, any module which is of finite length over its endomorphism ring (finite endolength for short) is algebraically compact. So the generic modules over Artin algebras – those indecomposables which are of finite endolength but not of finite length – are, together with the finite-length modules, examples of pure-injectives. Any

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injective module is pure-injective. Any functor between module categories which commutes with direct sums and direct products preserves pure-injectivity ([23, 7.35]), so the Prüfer modules ([56]) over tame hereditary (and other) algebras are pure-injective, being the images of injective modules under such functors. Similarly, rather dually, adic modules, being the images of suitably compactified modules, are pure-injective (see, e.g., [7]). Hom-duals of modules are pure-injective: if M is any right R-module and M D HomS .S M; S E/ is as in Section 2.3, then M is a pure-injective left R-module (see, e.g., [37, §4.3.4]). Moreover, the natural map from M to M is a pure embedding, so every module purely embeds in a pure-injective module. Indeed there is a unique minimal-to-isomorphism-fixing-M such embedding, M ! H.M /; we say that H.M / is the pure-injective hull (or pure-injective envelope) of M . Definable subcategories are closed under taking pure-injective hulls (see [37, 3.4.8]). The easiest way to get the existence of pure-injective hulls (various proofs are referenced at [37, 4.4.2]), and the general structure theorem, 3.2, for pure-injectives is to make use of the full embedding M 7! .M ˝R / of Mod-R into the functor category .R-mod; Ab/ ([18], see [23, B16], [37, §12.1.1]). Theorem 3.1. The right module N is pure-injective iff the functor N ˝ is injective. Because the functor M 7! .M ˝ / is full it follows, from the fact that this is true for injectives, that an indecomposable pure-injective has local endomorphism ring. The general decomposition theorem for injective objects in Grothendieck categories pulls back to the following theorem for pure-injective modules (the first proof seems to be that in [11, 7.21]). L 0 Theorem 3.2. Every pure-injective module has a decomposition H N ˚ N 0 where each N is indecomposable pure-injective and N is superdecomposable. It is the existence of superdecomposable pure-injectives in the category Mod-R and, more generally, in definable subcategories of Mod-R, that we are interested in here. For pure-injectivity in general, there are already surveys such as [22], [38] and, accounts with full proofs such as [23], [37]. In Section 5.1 we indicate what is known about existence of superdecomposable pure-injectives. 3.1 Pointed pure-injectives and pp-types. A key concept in the model theory of modules is that of a pp-type (a modification of the notion of type which is pervasive in model theory) but, just as pp formulas are equivalent to pointed finitely presented modules, pp-types are equivalent to pointed pure-injective modules. The actual definition of the pp-type, ppM .m/, of an element m in a module M is that it is the set of pp formulas such that m 2 .M / (similarly for pp-types of n-tuples) but below we will reformulate this algebraically. We say that a pp formula generates a pp-type p if p D f W g. We order pp-types so as to agree with the ordering on pp formulas,

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setting p q iff p q.6 The connection with pointed modules is as follows (see [37, 4.3.9]). Theorem 3.3 (e.g. [37], 4.3.9). Suppose that .M; m/ and .N; n/ are pointed modules and that N is pure-injective. Then ppM .m/ ppN .n/ iff there is a morphism .M; m/ ! .N; n/. Pure-injectivity of N is needed for this. Theorem 3.4. The poset of pointed pure-injective modules is naturally isomorphic to the poset (indeed, modular lattice) of pp-types which, in turn, is naturally isomorphic to the lattice of subfunctors of the forgetful functor .R R; / ' .R ˝ / in .R-mod; Ab/. The first statement is immediate from the previous result. For the other equivalence see, for instance, [37, 12.2.1]; it uses elementary duality (Section 2.3) to give that pp-types – that is filters in the lattice of finitely generated subfunctors of the forgetful functor .RR ; / 2 .mod-R; Ab/ – are in natural bijection with arbitrary subfunctors of the forgetful functor .R R; / 2 .R-mod; Ab/. We won’t really use this result but perhaps it makes the link between the model-theoretic and algebraic concepts clearer. Here is the algebraically reformulated definition of the pp-type of an element m in a module M : it is the collection of all pointed finitely presented modules .A; a/ such that there is a morphism .A; a/ ! .M; m/. We write ppM .m/ for this also (and whether you want to think of this as a set of pp formulas or a set of pointed finitely presented modules, or both, is up to you). This (rather this collection up to equivalence) is not just a set: it’s actually a filter7 in the lattice ppR , in the sense that if .A0 ; a0 / .A; a/ and .A; a/ 2 ppM .m/ then .A0 ; a0 / 2 ppM .a/ (trivial from the definitions) and if .A; a/ and .B; b/ are in ppM .m/ then so is their pushout/meet .A; a/ ^ .B; b/ (immediate). And conversely, given any set of pointed finitely presented modules closed under (finite) pushout we can take the direct limit of this directed system to get a module .M; m/ with pp-type the closure of this set under precomposition. We state this (the proof is from the fact that a module C is finitely presented iff the functor .C; / commutes with direct limits). Proposition 3.5 ([37], 3.2.5). Suppose that A is a set of pointed finitely presented modules closed under pushout (or even just closed under pushout up to equivalence). Let .M; m/ be the direct limit of this system of morphisms with domain R. Then, if .C; c/ is a pointed finitely presented module, there is a morphism .C; c/ ! .M; m/ iff there are .A1 ; a1 /; : : : ; .An ; an / 2 A and a morphism .C; c/ ! .A1 ; a1 / ^ ^ .An ; an /. In particular, given any set in ppR , the filter it generates (by taking finite meets and closing upwards) is the pp-type of some element in some module (which may be taken to be pure-injective). 6 The convention varies, over time as well as between people, for example this agrees with [37] but not [31]! 7 or, in more category-theoretically, a sieve closed under pushout

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For the last statement just replace M by its pure-injective hull. Corollary 3.6. If p is a pp-type then there is a (pure-injective) module N and an element n 2 N which realises p, that is, such that ppN .n/ D p. Note that if .A; a/ is a pointed finitely presented module then the pp-type ppA .a/ of a in A consists of all the pointed finitely presented modules which are greater than or equal to .A; a/ (just by definition). Let us state this. Remark 3.7. If .A; a/ is a pointed finitely presented module then ppA .a/ D f.B; b/ W B 2 mod-R and .B; b/ .A; a/g. A pp-type which consists of all the pointed modules above a single8 pointed finitely presented module .A; a/ is said to be finitely generated (by .A; a/). Of course everything said here for 1-pointed modules works for n-pointed modules. 3.2 Hulls of pp-types and Ziegler’s criterion. Arguments using the model theory of modules often work to produce a pp-type with some particular properties and then finish with an appeal to taking the hull of that type so as to get a pure-injective module whose (desired) properties are algebraic reflections of those properties of the pp-type. We describe what is happening here in terms of pointed modules. Suppose that p is a pp-type. By 3.6 there is a pointed pure-injective .N; n/ which realises p, that is, such that if .A; a/ is a pointed finitely presented module then there is a morphism from .A; a/ to .N; n/ iff .A; a/ 2 p. There is a minimal direct summand9 of N which contains n and this is unique-to-isomorphism over n. We can see this by considering the image of .n ˝ / W .R ˝ / ! .N ˝ /; by 3.1 the injective hull of this image has the form .N 0 ˝ / for some direct summand N 0 of N . We refer to this minimal pure-injective, N 0 , over n as the hull of n in N , writing H.n/ for (any copy of) this. Since the pp-type of n is p we also write H.p/ for this and call it the hull of p. That this is well-defined follows from the next result (and the fact that if m 2 M 0 and M 0 is pure in M , in particular if M 0 is a direct summand of M , then 0 ppM .m/ D ppM .m/). Proposition 3.8 (see [37], 4.3.42). Suppose that .N; n/ and .M; m/ are pointed pureinjective modules and that ppN .n/ D ppM .m/. Then (by 3.3) there is a morphism .N; n/ ! .M; m/; for any such morphism the image in M of any copy of the hull of n in N is a direct summand of M . Say that a pp-type is irreducible10 if its hull H.p/ is indecomposable. That is, the pp-type of an element m in a module M is irreducible iff the smallest direct summand of the pure-injective hull of M containing m is indecomposable. This can be characterised purely in terms of the lattice properties of p, as follows. 8

equivalently, by taking their pushout, finitely many pointed finitely presented modules That is, there is no direct summand of N which contains n and is properly contained in this. 10 the terminology “indecomposable” also is used 9

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Theorem 3.9 (Ziegler’s criterion, [67], 4.4). Let p D ppM .m/ be a pp-type. Then p is irreducible iff whenever .B; b/ and .B 0 ; b 0 / are pointed finitely presented modules not in p there is a pointed finitely presented module .A; a/ in p such that there is no morphism .C; c/ ˚ .C 0 ; c 0 / ! .M; m/ where .C; c/ and .C 0 ; c 0 / are the pushouts shown. a / a /A R@ A RB BB 0 @@ B c @@ BcB b b0 @@ BB / / C0 0 B C B That is, p is irreducible iff given .B; b/; .B 0 ; b 0 / … p there is .A; a/ 2 p such that ..A; a/ ^ .B; b// C ..A; a/ ^ .B 0 ; b 0 // … p. This criterion, which is an expression of the fact that the endomorphism ring of an indecomposable pure-injective is local, is extremely useful both in proving general results and in computations. We will see, in Section 5, that there is a derived criterion for detecting when the hull H.p/ of a pp-type contains no indecomposable summand, that is, is superdecomposable.

4 Dimensions on lattices There is a general process which assigns ordinal-valued dimensions to modular lattices modulo a specified class L of modular lattices which is closed under sublattices and quotient lattices. Two choices of L will be of interest to us. (Our convention will be that the requirement of having a top and a bottom element is part of our definition of modular lattice.) The first takes L to consist of the trivial, one-point, lattice together with the simple, two-point, lattice. The second takes L to be the class of all chains (totally ordered sets). The basic process is to take a lattice L, then collapse all intervals in L which are in L. More precisely, we define a congruence on L by setting a b if there is a chain a C b D a0 a1 an D a ^ b with each interval Œai ; aiC1 in L. We then form the quotient lattice L= which consists of the -equivalence classes. The quotient L= is again a modular lattice so we may apply the same process to L= . We continue this inductively, transfinitely (at limits we take the quotient of the original lattice L by the union of the pullbacks to L of the congruences on its successive quotients). Two things can happen. At some ordinal (necessarily of the form ˛ C 1 for some ordinal ˛) the top and bottom elements are identified and we obtain the trivial lattice, so we stop and set the L-dimension of L to be ˛. Or we reach a quotient of L which has no nontrivial interval in L, that is, in which the congruence is the identity relation, so the process stops but not at the trivial lattice. In that case we set the L-dimension of L to be 1 and say that the L-dimension of L is undefined (or that L does not have L-dimension). Let us look at the two special cases mentioned (the general treatment can be found in [31, §10.2] or [37, §7.1]). 4.1 m-dimension/Krull–Gabriel dimension. In the case where L consists of the trivial lattice and the two-point lattice, the congruence is that which identifies a and

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b iff the interval Œa C b; a ^ b has finite length; the resulting dimension is termed m-dimension. It was introduced (though not in this way) in [67, p. 191] and is an ambidextrous version of the dimension of Gabriel and Rentschler ([55]). These dimensions – m-dimension and Krull dimension in the sense of noncommutative ring theory (see [15]) – coexist: one is defined iff the other is, but m-dimension grows more slowly and also has the useful property that its value for a lattice is equal to its value for the opposite lattice. Proposition 4.1 (see, e.g., [37], 7.2.3). The following are equivalent for any modular lattice L: (i) the m-dimension of L is defined; (ii) the Krull dimension of L is defined; (iii) L contains no subset (with more than one point) which is densely ordered. Garavaglia had already proved that if the Krull dimension of the lattice of pp formulas is defined then there are no superdecomposable pure-injectives so we have the following (for the lattice of pp formulas/pointed finitely presented modules relativised to a definable subcategory see 2.2). Theorem 4.2 ([12], Theorem 1, see [37], 7.3.6). Suppose that D is a definable subcategory of Mod-R. If the lattice of pp formulas for D (in one free variable) has m-dimension then there is no superdecomposable pure-injective in D. It follows by elementary duality (Section 2.3) that the m-dimension for right modules equals that for left modules, so we may further conclude that there is no superdecomposable pure-injective left module in the elementary dual definable category (see, e.g., [37, §3.4.2]) of D which, in the case D D Mod-R, is R-Mod. Corollary 4.3. If M is an R-module and if the lattice of pp-definable subgroups of M has m-dimension then the definable subcategory of Mod-R generated by M contains no superdecomposable pure-injective. These results also may be phrased in terms of Krull–Gabriel dimension. This dimension, introduced in [13], see also [23, p. 197ff.], is defined on the functor category associated to D, which is just the localisation of the functor category .mod-R; Ab/ for Mod-R at the functors which are 0 on every member of D. This is defined like Gabriel dimension (see [16]), which inductively (and transfinitely if necessary) localises at the torsion class generated by the simple objects, except that at each stage it uses only the finitely presented simple objects in the current category to generate the torsion class11 We saw in 2.3 that the lattice of pp formulas is isomorphic to the lattice of pointed modules so, of course, the results above all can be said in terms of these. That was done in [35], using the notion of a factorisable system of morphisms in mod-R, that is, a 11 In fact, for locally coherent categories such as these, Gabriel dimension is defined iff Krull–Gabriel dimension is defined, [5, 5.1] (see [37, 13.2.9]).

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collection of pointed finitely presented modules .Aq ; aq /q2Q indexed by the rationals (or the rationals in Œ0; 1, as one prefers) such that .Aq ; aq / > .Ap ; ap / iff q < p (the (optional) reversal of ordering is to fit with the direction of morphisms). Corollary 4.4. If there is no factorisable system of pointed modules in mod-R then there is no superdecomposable pure-injective R-module. For more details, see that paper or [37, §7.2.2]. The implications in the corollaries above are definitely not reversible: if R is a nearly simple uniserial domain then its m-dimension is undefined but there is no superdecomposable pure-injective R-module ([46, 4.1, 6.3]). The above dimensions being defined is reflected to some extent in the category of pure-injectives in the sense that, over a countable ring, the m-dimension is defined iff there are only countably many indecomposable pure-injectives; whereas if it is undefined then there are continuum many ([67, 8.1, 8.4]). Over rings of arbitrary cardinality, if m-dimension is defined then so is the Cantor–Bendixson rank of the Ziegler spectrum; conjecturally this is an equivalence, and it has been proved under various additional assumptions (see [37, 5.3.60]). 4.2 Width/breadth. If we take L to be the class of chains then the corresponding Ldimension is termed breadth. This terminology was introduced in [31, §10.2] (where defining such dimensions by successive collapsing was introduced) in order to distinguish it from width which was defined in [67, p. 183] (more in the style of the usual definition of noncommutative Krull dimension). These coexist in the sense that if one of breadth or width is defined then so is the other ([31, 10.7]). In practice the term width is most used (but it has no connection with width in the sense of the size of a maximal antichain!). We write w.L/ D 1 if the width of L is undefined. Of course, if the width of L is undefined then so is the m-dimension of L. For width/breadth there is an analogue to 4.1. Here we will say that a lattice L is wide12 if, given any pair a > b in L there are incomparable c; d 2 L between a and b. In such a lattice the L-congruence is just equality, so it does not have width; indeed this is the obstruction to having width. Proposition 4.5 (see [37], 7.3.1). The following are equivalent for a modular lattice L: (i) the width (equivalently breadth) of L is defined; (ii) L has no wide (sub)quotient; (iii) L contains no wide subposet.13 Theorem 4.6 ([67], 7.1). Let D be a definable subcategory of Mod-R; if there is a superdecomposable pure-injective in D then the width of the lattice pp.D/ is undefined. If this lattice is countable, in particular if R is countable, then the converse is true. 12

Sometimes the term wide is used for any lattice which has width undefined. The definition of a poset, as opposed to lattice, being wide is slightly more complicated but we don’t need it here and I have added this equivalent only for completeness. 13

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It is unknown whether the converse of 4.6 is true in general, and that is currently a significant obstruction to proving existence of superdecomposable pure-injectives: one might be able to prove that the width of the lattice of pp formulas/pointed finitely presented modules is undefined but, to go beyond that to existence of superdecomposable pure-injectives, either one has to assume countability or find a direct construction (as Puninski has done, as discussed in Section 5.2). On the other hand, if one’s main concern is with finite-dimensional modules then showing that width is undefined is already showing that morphisms in the category of finite-dimensional modules contain a complex, continually branching, structure. For this dimension one could write down a definition analogous to that of factorisable system of pointed modules, just replacing reference to pp formulas by reference to pointed modules. A useable sufficient condition is described just before 5.6. As with m-dimension, by elementary duality, the width for right modules equals that for left modules. Corollary 4.7. If the lattice of pp formulas/pointed finitely presented modules for right R-modules has width D 1 then the same is true of the lattice for left modules. In particular, if R is a countable ring then there is a superdecomposable pureinjective right R-module iff there is a superdecomposable pure-injective left R-module. We give some idea of the proof of 4.6. First we consider the direction that needs no countability hypothesis. Recall that pp.N / denotes the lattice of pointed finitely presented modules modulo N (equivalently, as the notation indicates, the lattice of pp-definable subgroups of N ). Theorem 4.8 ([67], §7). Suppose that N is a pure-injective module and that .A; a/ ! .B; b/ is a pair of pointed finitely presented modules which is open on N , that is tr.A; a/.N / ‰ tr.B; b/.N /. Suppose that the interval Œ.A; a/; .B; b/ is totally ordered by the relation N . Then there is an indecomposable summand N0 of N (on which the pair .A; a/ .B; b/ is open). Proof. Our assumption is that if .A; a/ .C; c/ .B; b/ and .A; a/ .D; d / .B; b/, with C , D finitely presented then either tr.C; c/.N / tr.D; d /.N / or vice versa. Since the pair .A; a/ .B; b/ is open on N we can choose and fix some m 2 tr.A; a/.N / n tr.B; b/.N /. We use this element to define a cut on the interval with upper part U D f.C; c/ W .A; a/ .C; c/ .B; b/ and m 2 tr.C; c/.N /g and lower part L D f.C; c/ W .A; a/ .C; c/ .B; b/ and m … tr.C; c/.N /g. We then use Zorn’s lemma to extend U maximally to a pp-type p which contains no pointed module in L. It is then a standard use of Ziegler’s criterion, 3.9, to deduce that p is irreducible and so H.p/ is an indecomposable pure-injective. One does need to do more work, for which see the references, to show that a copy of H.p/ is a direct summand of N but what we have said here shows how we use the fact that some interval is a chain.

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Corollary 4.9 ([67], 7.8). If N is a superdecomposable pure-injective module then the lattice of pp-definable subgroups of N is wide. Proof. Otherwise there would be a non-trivial interval in that lattice which is a chain but then, by the result above, N would have an indecomposable direct summand, contradicting superdecomposability. It follows (by the comments in the last paragraph of Section 2.2) that if D is any definable category containing N then the width of pp.D/ is undefined. The current proof of the other direction of 4.6 requires countability and is quite technical. The original proof is [67, 7.8(2)] and is in terms of pp formulas. The proof was slightly reformulated in [31, 10.13] and then that version was cast into the form of an argument in the functor category in [37, §12.6] using the translation which replaces pure-injective modules by injective functors (cf. 3.1), pp formulas by finitely presented functors etc. but it becomes no less technical in the functor category, so I just describe the shape of that proof now. Suppose then that R is countable. Then there are just countably many finitely presented modules (up to isomorphism), each of which has just countably many elements. Therefore there are only countably many pointed finitely presented modules, so they can be listed using the natural numbers as index set: .A0 ; a0 /; .A1 ; aa /; : : : ; .An ; an /; : : : . We will construct a pp-type p by working through that list, deciding at each stage whether to put .An ; an / into p or whether to exclude it (permanently). The order is entirely immaterial; all we need is that each pointed finitely presented module occurs at some stage and that, at that stage, we have already had to consider only finitely many other pointed modules. In the limit we will have dealt with every pointed finitely presented module. We need to ensure that (1) the resulting set p of “included” pointed finitely presented modules is a pp-type and (2) no pointed finitely presented module is large for p (see Section 5 for the definition of “large”). For instance if, when we come to consider .An ; an /, it is the case that .An ; an / .Ai ; ai / for some i < n which has been put into p, then .An ; an / also must go into p. That shape of argument is used elsewhere in model theory (of modules and in general). In this case we also develop, through the stages, a set of non-trivial intervals in the lattice of pointed finitely presented modules, each of which relativised to N , has width 1 (by assumption we can start with the interval Œ.R; 1/; .0; 0/). At the stage n, when we consider .An ; an /, we take each of our current finitely many intervals: let Œ.Ck ; ck /; .Ck0 ; ck0 / be one of them; if the width of Œ.An ; an / C .Ck ; ck /; .An ; an / C .Ck0 ; ck0 /, relativised to N , still is undefined then this will go into our set of intervals to be used at stage n C 1 and .An ; an / will be put into p. Otherwise we exclude .An ; an / from p and we choose two incomparable intervals within Œ.Ck ; ck /; .Ck0 ; ck0 /, each still with undefined width, and put each of these into our set of intervals to be used at stage n C 1 (this is part of making sure that the excluded .An ; an / is not large for p). Then

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one shows that the conditions (1) and (2) above are indeed satisfied. (The obstacle to extending this argument to the uncountable case, by using transfinite induction, is in trying to carry the set of intervals through any limit stage.)

5 Superdecomposable pure-injectives Sometimes superdecomposable (pure-)injective modules can be produced directly: injective hulls of modules without uniform submodules are superdecomposable and these can be moved, using functors which commute with direct products and direct sums, hence preserve pure-injectivity, and are full, hence preserve superdecomposability, to produce superdecomposable pure-injectives in other categories. This method is used in [23, p. 211ff.] and shows, for example, that strictly wild algebras have superdecomposable pure-injectives. Beyond that, our main method of obtaining superdecomposable pure-injectives is to produce a pp-type p, the hull of which has no indecomposable direct summand. As in 3.9 this can be detected just from the lattice structure of pointed finitely presented modules. Say that a pointed finitely presented module .B0 ; b0 / is large for14 p if it is not in p and the condition of 3.9 holds above .B0 ; b0 /; that is, if for every .B; b/; .B 0 ; b 0 / … p with .B; b/; .B 0 ; b 0 / .B0 ; b0 / there is .A; a/ 2 p such that ..A; a/ ^ .B; b// C ..A; a/ ^ .B 0 ; b 0 // … p. Theorem 5.1 ([67], 7.6). A pp-type has superdecomposable hull iff it has no large pointed finitely presented module. As one would expect, if there is a pointed finitely presented module .B0 ; b0 / which is large for p then one can say something about the relationship between this and a corresponding indecomposable direct summand of H.p/ (see e.g. [37, 4.3.79]) but we don’t need that detail here. If the lattice pp has width 1 and is countable then, as discussed in the previous section, there is an argument of Ziegler which produces a pp-type with no large formula. Of course, one has to have a method for showing that the lattice pp has width 1. It is also desirable to avoid having to use the countability hypothesis. For example, Puninski characterised those commutative valuation rings which have superdecomposable pureinjective modules; for domains this is as follows. Theorem 5.2 ([41], 4.2, also see [44], 12.12). Let R be a commutative valuation domain. Then there is a superdecomposable pure-injective R-module iff the Krull dimension of R is 1 iff the value group of R contains a densely ordered subset. In this case, a densely ordered chain of (right) ideals gives rise to a similarly-ordered chain of divisibility pp formulas and, in the opposite direction, a densely ordered chain 14 The usual terminology is “large in” p but that doesn’t sit very well with the fact that the pointed module is definitely not in p; the usual terminology makes more sense in the original context where a distinction is made between types and pp-types.

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of annihilation pp formulas. It turns out that if two such chains freely generate a modular (indeed, distributive) lattice within the lattice ppR then that lattice with width D 1. This underlying lattice theory has been developed into a method which we describe in Section 5.2. This method has proved to be effective in a number of contexts, both in showing that the lattice of pointed finitely presented modules does not have width and, with a stronger hypothesis, in directly producing pp-types without large pointed modules, hence superdecomposable pure-injectives, in the absence of any countability hypothesis. 5.1 Existence of superdecomposable pure-injectives. Here we summarise much of what is known about existence of superdecomposable pure-injectives over various kinds of ring. When the results apply to more general definable subcategories we state them in that way but one may always take the definable subcategory to be Mod-R itself. • If we ask about the existence of superdecomposable, not necessarily pure-injective, modules then these exist even among abelian groups (see [14, p. 606]). This reflects the fact that the general infinitely-generated representation theory of most rings is wild (see e.g. [60]). • Rings of finite representation type, and the possibly more general pure-semisimple rings, have no superdecomposable pure-injectives, although every module is pure-injective (for a proof and references, see, e.g. [37, §4.5.1]). • Tame hereditary Artin algebras have no superdecomposable pure-injectives (this follows from [13] and 4.2 see also [36], [59]). • It is conjectured that all domestic string algebras have Krull–Gabriel dimension (for which see Section 4.1) and hence (4.3) have no superdecomposable pureinjectives. • There are results in [43], [42] for differential polynomial rings over fields of characteristic 0, somewhat analogous to those for hereditary finite-dimensional algebras, in particular implying that there are no superdecomposable pure-injectives for “tame” such rings. • Non-domestic string algebras have width undefined hence, if countable, have superdecomposable pure-injectives ([49, 3.2]). The countability hypothesis is conjecturally unnecessary and has been bypassed in some cases ([51]). • Tubular algebras have width undefined hence, if countable, superdecomposable pure-injectives, indeed, for each positive irrational there is a superdecomposable pure-injective with that slope (see Section 6). • Tame strongly simply connected algebras of non-polynomial growth, over a field of characteristic ¤ 2 have width undefined, hence superdecomposable pureinjectives if countable ([25]). • The free associative algebra KhX; Y i in two generators has superdecomposable injective hull (since RR has no uniform submodule).

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• Strictly wild algebras have superdecomposable pure-injectives (the image of the injective hull of KhX; Y i will be pure-injective and, by fullness of the representation embedding, superdecomposable). • PI Dedekind domains, in particular Z and KŒX , more generally, PI hereditary noetherian prime rings, have no superdecomposable pure-injectives (the commutative case goes back to Kaplansky [24], the general PI case is, noting 4.3, [36, 1.6, 3.3]). • The polynomial ring KŒX; Y has superdecomposable pure-injectives, as does the power series ring KŒŒX; Y and any regular local ring of dimension 2 ([23, p. 218]). • For many generalised Weyl algebras (in the sense of [3]), including the first Weyl algebra A1 and its localisation B1 , the quantum Weyl algebra Aq (q ¤ 0; 1) and the universal enveloping algebra Usl2 , there is a wide poset of pointed finitely presented modules, hence if the field is countable, a superdecomposable pureinjective ([40, 7.5, §6], using a construction from [26]). • If G is a finite non-trivial group then there is a superdecomposable ZG-module ([52]). • If R is the pullback (in the sense of [30]) of two commutative Dedekind domains, neither of which is a field, then the width of ppR is undefined so, if the Dedekind domains are countable, there is a superdecomposable pure-injective R-module ([52, 5.8]). • If R is von Neumann regular then there is a superdecomposable pure-injective R-module iff R is not semiartinian ([65, §1]); in the case that R is commutative this is iff the boolean algebra of idempotents of R is superatomic ([31, 16.26]). • Over serial rings, the model theory of modules and the structure of pure-injectives (as well as pure-projectives, that is, direct summands of direct sums of finitely presented modules) has been extensively studied by Puninski in a series of papers, see especially [45], [46] and the book [44]. If R is a serial ring Puninski gives a criterion, purely in terms of the structure of the lattices of right and left ideals of R, equivalent to existence of a superdecomposable pure-injective ([50, 5.2]). The criterion is right/left symmetric, so it follows that there is a superdecomposable pure-injective right module iff there is such a left module [50, 6.1]. It is also the case, [50, 6.2], that if the lattice of two-sided ideals of a serial ring R has Krull dimension (in the Gabriel–Rentschler sense) then there is no superdecomposable pure-injective R-module. Furthermore, [50, 6.4], in the case of serial rings the countability restriction in 4.6 can be dropped: width being undefined is equivalent to existence of a superdecomposable pure-injective. • Those group rings KG where K is a field and G is a finite group such that the category of KG-modules does not have width are almost completely characterised in [53], the only unresolved case being that where G is the quaternion group

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and K has characteristic 2 and does not contain a primitive cube root of 1 (the conjecture is that the width is undefined in this case). 5.2 Lattices freely generated by chains. Suppose that L1 and L2 are chains with top and bottom. Denote by L1 ˝ L2 the modular lattice freely generated by L1 and L2 subject to identifying their respective tops and bottoms. This is a distributive lattice (see [17, Theorem 13]) with a canonical form (indeed, two canonical forms) for its elements: each element of L1 ˝ L2 has a unique representation of the form .an ^ b1 / C .an1 ^ b2 / C C .a1 ^ bn / with a1 < < an in L1 and b1 < < bn in L2 ; by distributivity this is equal to an ^ .b1 C an1 / ^ ^ .bn1 C a1 / ^ bn and that is the second canonical form. Lemma 5.3 ([52], p. 61). If a1 < a2 in the chain L1 and b1 < b2 in the chain L2 then, in L1 ˝ L2 , the elements a1 C b1 and a2 ^ b2 are not comparable. Proof. The meet of these two elements is .a1 Cb1 /^a2 ^b2 D b2 ^.a1 Cb1 /^a2 which is a canonical form of the second type and different from the (second-type) canonical form b2 ^ a2 , hence these elements are not equal and so a2 ^ b2 — .a1 C b1 / ^ a2 ^ b2 . Also, by distributivity, .a1 C b1 / ^ a2 ^ b2 D .a1 ^ b2 / C .b1 ^ a2 / which is in first canonical form and different from a1 C b1 , which is also in first canonical form. So we deduce that a1 C b1 — .a1 C b1 / ^ a2 ^ b2 and the claim is established. It is useful to have a description of the ordering in L1 ˝ L2 , which is as follows (see [51, p. 708]). To check whether two elements of this lattice satisfy l Pl 0 , write the first as a sum of meets (for example, using theVfirst canonical form) l D i .ai ^ bi /, and write the second as a meet of sums l 0 D j .aj0 C bj0 / (for example, using the second canonical form). Then l l 0 iff for each i and j we have ai ^ bi aj0 C bj0 and that, by the lemma above, is the case iff ai aj0 or bi bj0 . With this clear description of L1 ˝ L2 , Puninski proves the following [49, 3.1, 3.2] (he also gives a visual version of the proof, using a graphical representation for the elements of L1 ˝ L2 ). Proposition 5.4. Suppose that L1 and L2 are chains, each containing a densely ordered subset. Then L1 ˝ L2 has width 1. In fact, the result proved in [49] is more general, being an estimate of the mdimension (rather, a slightly modified version of that) of L1 ˝ L2 in terms of the m-dimensions of L1 and L2 . To use this in practice, one must be able to come up with two candidate chains of pointed finitely presented modules/pp formulas for L1 and L2 and then show that the lattice they generate within ppR is freely generated by them. Densely ordered chains, for many types of algebra, are not hard to find (see [35]). For instance in [52] the modules are string modules over a nondomestic string algebra so one may fix a string, extend it to the left to get a densely ordered chain L1 , alternatively extend it to the right to get L2 . But how do we show that these chains really are independent? The following lemma is the key.

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Lemma 5.5 ([52], 5.4). If L1 and L2 are chains then a factor L0 of the lattice L1 ˝ L2 is a proper factor iff there are a1 < a2 2 L1 and b1 < b2 2 L2 such that the images of the elements a1 C b1 and a2 ^ b2 in L0 are comparable. 0 Proof. One direction is 5.3. For the other, if L0 is a proper factor then there V are0 l aj0 and bi > bj0 for some i , j then we have the desired configuration in L0 . Otherwise, for every pair i , j we have either ai aj0 or bi bj0 and hence ai ^ bi aj0 C bj0 for all i, j , and hence l l 0 , so l and l 0 already were equal, contradiction.

In practice so far, in particular in [53] and [25], the following special case of the notion of a factorisable/densely ordered system of morphisms/pointed modules has been used. The extra requirement is that the modules Aq in the factorisable system should be indecomposable (and that no ap should be 0). To keep the terminology reasonably consistent and brief, let us refer to what we define below as a dense chain of indecomposable pointed modules (the “finitely presented” being understood). The key definition, from [53, 5.4], is that of an independent pair of such dense chains, that is, two dense chains .Ap ; ap /p2Q , .Bs ; bs /s2Q 15 of indecomposable pointed modules such that: • there is no pointed map between any .Ap ; ap / and any .Bs ; bs /; • for all p and s, in the pushout diagram ap / Ap RB BB BBcps bs BB B / Cps Bs

giving the pointed module .Cps ; cps / which is the meet of .Ap ; ap / and .Bs ; bs /, the module Cps is indecomposable; • for fixed p and s, for all q 2 QC , q ¤ p we have .Cqs ; cqs / 6' .Cps ; cps / and for all t 2 Q , t ¤ s, we have .Cpt ; cpt / 6' .Cps ; cps /. Theorem 5.6 ([53], 5.4). Let R be any ring. Suppose that there is an independent pair of dense chains of indecomposable pointed modules. Then the sublattice of ppR generated by these is freely generated by them, hence ppR has width 1. 15 In the source, one set is ordered by the positive rationals and the other inversely ordered by the negative rationals but this, and the orderings I have used here, are just matters of taste or of suitability to a particular situation.

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Proof. To show free generation we check the condition of 5.5 (and then the second statement will follow from 5.4). So suppose that there are p < q and s < t such that .Ap ; ap / ^ .Bs ; bs / .Aq ; aq / C .B t ; b t / (recall, re checking the criterion of 5.5, that our ordering of indices is opposite to that in the lattice of pointed finitely presented modules). Using distribu tivity we have .A ; a / ^ .B ; b / D .A ; a / ^ .B ; b / ^ .A ; a / C .B ; b / D p p s s p p s s q q t t .Ap ; ap / ^ .B s ; bs / ^ .Aq ; aq / C .Ap ; ap / ^ .Bs ; bs / ^ .B t ; b t / D .Bs ; bs / ^ .Aq ; aq / C .Ap ; ap / ^ .B t ; b t / . This last is represented by the direct sum of the pushouts .Cqs ; cqs / and .Cpt ; cpt /, so there is a morphism from that direct sum to .Cps ; cps / with cps D(say) dqs C dpt (in reasonably obvious notation). Since the endomorphism ring of Cps is, by assumption, local, and we have maps from .Cps ; cps / to each of those direct summands, this implies that we have, say, cps D dqs and so, in fact, .Cps ; cps / is equivalent to .Cqs ; cqs / in the ordering on pointed modules. ap

/ Ap

/ Aq

Bs

/ Cps

/ Cqs

Bt

/ Cpt

R bs

That contradicts the incomparability assumption (the last of the bulleted conditions), as required. The methods above get us lattices with width 1, but not superdecomposable pureinjectives directly. There is another version which was introduced by Puninski in [49] which bypasses the countability restriction (cf. 4.6) but assumes that we know an interval essentially completely. Theorem 5.7 ([49], 2.3). Suppose that M is an R-module such that there is an interval in the lattice pp.M / which is freely generated by two chains L1 , L2 each of which contains a dense subchain. Then there is a superdecomposable pure-injective module in the definable subcategory generated by M . Since the hypothesis is self-dual, the dual definable category to that generated by M also contains a superdecomposable pure-injective (left) module. The proof constructs a pp-type with no large pointed finitely presented module, making use of the control given by having a canonical form for all points of that interval in pp.M /.

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6 Superdecomposables over tubular algebras In this long section I will outline Harland’s proof of existence of superdecomposable pure-injectives over tubular algebras. This is given in the first part of his thesis [20] (the second part is about pure-injective modules over string algebras), see also [21]. Some parts of the argument are written making extensive use of the terminology and techniques around pp formulas so what I have done here is to outline the whole proof and give details of those particular parts but phrased in terms of pointed modules. The major terminological replacement is as follows. In the pp-versions of arguments, one takes a pp formula and chooses a pointed module .A ; a / of which it is a free realisation (Section 2.1) but one continues to use the notation .M / for the solution set of in a module M . In order to bias the emphasis and terminology of the proofs towards pointed modules, we can start instead with the pointed module .A; a/ and, rather than go on to introduce the corresponding pp formula , as described in Section 2.1, refer to the trace of .A; a/ in an arbitrary module M . This, recall, is tr.A; a/.M / D fm 2 M W m D f a for some f W A ! M g; as discussed in Section 2.1, this is exactly the pp-definable subgroup .M /. This is really just a translation – I have not made any significant changes to the actual proofs, indeed I follow [21] very closely – but perhaps it will make these arguments more accessible. Throughout this section R is a tubular algebra; these algebras are defined in [57, Chapter 5], also see [64, XIX.3.19]. For simplicity we assume that R is basic, connected and that the field K is algebraically closed. Let S1 ; : : : ; Sn denote the simple modules, so dimension vectors of finite-dimensional modules live in Zn . By ind-R we denote the set of (isomorphism types of) indecomposable finite-dimensional R-modules. We will use the standard bilinear form h; i for R-modules. Tubular algebras have global dimension 2 ([57, 3.1.5]) so the usual formula (see, e.g., [1, III.3.13]) becomes hdim.M /; dim.N /i D dim .M; N / dim Ext 1 .M; N / C dim Ext 2 .M; N /. Indeed, when we use it, either M will have projective dimension 1 or N will have injective dimension 1, so only the Hom and Ext1 terms will be non-zero. The corresponding quadratic form is denoted R .x/ D hx; xi, the radical of R is the subgroup rad.R / D fx W R .x/ D 0g of Zn and its elements are the radical vectors; the roots of R are those x 2 Zn such that R .x/ D 1. There is ([57, §5.1]) a pair h0 ; h1 of canonical linearly independent radical vectors. These generate a subgroup of rad.R / of finite index. Define the slope of A 2 ind-R to be the ratio hh0 ; dim.A/i I .dim.A// D hh1 ; dim.A/i then, if A is in neither P0 nor Q1 , we have .dim.A// D q iff A 2 Tq (notation as just below). The basic structure theorem for mod-R fibres ind-R over the values taken by slope. Theorem S 6.1 ([57], §5.2, ([57], 3.1.5). Let R be a tubular algebra; then ind-R D P0 [ fTq W q 2 Q1 0 g [ Q1 (disjoint union) where each Tq is a tubular family

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S S separating Pq D P0 [ fTq 0 W q 0 < qg from Qq D fTq 0 W q < q 0 g [ Q1 . S Every module in fTq W q 2 QC g has both injective and projective dimension 1. Note that the finite-dimensional modules A of slope q, meaning the modules in add.Tq /, are characterised by the conditions .A; Pq / D 0 D .Qq ; A/. More generally, we say that the slope of any module M 2 Mod-R is r 2 RC if .M; Pr / D 0 D .Qr ; M / where Pr and Qr are defined as in 6.1 but with r in place of the rational value q. By the Auslander–Reiten formula (see, e.g., [1, IV.2.15]) and the fact that Pr and Qr , being unions of Auslander–Reiten components, are closed under ˙1 , this is equivalent to Ext1 .Pr ; M / D 0 D .Qr ; M /. Now, if B is finitely presented and C is FP2 (i.e., has a projective presentation the first three terms of which are finitely generated) then each of the conditions .B; M / D 0 and Ext1 .C; M / D 0 on M is expressible in the form .M / D .M / for certain pp formulas , ([33, p. 211–12], see [37, 10.2.35]). Therefore (see, e.g., [37, §3.4.1]) the modules which satisfy any collection of such conditions form a definable subcategory of Mod-R. In particular we will denote by Dr the definable category consisting of all modules of slope r 2 RC . Note that if r is irrational then Dr contains no finitedimensional module apart from 0. By 6.1 every finite-dimensional module is a direct sum of modules with slope, but what about the infinitely generated modules? For indecomposables the answer is surprisingly strong (and implies, by [67, 6.9], that every module over a tubular algebra is elementarily equivalent to a direct sum of modules with slope). Theorem 6.2 ([54], 13.1). Every indecomposable module over a tubular algebra has a slope. So what lives at irrationals r? There’s something there in Dr , apart from 0, because the observations above, combined with an application of the Compactness Theorem from model theory. The argument is, in brief, as follows. If all the conditions .B; M / D 0 with B of slope < r and Ext1 .C; M / D 0 with C of slope > r implied M D 0 then, by the Compactness Theorem, some finite number of them would imply M D 0 but these finitely many conditions would involve only finitely many modules B, C , so then we just appeal to 6.1 to find a non-zero finitedimensional module satisfying these finitely many Hom and Ext conditions – giving a contradiction. So now we know that the modules of positive irrational slope r form a nonzero definable subcategory of Mod-R and it follows that there are indecomposable pureinjectives of slope r. This much, but essentially nothing else about Dr , had been known for a long time. Harland showed that the width of each definable category Dr for r positive irrational is 1 hence, if K is countable, there is a superdecomposable pure-injective in each Dr . This is achieved by showing that the lattice of pp-formulas for Dr is wide (in fact, in the strong sense of the term as used in Section 4.2).

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Before outlining the proof we mention that if q is a positive rational then the definable category Dq generated by the finite-dimensional indecomposables of slope q is well known (see, e.g., [20, §3.5]), with structure essentially like that seen in the tame hereditary case ([36], [59], also see [27]). Namely, the infinite-dimensional indecomposable pure-injectives of slope q are of adic and Prüfer types, parametrised by the quasisimple modules of slope q, plus a generic module. There are, moreover, no more generic R-modules apart from these ([29]), in particular none of irrational slope. Fix a module M and recall, from Section 2.2, the relative ordering .A; a/ M .B; b/ on pointed finitely presented modules. Note that, if every indecomposable summand of A and B has slope strictly less than that of every indecomposable summand of C in the pushout shown, then, if M is any module with slope strictly between those of summands of A, B and those of summands of C , it must be that tr.A; a/.M / \ tr.B; b/.M / D 0. a

/A R@ @@ c @@ b @@ ˇ B ˛ /C

For any non-zero element in the intersection of the traces would induce a non-zero morphism from C to M . We will say that a finite-dimensional module X is in an interval of R1 0 if each of its indecomposable summands has slope in that interval. Here is a key technical result from [20]. It shows that each pointed finitely presented module has (indeed, the proof produces) a “good” representative in a neighbourhood of any specified irrational. Theorem 6.3 ([20], Proposition 2, p. 79). Let .A; a/ be a pointed finitely presented module and let r be a positive irrational. Then there is a pointed finitely presented module .A0 ; a0 / with .A; a/ .A0 ; a0 / and there is > 0 such that: • • • •

A0 2 add.Pr /; coker.a0 / 2 add.QrC /; .A; a/ X .A0 ; a0 /, that is, tr.A; a/.X / D tr.A0 ; a0 /.X /, for all X in .r ; r C/; the map f 7! f a0 induces a bijection .A0 ; X / ' tr.A0 ; a0 /.X / for all X in .r ; r C /.

In fact, the conclusion also holds for infinite-dimensional (indecomposable) modules in place of X. The proof given in [20] and [21] already is in terms of pointed modules, so I don’t repeat or explicate it here. Using this, the following result, extending the above to pairs (“pp-pairs” in the model-theoretic terminology) and extracting the information about dimensions is derived. Corollary 6.4 ([20], Proposition 3, p. 82). Let .A; a/ .B; b/ be a pair of pointed finitely presented modules and let r be a positive irrational. Then there is > 0 and

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a vector v 2 Zn such that dim.tr.A; a/.X /=tr.B; b/.X // D v dim.X / for all X in .r ; r C /. Say that a pair .A; a/ .B; b/ of pointed finitely presented modules is closed on a module M if tr.A; a/.M / D tr.B; b/.M /, otherwise the pair is open on M . Say that this pair is closed near the left of r if there is > 0 such that .A; a/ .B; b/ is closed on every indecomposable X in .r ; r/; otherwise say that .A; a/ .B; b/ is open near the left of r. The latter says only that .A; a/ .B; b/ is open on “cofinally many” modules near to, and to the left of, r but it is proved eventually (6.5 then 6.7) that, in this case, .A; a/ .B; b/ is open on every module in some interval .r 0 ; r/. Similarly say that .A; a/ .B; b/ is closed near the right of r if there is > 0 such that .A; a/ .B; b/ is closed on every indecomposable X in .r; r C /; otherwise say that .A; a/ .B; b/ is open near the right of r. Corollary 6.5 ([20], Corollary 9, p. 81, Theorem 30, p. 83). Let .A; a/ .B; b/ be a pair of pointed finitely presented modules and let r be a positive irrational. If .A; a/ .B; b/ is open near the left of r then there is > 0 such that .A; a/ .B; b/ is open on every module in .r ; r/ which lies in a homogeneous tube. Similarly, if .A; a/ .B; b/ is open near the right of r then there is > 0 such that .A; a/ .B; b/ is open on every module in .r; r C / which lies in a homogeneous tube. Proof. Apply 6.3 to obtain pointed finitely presented modules .A0 ; a0 / .A; a/ and .B 0 ; b 0 / .B; b/, and 1 ; 2 satisfying the conclusions of that result. Set 0 D min.1 ; 2 /. Suppose that there is 2 .r 0 ; r/ \ Q and a module EŒk in a homogeneous tube T . / in T such that .A; a/ .B; b/ is closed on EŒk. Here, if E is a quasisimple module (that is, a module at the mouth of a tube) then we will denote by EŒk the module in the same tube which has quasisimple length k and quasisimple socle E. We shall prove that .A; a/ .B; b/ must be closed near the left of r. We have tr.A0 ; a0 /.EŒk/ D tr.A; a/.EŒk/ D tr.B; b/.EŒk/ D tr.B 0 ; b 0 /.EŒk/ and so, by 6.3, dim.A0 ; EŒk/ D dim.B 0 ; EŒk/. By considering almost split sequences in T . / and induction, it is easy to check that for all positive integers m, dim.A0 ; EŒm/ D dim.B 0 ; EŒm/. That is .A0 ; a0 / .B 0 ; b 0 / is closed on every module in T . / and hence on every module in add.T . //. Now, given X in . ; r/ and any f 2 .A; X / there is, by 6.3, f 0 2 .A0 ; X / with f 0 a0 D f a. By 6.1, f 0 factors through a module Y 2 add.T . // (this being part of the definition of a separating tubular family). f0

A0

9h 9g

Y

/X ?

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Since .A0 ; a0 / .B 0 ; b 0 / is closed on Y it follows that g.a0 / 2 tr.B 0 ; b 0 /.Y / and so, composing with h, f a D f 0 a0 2 tr.B 0 ; b 0 /.X / D tr.B; b/.X /. Thus .A; a/ .B; b/ is closed on every module in . ; r/ – as required (take D r ). The proof of the second statement is similar. The next result shows how the lattice of pp formulas/pointed finitely presented modules at an irrational r is completely determined by and reflected in what happens near r. By saying that a pair .A; a/ .B; b/ is open at r we mean that it is open on some module of slope r 16 . Corollary 6.6. Let .A; a/ .B; b/ be a pair of pointed finitely presented modules and let r be a positive irrational. Then the following are equivalent: (i) .A; a/ .B; b/ is open near the left of r; (ii) .A; a/ .B; b/ is open near the right of r; (iii) .A; a/ .B; b/ is open at r. Proof. If we have (i) then we can use a compactness argument, very similar to that used earlier, to get (iii). Namely, given finitely many of the (Hom and Ext) conditions cutting out the subcategory Dr , there is > 0 such that every indecomposable finitedimensional module with slope in .r ; r/ satisfies them. By assumption there is such a module on which .A; a/ .B; b/ is open and that can be expressed by a suitable sentence of the formal language for R-modules. Thus the conditions cutting out Dr are finitely consistent with the condition that .A; a/ .B; b/ is open so, by the Compactness Theorem, there is a module which satisfies all these conditions, in particular is of slope r, as required. This argument also shows (ii) ) (iii)17 . For the converse, suppose we have (iii), say M is a module of slope r on which .A; a/ .B; b/ is open, say m 2 tr.A; a/.M / n tr.B; b/.M /. Choose f W A ! M such that m D f a. Given > 0, M is, by [54, Lemma 11], the directed union of its finite-dimensional submodules in .r ; r/, so there is a finite-dimensional submodule X of M which contains fA and is in .r ; r/. Then certainly m 2 tr.A; a/.X / and, since m … tr.B; b/.M / it must be that m … tr.B; b/.X /. Therefore .A; a/ .B; b/ is 16 Here we follow [21] in saying that a pair is closed at r if it is not open at r; this is not the relation r in [20] which is defined in terms of neighbourhoods of r. 17 Appeals to the Compactness Theorem can be replaced by using the (algebraic) ultraproduct construction but this is usually more work. To indicate how this might go in this case: we would take a sequence, .qi /i , of rationals approaching r from below and for each of these choose a module Xi with slope in .qi ; r/ such that there is fi W A ! Xi such that there is no g W B ! Xi with gb D fi a. By assumption we can Q do this. Then we would choose an ultrafilter U on the index set fi gi and form the ultraproduct M D i Xi =U. The morphisms fi together induce a morphism f W A ! M which is such that there is no g W B ! M with gb D f a – because this equality would imply equality on some components in the product – indeed on a set of indices in U. So .A; a/ .B; b/ is open on M . But we have yet to show that M is, or can be taken to be, of slope r. One way would be to replace M by an elementarily equivalent direct sum of indecomposable pure-injectives and show that any of slope smaller than r are irrelevant, but again the obvious route appeals to model theory!

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open on X and hence is open on some indecomposable summand of X , and we have proved (i). Finally, assume (iii) and, in order to prove (ii), continue with notations and assumptions as in the previous paragraph. Again using representation of M as a directed union of submodules, there is a submodule Z of M into which X embeds, such that every indecomposable summand of Z has slope greater than the maximum slope of any indecomposable summand of X . By 6.1 the embedding X ! Z factors through a module X 0 whose indecomposable summands all lie in some homogeneous tube of slope, say, between that of any direct factor of X and r. Note that if m0 denotes the image in X 0 of m regarded as an element of X, then m0 2 tr.A; a/.X 0 /ntr.B; b/.X 0 /. If the vector v is chosen for .A; a/ .B; b/ as in 6.4 then we therefore have v:dim.X 0 / > 0. From the definition of slope, it follows that the dimension vector of X 0 has the form c.h0 C h1 / where is the slope of X 0 and c is a positive integer. Therefore v:.h0 C h1 / > 0. Repeating this whole argument (but noting that v can be taken to be fixed), we produce an increasing sequence of rationals (the various ) with limit r, each satisfying v:h0 C v:h1 > 0. Since r is irrational it cannot be that v:h0 C rv:h1 D 0, so v:h0 C rv:h1 > 0 and hence there is a rational 0 > r, which we may take in the interval .r; r C 00 / for any given 00 > 0, with v:h0 C 0 v:h1 > 0. By 6.4, .A; a/ .B; b/ is open on the homogeneous modules of slope 0 . Thus, .A; a/ .B; b/ is open near the right of r. We need to remove the restriction in 6.5 to modules in homogeneous tubes. This is done for certain tubular algebras; the consequences, in particular undefined width for the relativised pp/pointed-module lattice (Section 2.2) at irrational r, are derived over these algebras and then extended by tilting functors to arbitrary tubular algebras. Here are some, but by no means most, of the details. The particular algebras which are dealt with are those appearing in [57, §5.6] and the result proved is the following. Corollary 6.7 ([20], Lemma 63, p. 102). Let R be one of the algebras C.4; /, C.6/, C.7/, C.8/ from [57, §5.6]. Let .A; a/ .B; b/ be a pair of pointed finitely presented R-modules and let r be a positive irrational. If .A; a/ .B; b/ is open at r then there exists > 0 such that .A; a/ .B; b/ is open on every module in .r ; r C /. The proof uses 6.4 to relate what happens on homogeneous tubes to modules in inhomogeneous tubes, computing relevant estimates on slopes and dimensions for these particular algebras. Further computations (none of this part uses pp formulas or model theory) yield the following key result. Theorem 6.8 ([20], Lemma 66, p. 106). Let R be one of the algebras C.4; /, C.6/, C.7/ or C.8/. Given any positive irrational r, any > 0 and any d 1, there exists an inhomogeneous tube T . / of rank hh0 ; h1 i and with slope in .r ; r/ such that, if E is a quasisimple of T . / and X is an indecomposable finite-dimensional module in . ; r/, then dim.X / dim.E/ C d . In particular, any nonzero morphism from E to a module in . ; r/ is an embedding.

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Now we have the ingredients we need for the proof that over these algebras width at an irrational is undefined. Let M.r/ be any module which generates the definable category Dr , so we have the lattice pp.M.r// of pointed finitely presented modules at r. (We remark that, by 6.6, this can be defined in terms of modules with slope near r, hence purely in terms of mod-R.) Theorem 6.9 ([20], Theorem 31, p. 107). Let R be one of the algebras C.4; /, C.6/, C.7/ or C.8/. For each positive irrational r the lattice pp= r of pointed finitely presented modules relativised to Dr has width 1. Indeed, every non-trivial interval in this lattice contains incomparable elements. Proof. Take any pp-pair .A; a/ .B; b/ which is open at r, hence open in a neighbourhood of r. By 6.3 we may replace each of these pointed finitely presented modules by one to which it is r -equivalent and with properties as in 6.3. We assume that we have made these replacements from the outset. Let d D dim.B/ and apply 6.8 to obtain 2 .r ; r/ \ Q and a tube T . / of index as in the statement of that result. Pick any quasisimple module E in T . / and let E 0 be any other quasisimple module in that tube. Fix f W A ! E such that there is no g W B ! E with gb D f a and similarly fix f 0 W A ! E witnessing that tr.A; a/.E 0 / > tr.B; b/.E 0 /. We shall show that the images of .B; b/ C .E; x/ and .B; b/ C .E 0 ; x 0 / in pp= r are incomparable. So suppose, for a contradiction, that, say, .B; b/ C .E; x/ r .B; b/ C .E 0 ; x 0 /, that is, .E; x/ r .E; x/ ^ ..B; b/ C .E 0 ; x 0 //. Therefore, by 6.7, there is ı > 0 such that for all X in .r ı; r/ we have tr.E; x/.X / D tr .E; x/ ^ ..B; b/ C .E 0 ; x 0 // .X /; we may take ı < . Recalling what the operations in the lattice of pointed finitely presented modules are, the right-hand side of this equation is tr.L; l/.X / where L is the pushout module shown and l D g.x/ D g 0 .b; x 0 /. R

x

/E

g0

/L

.b;x 0 /

B ˚ E0

g

Note that dim.L/ < dim.E/ C dim.E 0 / C dim.N /. Write L as L0 ˚ L00 where each summand of L0 has slope < r and each summand of L00 has slope > r. So, by 6.1, there are nonzero morphisms, embeddings by 6.8, from E to modules X in .r ı; r/, .L; X / ¤ 0. Therefore L0 ¤ 0 and so, if we set f 0 D 0 g where 0 W L ! L0 is the induced projection, then f 0 .x/ ¤ 0. We will show that L0 has no summand of slope > ; we do this by showing that this is also true of coker.f 0 /, noting that by choice of and since L0 is in . ; r/, f 0 is, by 6.8, an embedding, so we have the exact sequence f0

0 ! E ! L0 ! coker.f 0 / ! 0:

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Pick 0 2 . ; r/ such that L0 is in .0; 0 / (indeed, in Œ ; 0 /). First we show that coker.f 0 / is in Œ0; 0 /. Suppose that h W E ! Z where Z is in .r ı; r/; then h factors through g. Indeed, since h.x/ 2 tr.E; x/.Z/ tr..B; b/ C .E 0 ; x 0 //.Z/, there must exist a map h0 W B ˚ E 0 ! Z with h0 .b; x 0 / D h.x/. The pushout property then gives a factorisation of h through g, hence through f 0 , as claimed. Pick Z of slope 0 ; then we have the long exact sequence coker.f 0 ;Z/

0 ! .coker.f 0 /; Z/ ! .L0 ; Z/ ! .E; Z/ ! Ext.coker.f 0 /; Z/ ! Ext.L0 ; Z/ D 0 (the last term is 0 since, by the Auslander–Reiten formula, it has the same dimension as .Z; L0 / D 0). We have just seen that coker.f 0 ; Z/ is surjective so it follows that Ext.coker.f 0 /; Z/ D 0 and then, by the Auslander–Reiten formula and 6.1, that .Z; coker.f 0 // D 0. This is so for every Z of slope 0 so, by 6.1, coker.f 0 / must be in Œ0; 0 /. Now, notice that dim.coker.f 0 // D dim.L0 / dim.E/ dim.L/ dim.E/ < dim.B/ C dim.E 0 / C dim.E/ dim.E/ D dim.N / C dim.E 0 /. So, by choice of , T . / to satisfy 6.8, and the fact that E 0 is in T . /, it must be that every summand of coker.f 0 / with slope > has slope > r. We saw above that every summand of coker.f 0 / has slope < 0 < r, so we deduce that coker.f 0 / has slope . It follows that L0 also has slope and hence that the exact sequence f0

0 ! E ! L0 ! coker.f 0 / ! 0 lies in add.T . //. If L0 is not indecomposable then we can replace it by a direct summand to which the induced map from E is nonzero hence still an embedding, and that sequence will, up to isomorphism, have the form f 00

p

! 1 EŒk 1 ! 0: 0 ! E ! EŒk Since EŒk has quasisimple socle E, it follows that 1 g 0 E 0 D 0, where 1 W L ! EŒk is the projection, and hence 1 g 0 .b/ D f 00 .x/, so p1 g 0 .b/ D 0. Therefore p1 g 0 factors through coker.b/ which, by choice of B, b and to satisfy 6.3, must be in .r; 1. Therefore p1 g 0 D 0 and 1 g takes B to f 00 E ' E. We deduce that f 00 .x/ 2 tr.B; b/.f 00 E/ and hence that x 2 tr.B; b/.E/ – a contradiction to the choice of x. Thus .B; b/ C .E; x/ and .B; b/ C .E 0 ; x 0 / have incomparable images in the lattice pp= r as claimed. This is true for arbitrary pairs .A; a/ .B; b/ as at the start of the proof, so we have shown that this lattice is wide. The final stage is to use tilting functors to move the conclusion of the result above to other tubular algebras. This is done in [20] and [21] using the language of pp formulas but the proofs actually use pointed modules (in fact n-pointed modules since the

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generator RR will in general be tilted to a module which is not cyclic). We don’t repeat the arguments here. An alternative would be to use general results about interpretation functors [37, §18.2.1] or [39, §25] and the fact that tilting functors are such ([33, 4.8], see [37, 18.2.22]). In any case the conclusion is as follows. Theorem 6.10 ([20], Theorem 34, p. 119). Let R be a tubular algebra and let r be a positive irrational. Then the width of the lattice of pointed finitely presented modules for the definable category Dr is undefined. If R is countable then there is a superdecomposable pure-injective module of slope r. Corollary 6.11. Let R be a tubular algebra and let r be a positive irrational. Then the Krull–Gabriel dimension of the definable category Dr is undefined. If R is countable then there are continuum many indecomposable pure-injective modules of slope r. For the second assertion see the comment at the end of Section 4.1.

7 Final remarks Here we have stopped at the point of proving existence of a superdecomposable pureinjective and we haven’t considered the structure of such a module. For example if N is superdecomposable it may or may not be the case that N ' N 2 . There is a structure theory for injective modules over von Neumann regular rings and that can be reflected into the structure of pure-injective modules, as in [9]. Puninski, see e.g. [47], also has developed some ways of understanding the structure of these modules and has formulated conjectures about their structure. There seems, however, to have been rather limited development in this direction. A specific question we can ask, in the context of modules over tubular algebras, is whether every (superdecomposable) pure-injective is the pure-injective hull of a direct sum of modules with slope. It would be good to know whether the countability hypothesis currently needed to deduce from having width undefined that there is a superdecomposable pure-injective can be removed. An obvious approach, if R is an uncountable ring with ppR having width 1, is to take a countable elementary subring R0 (which, by the downwards Löwenheim–Skolem Theorem of model theory, may be chosen to contain any specified countably many elements of R) and which is such that ppR0 also has width 1 (this can be done: see [34, Proposition 10]). Then there will be a superdecomposable pure-injective R0 -module N0 . Can one, for example, obtain a superdecomposable pure-injective R-module from N0 ˝R0 R?

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[2] M. Auslander, Functors and morphisms determined by objects. In Representation theory of algebras, R. Gordon (ed.), Lecture Notes in Pure Appl. Math. 37, Marcel Dekker, New York 1978, 1–244. [3] V. Bavula, Generalised Weyl algebras and their representations. Algebra i Analiz 4 (1992), no. 1, 75–97; English transl. St. Petersburg Math. J. 4 (1993), no. 1, 71–92. [4] K. Burke, Some model-theoretic properties of functor categories for modules. Doctoral Thesis, University of Manchester, 1994. [5] K. Burke, Co-existence of Krull filtrations. J. Pure Appl. Algebra 155 (2001), no. 1, 1–16. [6] K. Burke and M. Prest, The Ziegler and Zariski spectra of some domestic string algebras. Algebr. Represent. Theory 5 (2002), no. 3, 211–234. [7] W. Crawley-Boevey, Infinite-dimensional modules in the representation theory of finitedimensional algebras. In Algebras and modules I, I. Reiten, S. Smalø and Ø. Solberg (eds.), CMS Conf. Proc. 23, Amer. Math. Soc., Providence, RI, 1998, 29–54. [8] S. Ebrahimi Atani, On pure-injective modules over pullback rings. Comm. Algebra 28 (2000), no. 9, 4037–4069. [9] A. Facchini, Decompositions of algebraically compact modules. Pacific J. Math. 116 (1985), no. 1, 25–37. [10] A. Facchini, Module theory: Endomorphism rings and direct sum decompositions in some classes of modules. Progr. Math. 167, Birkhäuser, Basel 1998. [11] E. Fisher, Abelian structures. Yale University, 1974/5; partly published as “Abelian structures I”, in Abelian group theory, Lecture Notes in Math. 616, Springer, Berlin 1977, 270–322. [12] S. Garavaglia, Dimension and rank in the model theory of modules. Preprint, University of Michigan, 1979, revised 1980. [13] W. Geigle, The Krull-Gabriel dimension of the representation theory of a tame hereditary Artin algebra and applications to the structure of exact sequences. Manuscripta Math. 54 (1985), no. 1–2, 83–106. [14] R. Göbel and J. Trlifaj, Approximations and endomorphism algebras of modules. De Gruyter Exp. Math. 41, Walter de Gruyter, Berlin 2006. [15] R. Gordon and J. C. Robson, Krull dimension. Mem. Amer. Math. Soc. 133, Amer. Math. Soc., Providence, RI, 1973. [16] R. Gordon and J. C. Robson, The Gabriel dimension of a module. J. Algebra 29 (1974), no. 3, 459–473. [17] G. Grätzer, General lattice theory. Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 52, Birkhäuser, Basel 1978. [18] L. Gruson and C. U. Jensen, Modules algébriquement compact et foncteurs lim.i / . C. R. Acad. Sci. Paris 276 (1973), 1651–1653. [19] L. Gruson and C. U. Jensen, Dimensions cohomologiques reliées aux foncteurs lim.i / . In Séminaire d’algèbre Paul Dubreil et Marie-Paule Malliavin, 33ème Année, Lecture Notes in Math. 867, Springer, Berlin 1981, 234–294.

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[20] R. Harland, Pure-injective modules over tubular algebras and string algebras. Doctoral Thesis, University of Manchester, 2011, available at www.maths.manchester.ac.uk/~mprest/publications.html [21] R. Harland and M. Prest, Modules with irrational slope over tubular algebras. Preprint, arXiv:1302.6763 [math.RT]. [22] B. Huisgen-Zimmermann, Purity, algebraic compactness, direct sum decompositions and representation type. In [28], 331–367. [23] C. U. Jensen and H. Lenzing, Model theoretic algebra with particular emphasis on fields, rings and modules. Algebra, Logic and Applications 2, Gordon and Breach, New York 1989. [24] I. Kaplansky, Infinite abelian groups. Univ. of Michigan Press, Ann Arbor 1954; revised edition, Ann Arbor 1969. [25] S. Kasjan and G. Pastuszak, On two tame algebras with super-decomposable pure-injective modules. Colloq. Math. 123 (2011), no. 2, 249–276 [26] L. Klingler and L. Levy, Wild torsion modules over Weyl algebras and general torsion modules over HNPs. J. Algebra 172 (1995), no. 2, 273–300. [27] H. Krause, Generic modules over artin algebras. Proc. London Math. Soc. (3) 76 (1998), no. 2, 276–306. [28] H. Krause and C. M. Ringel (eds.), Infinite length modules. Trends Math., Birkhäuser, Boston, Mass., 2000. [29] H. Lenzing, Generic modules over tubular algebras. In Advances in algebra and model theory, M. Droste and R. Göbel (eds.), Algebra, Logic and Applications 9, Gordon and Breach, Amsterdam 1997, 375–385. [30] L. S. Levy, Modules over pullbacks and subdirect sums. J. Algebra 71 (1981), no. 1, 50–61. [31] M. Prest, Model theory and modules. London Math. Soc. Lecture Notes Ser. 130, Cambridge University Press, Cambridge 1988. [32] M. Prest, Duality and pure-semisimple rings. J. London Math. Soc. (2) 38 (1988), no. 2, 403–409. [33] M. Prest, Interpreting modules in modules. Ann. Pure Applied Logic 88 (1997), no. 2–3, 193–215. [34] M. Prest, The representation theories of elementarily equivalent rings. J. Symbolic Logic 63 (1998), no. 2, 439–450. [35] M. Prest, Morphisms between finitely presented modules and infinite-dimensional representations. In Algebras and modules, II, CMS Conf. Proc. 24, Amer. Math. Soc., Providence, RI, 1998, 447–455. [36] M. Prest, Ziegler spectra of tame hereditary algebras. J. Algebra 207 (1998), no. 1, 146–164. [37] M. Prest, Purity, spectra and localisation. Encyclopedia Math. Appl. 121, Cambridge University Press, Cambridge 2009 [38] M. Prest, Pure-injective modules. Arab. J. Sci. Eng. ASJE. Math. 1 (2009), no. 1, 175–191. [39] M. Prest, Definable additive categories: purity and model theory. Mem. Amer. Math. Soc. 210 (2011), no. 987.

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Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves Jan Št’ovíˇcek

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Quasi-coherent modules . . . . . . . . . . . . . . . . . . . . . . 3 Exact categories of Grothendieck type . . . . . . . . . . . . . . 4 Weak factorization systems . . . . . . . . . . . . . . . . . . . . 5 Complete cotorsion pairs . . . . . . . . . . . . . . . . . . . . . 6 Exact and hereditary model categories . . . . . . . . . . . . . . 7 Models for the derived category . . . . . . . . . . . . . . . . . 8 Monoidal model categories from flat sheaves and vector bundles References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction The aim of the present notes, partly based on the doctoral course given by the author at the University of Padova in March 2012, is twofold: (1) to give a rather complete account on the construction of exact model structures and describe the link to cotorsion pairs and approximation theory; (2) to generalize the theory so that it applies to interesting recently studied classes of examples. The parts of the paper related to (1) are mostly not new, except for the presentation and various improvements. However, there does not seem to be a suitable reference containing all the story and, as the adaptation to the algebraic setting sometimes requires small changes in the available definitions related to model categories, it seemed desirable to write up the construction at a reasonable level of details. Some results related to (2), on the other hand, are to our best knowledge original. The concept of a model category [60], [41], [38] has existed for half a century. Despite being intensively studied by topologists, it has not attracted much attention in the theory of algebraic triangulated categories. There are probably two reasons for this development: The foundation of the theory of abelian and exact model categories has been only given a decade ago by Hovey [42] (see also [43] for a nice overview), and the “implementation details” for their construction are rather recent, see [23], [27], [28], [29], [30], [65], [71]. In the meantime, a successful theory for algebraic triangulated categories has been developed, based on dg algebras and dg categories.

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The exact model categories give in many respects a complementary approach to that of dg algebras, with different advantages and weaknesses, and it has a good potential for instance in Gorenstein homological algebra [39], homological algebra in the category of quasi-coherent sheaves [22], [23], [28], or in connection with recent developments regarding the Grothendieck duality [45], [56], [57], [58], [59]. Interestingly, although model structures in connection with singularity categories are first explicitly mentioned in [6] by Becker, Murfet [56] and Neeman [57], [58], [59] still implicitly use parts of the theory which we are going to present. While dg algebras provide perfect tools for constructing functors from single objects (say tilting or Koszul equivalences), the approach via models gives advantage in several theoretical questions. It may for example happen as in [45], [57] that a dg model for a given triangulated category is too complicated to understand, but the category itself has a rather easy description. There is, however, another important aspect – the model theoretic approach links the theory of triangulated categories to approximation theory [32], allowing deep insights on both sides. This is by no means to say that the dg and model techniques exclude each other – Keller in his seminal paper [47] in fact constructed two model structures for the derived category of a small dg category, and this point of view has been for example used to prove non-trivial results about triangulated torsion pairs in [68, §3.2]. Approximation theory is roughly speaking concerned with approximating general objects (modules, sheaves, complexes) by objects from special classes. Cofibrant and fibrant replacements in model categories are often exactly this kind of approximations. The central notion in that context is that of a cotorsion pair [64], whose significance has been recognized both in abstract module theory [32] and representation theory of finite dimensional algebras [4, Chapter 8]. The approach to construct approximations which we follow here started in [15], and the connection to model categories and Quillen’s small object argument have been noticed in [42], [62] and in some form also in [7]. It was soon realized that similar results hold also for sheaf categories, for instance in [17], [18], [19]. It is fair to remark that there is an alternative approach to approximation theory, namely Bican’s and El Bashir’s proof of the Flat Cover Conjecture in [8] and its follow ups [16], [20], [40], [63], which does not seem to fit in our framework. The first aim of ours, partly inspired by [21], but at a more advanced level, is to collect the essentials of the theory in one place together with a motivating and guiding example from [28], [23]: to construct for an arbitrary scheme X a model structure for D.Qcoh.X//, the unbounded derived category of quasi-coherent sheaves, which is compatible with the tensor product. Of course, D.Qcoh.X // may not be the category we wish to work with as we also have the subcategory of the derived category of all sheaves consisting of objects with quasi-coherent cohomology, but for many schemes the two categories are equivalent by [9, Corollary 5.5]. In order to achieve the goal, we also discuss in detail an equivalent description of Qcoh.X/ as the category of certain modules over a representation of a poset in the

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category of rings. The description is due to Enochs and Estrada [17] and, although not very well suited for direct computations with coherent sheaves, it is excellent for theoretical questions regarding big sheaves. For example, it is a relatively straightforward task to prove that Qcoh.X / always is a Grothendieck category – compare to [69, B.2, p. 409]! This presentation is also quite accessible to the readers not acquainted with algebraic geometry. As mentioned above, the other goal of the paper is to generalize the theory so that it is strong enough to apply to model structures in exact categories “appearing in the nature.” Our motivation involves in particular an interpretation of recent results about singularity categories [45], [56], [57], [58], [59] and using models in conjunction with dg categories [68]. This program has been started by Saorín and the author in [65], [67] and it follows the spirit of [31]. It is also, in a way, not a compulsory part for the reader, as it should be manageable to read the paper as if it were written only for, say, module categories instead of more general exact categories. Even in this restriction the presented results are relevant. The main problem which we address here is a suitable axiomatics for exact categories which allows to use Quillen’s small object argument and deconstructibility techniques to construct cotorsion pairs and model structures. The best suited concept so far seems to be an exact category of Grothendieck type defined in this text, although the theory is not optimal yet. The main problem is that we do not know whether the important Hill Lemma (Proposition 3.14) holds for these exact categories or in which way we should adjust the axioms to make it hold. As a consequence, some of our results including Proposition 3.19, Corollary 5.18 or Theorem 7.11 cannot be stated in as theoretically clean way as we would have wished. This is left as a possible direction for future research, where the promising directions include Enochs’ filtration shortening techniques [20], or Lurie’s colimit rearrangements from [49, §A.1.5] or [52]. ˇ P201/12/G028 Acknowledgments. This research was supported by grant GA CR from Czech Science Foundation.

2 Quasi-coherent modules In order to have classes of examples at hand, we start with describing the categories of quasi-coherent modules over schemes and diagrams of rings. 2.1 Grothendieck categories. Although the construction of model structures described later in this text has been motivated from the beginning by homological algebra in module and sheaf categories, several constructions work easily more abstractly for Grothendieck categories and, as we will discuss in Section 3, even for nice enough exact categories. Thus we start with the definition and basic properties of Grothendieck categories.

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Definition 2.1 ([34]). An abelian category G is called a Grothendieck category if (Gr1) G has all small coproducts (equivalently: G is a cocomplete category). (Gr2) G has exact direct limits. That is, given a direct system pj

ij

.0 ! Xj ! Yj ! Zj ! 0/j 2I of short exact sequences, then the colimit diagram 0 ! lim Xj ! lim Yj ! lim Zj ! 0 ! ! ! j 2I

j 2I

j 2I

is again a short exact sequence in G . This is sometimes called theAB5 condition following an equivalent requirement in [34, p. 129]. (Gr3) G has a generator. That is, there is an object G 2 G such that every X 2 G admits an epimorphism G .I / ! X ! 0. Here, G .I / stands for the coproduct ` j 2I Gj of copies Gj of G. An important property of a Grothendieck category is that it always has enough injective objects, which is very good from the point of view of homological algebra. This is in fact a good reason to consider infinitely generated modules or sheaves of infinitely generated modules: injective objects are often infinitely generated in any reasonable sense. We summarize the comment in a theorem: Theorem 2.2. Let G be a Grothendieck category. Then each X 2 G admits an injective envelope X ! E.X /. Moreover, G admits all small products (equivalently: it is complete) and has an injective cogenerator C . That is, C is injective in G and each X 2 G admits a monomorphism of the form 0 ! X ! C I . Proof. The fact that every object X 2 G admits a monomorphism 0 ! X ! E with E injective was shown already in [34, Théorème 1.10.1]. The existence of injective envelopes and an injective cogenerator is proved in [55, Theorem 2.9] and [55, Corollary 2.11], respectively. The fact that G has products and many other properties of G are clear from the Popescu–Gabriel theorem, see e.g. [66, Theorem X.4.1]. 2.2 Quasi-coherent modules over diagrams of rings. The simplest examples of Grothendieck categories are module categories G D Mod-R. In this section we construct more complicated examples, involving diagrams of rings and diagrams of modules over these rings. In fact, for suitable choices we obtain a category equivalent to the category of quasi-coherent sheaves over any given scheme. The presentation here is an adjusted version of [17, §2]. Since the discussion in [17, §2] is rather brief and many details are omitted, we will also discuss in §2.3 the translation between quasi-coherent sheaves and the Grothendieck categories which we describe here.

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Definition 2.3. Let .I; / be a partially ordered set. Then a representation R of the poset I in the category of rings is given by the following data: (1) for every i 2 I , we have a ring R.i /, (2) for every i j , we have a ring homomorphism Rji W R.i / ! R.j /, and (3) we require that for every triple i j k the morphism Rki W R.i / ! R.k/ coincides with the composition Rkj B Rji , and also that Rii D 1R.i/ . Remark 2.4. If we view I as a thin category in the usual way, then R is none other than a covariant functor R W I ! Rings: Remark 2.5. Although all of our examples and the geometrically minded motivation will involve only representations of posets in the category of commutative rings, noncommutative rings can be potentially useful too. For instance, one can consider sheaves of algebras of differential operators and ring representations coming from them. In any case, the commutativity is not necessary for the basic properties which we discuss in this section, so we do not include it in our definition. Having defined representations of I in the category of rings, we can define modules over such representations in a straightforward manner. Definition 2.6. Let R be a representation of a poset I in the category of rings. A right R-module is (1) a collection .M.i //i2I , where M.i / 2 Mod-R.i / for each i 2 I , (2) together with morphisms of the additive groups Mji W M.i / ! M.j / for each i j (3) satisfying the compatibility conditions Mki D Mkj B Mji and Mii D 1M.i/ for every triple i j k, and such that (4) the ring actions are respected in the following way: Given x 2 R.i / and m 2 M.i / for i 2 I , then for any j i we have the equality Mji .m x/ D Mji .m/ Rji .x/: All our modules in the rest of the text are going to be right modules unless explicitly stated otherwise, so we will omit usually the adjective “right”. In order to obtain a category, it remains to define morphisms of R-modules. The definition is the obvious one.

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Definition 2.7. Let R be a representation of a poset I in the category of rings and M; N be R-modules. A morphism f W M ! N is a collection of .f .i / W M.i / ! N.i //i2I , where f .i/ is a morphism of R.i /-modules for every i 2 I , and the square M.i /

f .i/

/ N.i /

Mji

M.j /

Nji

f .j /

/ N.j /

commutes for every i < j . Let us denote the category of all R-modules by Mod-R. As we quickly observe: Proposition 2.8. Let .I; / be a poset and R a representation of I in the category of rings. Then Mod-R is a Grothendieck category. Moreover limits and colimits of diagrams of modules are computed component wise – we compute the corresponding (co)limit in Mod-R.i / for each i 2 I and connect these by the (co)limit morphisms. Proof. Everything is very easy to check except for the existence of a generator in Mod-R. In fact, there is a generating set fPi j i 2 I g of projective modules described as follows: ´ R.j / if j i; Pi .j / D 0 otherwise and the homomorphism Pi .j / ! Pi .k/ for j k either coincides with R.j / ! R.k/ if i j k or vanishes otherwise. One directly checks that there is a isomorphism HomR .Pi ; M / Š M.i / for each i 2 I and M 2 Mod-R which assigns to every f W Pi ! M the element f .i /.1R.i/ / 2 M.i /. Moreover, the canonical homomorphism a .M.i// Pi ! M i2I

is surjective for every M 2 Mod-R, so G D

` i2I

Pi is a projective generator.

Although being valid Grothendieck categories, the categories Mod-R as above are not the categories of our interest yet. In order to get a description of categories of quasicoherent sheaves as promised, we must consider certain full subcategories instead. In order for this to work, we need an extra condition on R: Definition 2.9 ([17]). Let R be a representation of a poset I in rings. We call R a flat representation if for each pair i < j in I , the ring homomorphism Rji W R.i / ! R.j / gives R.j / the structure of a flat left R.i /-module. That is, ˝R.i/ R.j / W Mod-R.i / ! Mod-R.j / is an exact functor.

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As discussed later in §2.3, the representations coming from structure sheaves of schemes always satisfy this condition. For such an R, we can single out the modules we are interested in: Definition 2.10. Let R be a flat representation of I in rings. A module M 2 Mod-R is called quasi-coherent if, for every i < j , the R.j /-module homomorphism M.i / ˝R.i/ R.j / ! M.j /; m ˝ x 7! Mji .m/ x; is an isomorphism. Denote the full subcategory of Mod-R formed by quasi-coherent R-modules by Qcoh.R/. Again, we obtain a Grothendieck category. Theorem 2.11. Let .I; / be a poset and R be a flat representation of I in the category of rings. Then Qcoh.R/ is a Grothendieck category. Moreover colimits of diagrams and limits of finite diagrams are computed component wise – that is, for each i 2 I separately. Proof. Again, the main task is to prove that Qcoh.R/ has a generator and the rest is rather easy, since taking colimits and kernels (hence also finite limits) commutes with the tensor products ˝R.i/ R.j /, where i; j 2 I and i j . We omit the proof of the existence of a generator as it is rather technical, and refer to [17, Corollary 3.5] instead. Remark 2.12. Every Grothendieck category has small products by Theorem 2.2, and so must have them Qcoh.R/. However, these are typically not computed component wise and do not seem to be well understood. Since Qcoh.R/ is a cocomplete category with a generator and the inclusion functor Qcoh.R/ ! Mod-R preserves small colimits, the inclusion Qcoh.R/ ! Mod-R has a right adjoint Q W Mod-R ! Qcoh.R/ by the special adjoint functor theorem [51, §5.8] (compare to [69, Lemma B.12]!) Following [69], we call such a Q the coherator. Clearly, if .Mk /k2K isQ a collection of quasi-coherent R-modules, the product in Qcoh.R/ is computed as Q. Mk /, where Q Mk stands for the (component wise) product in Mod-R. Q However, the abstract way of constructing Q gives very little information on what Q. Mk / actually looks like. Some more information on this account is given in [69, B.14 and B.15]. Before discussing a general construction in the next section, we exhibit particular examples of flat representations of posets of geometric origin and quasi-coherent modules over them.

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Example 2.13. Consider the three element poset be given by the Hasse diagram /o

and a representation in the category of rings of the form RW

kŒx

/ kŒx; x 1 o

kŒx 1 ;

where k is an arbitrary commutative ring. Clearly R is a flat representation since the inclusions are localization morphisms. For each n 2 Z, we have a quasi-coherent R-module O.n/ W

kŒx

/ kŒx; x 1 o

x n

kŒx 1 :

One can easily check that O.m/ 6Š O.n/ whenever m ¤ n, since by direct computation HomR .O.m/; O.n// D 0 for m > n. In fact, the category Qcoh.R/ is equivalent to the category of quasi-coherent sheaves over Pk1 , the projective line over k. Example 2.14. Given a commutative ring k, let us now show a flat representation of a poset corresponding to the scheme Pk2 , the projective plane over k. The Hasse diagram of the poset has the following shape: N NNN NNN ppppp NNNNN ppppp NNpN p pppNNNNN pp NNN N' wpppp N' wpppp NNN p NNN pp p p NNN p NNN ppp N' wpppp To describe the representation R corresponding to Pk2 , it is enough to define the ring homomorphisms corresponding to arrows in the Hasse diagram. Such a description is given in the following diagram, where all the rings are subrings of kŒx0˙1 ; x1˙1 ; x2˙1 , the ring of Laurent polynomials in three indeterminates over k, and all the ring homomorphisms are inclusions: kŒ xx10 ; xx20

kŒ xx01 ; xx21 kŒ xx02 ; xx12 QQQ Q QQQ m QQQ mmmmmm QQ mmmmm QQQ Q m Q m m QQQ QQQ m mm Q( ( vmmm vmmm x2 x1 ˙1 x1 x0 ˙1 kŒ x0 ; . x0 / kŒ x2 ; . x2 / kŒ xx01 ; . xx21 /˙1 QQQ QQQ mmm QQQ mmm m m QQQ m ( vmmm x1 ˙1 x2 ˙1 kŒ. x / ; . x / 0

0

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2.3 Realizing modules over schemes as modules over diagrams. The aim of this section is to make precise how the category of quasi-coherent sheaves of modules over a given scheme can be described in terms of quasi-coherent modules over a flat ring representation of a poset. Here we assume some familiarity with the basic notions in place: those of a scheme and of a quasi-coherent sheaf of modules over a scheme [37], [33]. On the other hand, understanding this part is not necessary for understanding most of the text which follows, so the reader can skip it and continue with Section 3. Suppose that .X; OX / is a scheme, that is a ringed space which is locally isomorphic to .Spec R; OSpec R / for a commutative ring R. Given this data, we first construct a representation of a poset in the category of commutative rings. Construction 2.15. Let U be a collection of open affine sets of X satisfying the following two conditions: S (1) U covers X ; that is X D U. S (2) Given U; V 2 U, then U \ V D fW 2 U j W U \ V g. It is always a safe choice to take the collection of all affine open sets, but often much smaller sets U will do. For projective schemes for example, we can always choose U to be finite. Now U is a poset with respect to inclusion and we put I D Uop , the opposite poset. Since OX is a sheaf of commutative rings, we in particular have a functor Uop ! CommRings which sends a pair U V of sets in U to the restriction resU V W OX .U / ! OX .V /. By the very definition of I , this is the same as saying that we have a covariant functor R W I ! CommRings such that, in the notation of Definition 2.3, we have R.U / D O.U / and RVU D resU V. A standard fact is that the representation of I we get in this way is flat: Lemma 2.16. Let R be the representation of I in the category of rings as in Construction 2.15. Then R is flat. Proof. This is proved for instance in [33, Proposition 14.3 (1) and (4)]. Upon unraveling the definitions, the statement relies on the following fact from commutative algebra, [54, Theorem 7.1]: Given a homomorphism ' W R ! S of commutative rings, then S is flat over R if and only if Sq is flat over R' 1 .q/ for every q 2 Spec S . It is now easy to construct a functor from the category Qcoh.X / of quasi-coherent sheaves of OX -modules to the category Qcoh.R/ of quasi-coherent modules over R. Construction 2.17. Let us adopt the notation from Construction 2.15. Given M 2 Qcoh.X/ and two affine open sets U V , then canonically M.U / ˝OX .U / O.V / Š

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M.V / – here we just apply [33, Remark 7.23 and Proposition 7.24 (2)] to the open immersion .Spec R.V /; OSpec R.V / / Š .V; OXjV / ! .U; OXjU / Š .Spec R.U /; OSpec R.U / / and the corresponding ring homomorphism RVU W R.U / ! R.V /. Thus, viewing the sheaf M as a contravariant functor from the poset of open sets of X to Ab, we may restrict the functor to I D Uop . This way we assign to M 2 Qcoh.X/ an R-module F .M / and F .M / is quasi-coherent by the above discussion. This assignment is obviously functorial, so that we get an additive functor F W Qcoh.X / ! Qcoh.R/: Seemingly, there is much more structure in M 2 Qcoh.X / than in F .M /. While the former is a sheaf of modules over a possibly complicated topological space X, the latter is only a collection of modules satisfying a certain coherence condition. However, the fact that M is quasi-coherent is itself very restrictive and we have the following crucial result; see [17, §2]. Theorem 2.18. The functor F from Construction 2.17 (which depends on the choice of U in Construction 2.15) is an equivalence of categories. Proof. Note that a quasi-coherent sheaf of modules M 2 Qcoh.X / is determined up to a canonical isomorphism by its image under F . Indeed, this follows from conditions (1) and (2) in Construction 2.15, [33, Theorem 7.16 (1)] and the sheaf axiom. Similarly a morphism f W M ! N in Qcoh.X / is fully determined by F .f /. In particular F is a faithful functor. In order to prove that F is dense, fix a module A 2 Qcoh.R/ and let us introduce op some notation. S Given an upper subset L I D U with respect to the partial order on I , then SL X is an open subset of X , so that we can consider quasi-coherent sheaves over L. We can also restrict the representation R W I ! CommRings to the representation L ! CommRings, which we denote by RL . Clearly RL is a flat representation and Construction 2.17 provides us with a functor [ FL W Qcoh. L/ ! Qcoh.RL /: Finally, we have the restriction functor Qcoh.R/ ! Qcoh.RL / and we will denote the image of A under this functor by AjL . op Now we shall consider the collection ƒ of all upper subsets S L I D U such that there is a quasi-coherent sheaf of modules ML 2 Qcoh. L/ with FL .ML / D AjL . As such ML is unique up to a canonical isomorphism, the collection ƒ is closed under unions of chains. Hence by Zorn’s lemma there is an upper subset L I which belongs to ƒ and is maximal such with respect to inclusion. We claim that L D I . Suppose by S op way of contradiction that L ¤ I . Then there is U 2 U D I such that U 6 L and we consider the unique quasi-coherent sheaf A.U / 2 Qcoh.U / whose global section

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module is A.U /. Now we invoke condition (2) of Construction 2.15 which, together with [33, Theorem 7.16 (1)] and the sheaf axiom, allows us to construct a canonical S isomorphism .ML /jV Š A.U /jV of sheaves over the open set V D L \ U . Thus S we can glue ML and A.U / to a quasi-coherent sheaf over L [ U , showing that L [ fW 2 I j W U in I g belongs to ƒ, in contradiction to the choice of L. This proves the claim and the density of F . The fact that F is full is proved in a similar way. Given a morphism g W A ! B in Qcoh.R/, we denote by ƒ0 the collection of all upper subsets L SI such that gjL lifts to a morphism of sheaves of modules over the open subscheme L X . We ought to prove that L D I and we again do so using Zorn’s lemma.

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3 Exact categories of Grothendieck type In various contexts (see [65], [68] for example), it is useful to consider more general categories than Grothendieck categories. The rest of the text, however, is perfectly relevant when read as if it were written for Grothendieck categories or even for module categories. Thus, the reader who wishes to avoid the related technicalities may skip the section and read further from Section 4. 3.1 Efficient exact categories. In order describe our object of interest, we recall some terminology. The central concept is that of an exact category, which is originally due to Quillen, but the common reference for a simple axiomatic description is [46, Appendix A] and an extensive treatment is given in [10]. An exact category is an additive category E together with a distinguished class of diagrams of the form i

d

0 ! X ! Y ! Z ! 0; called conflations, satisfying certain axioms which make conflations behave similar to short exact sequences in an abelian category and allow to define Yoneda Ext groups with usual properties (see Section 5). Adopting the terminology from [46], the second map in a conflation (denoted by i ) is called inflation, while the third map (denoted by d ) is referred to as deflation. Morally, an exact category is an extension closed subcategory of an abelian category, which is made precise in the following statement. Proposition 3.1 ([46], [10]). (1) Let A be an abelian category. Then A considered together with all short exact sequences as conflations is an exact category. (2) Let E be an exact category and E 0 E be an extension closed subcategory (i.e. if 0 ! X ! Y ! Z ! 0 is a conflation and X; Z 2 E 0 , then Y 2 E 0 ). Then E 0 , considered together with all conflations in E whose all terms belong to E 0 , is again an exact category. (3) Every small exact category arises up to equivalence as an extension closed subcategory of an abelian category in the sense of (1) and (2).

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For the results presented later to work, we need to impose extra conditions on our exact categories. Long story short – we need to impose requirements on the exact category which make it sufficiently resemble a Grothendieck category. As the requirement that E be a cocomplete category seems too restrictive in practice, we need to specify first which direct limits we are interested in connection with the analogue of the left exactness property (Definition 2.1 (Gr2)). In order to do so, the following definition is handy. Definition 3.2. Let C be an arbitrary category, let be an ordinal number, and let .X˛ ; f˛ˇ /˛