The study of determinantal ideals and of classical determinantal rings is an old topic of commutative algebra. As in mos

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*Table of contents : Introduction iii1 Preliminaries 12 Divisor class group and canonical class of determinantal rings 102.1 Integrity and normality of determinantal rings . 102.2 Divisor class group of determinantal rings 172.3 The canonical class of determinantal rings 203 Gröbner bases and determinantal ideals 313.1 Bitableaux, combinatorial algorithms, and the Knuth-Robinson-Schensted correspondence 313.2 Schensted and Greene’s theorems 343.3 KRS and generic matrices 353.4 Gröbner bases of determinantal ideals 434 Powers and products of determinantal ideals 514.1 Primary decomposition of the (symbolic) powers of determinantal ideals 514.2 Powers of ideals of maximal minors 594.3 Gröbner bases of powers of determinantal ideals 625 Simplicial complexes associated to determinantal ideals 715.1 Simplicial complexes 715.2 Multiplicity of determinantal rings 745.3 Hilbert series of determinantal rings 885.4 Thea-invariant of determinantal rings 926 Algebras of minors6.1 Cohen-Macaulayness and normality of algebras R(It) and At6.2 Divisor class group of R(It) and At 1036.3 Canonical modules of inτ(R(It)) and inτ(At) 1106.4 Canonical classes of R(It) and At 1147 F-rationality of determinantal rings 121 7.1 F-regularity of determinantal rings 121 7.2 F-rationality of Rees algebras 129References 132*

COMBINATORICS OF DETERMINANTAL IDEALS Cornel B˘aet¸ica

Contents Introduction

iii

1 Preliminaries

1

2 Divisor class group and canonical class of determinantal rings 2.1 Integrity and normality of determinantal rings . . . . . . . . . 2.2 Divisor class group of determinantal rings . . . . . . . . . . . 2.3 The canonical class of determinantal rings . . . . . . . . . . .

10 10 17 20

3 Gr¨ obner bases and determinantal ideals 3.1 Bitableaux, combinatorial algorithms, and the Knuth-RobinsonSchensted correspondence . . . . . . . . . . . . . . . . . . . . 3.2 Schensted and Greene’s theorems . . . . . . . . . . . . . . . . 3.3 KRS and generic matrices . . . . . . . . . . . . . . . . . . . . 3.4 Gr¨obner bases of determinantal ideals . . . . . . . . . . . . . .

31

4 Powers and products of determinantal ideals 4.1 Primary decomposition of the (symbolic) powers of determinantal ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Powers of ideals of maximal minors . . . . . . . . . . . . . . . 4.3 Gr¨obner bases of powers of determinantal ideals . . . . . . . .

51

5 Simplicial complexes associated to determinantal ideals 5.1 Simplicial complexes . . . . . . . . . . . . . . . . . . . . . 5.2 Multiplicity of determinantal rings . . . . . . . . . . . . . 5.3 Hilbert series of determinantal rings . . . . . . . . . . . . . 5.4 The a-invariant of determinantal rings . . . . . . . . . . .

. . . .

71 71 74 88 92

6 Algebras of minors 6.1 Cohen-Macaulayness and normality of algebras R(It ) and At . 6.2 Divisor class group of R(It ) and At . . . . . . . . . . . . . . . 6.3 Canonical modules of in τ (R(It )) and in τ (At ) . . . . . . . . .

97 97 103 110

i

. . . .

31 34 35 43

51 59 62

ii

Contents 6.4

Canonical classes of R(It ) and At . . . . . . . . . . . . . . . .

114

7 F -rationality of determinantal rings 121 7.1 F -regularity of determinantal rings . . . . . . . . . . . . . . . 121 7.2 F -rationality of Rees algebras . . . . . . . . . . . . . . . . . . 129 References

132

Introduction The study of determinantal ideals and of classical determinantal rings is an old topic of commutative algebra. As in most of the cases, the theory evolved from algebraic geometry, and soon became an important topic in commutative algebra. Looking back, one can say that it is the merit of Eagon and Northcott [49], [50] to be the ﬁrst who brought to the attention of algebraists the determinantal ideals and investigated them by the methods of commutative and homological algebra. Later on, Buchsbaum and Eisenbud, in a series of articles, went further along the way of homological investigation of determinantal ideals, while Eagon and Hochster [48] studied them using methods of commutative algebra in order to prove that the classical determinantal rings are normal Cohen-Macaulay domains. Subsequently, L. Avramov, T. J´ozeﬁak, H. Kleppe, R. E. Kutz, D. Laksov, V. Marinov, P. Pragacz et al. have turned the attention to the study of determinantal (pfaﬃan) ideals, respectively of classical determinantal (pfaﬃan) rings associated to symmetric (alternating) matrices; see [3], [74], [75], [76], [83], [84], [85]. As shown later by C. De Concini, D. Eisenbud, and C. Procesi [41], [42], the appropriate framework including all three types of rings is that of algebras with straightening law, and the standard monomial theory on which these algebras are based yields computationally eﬀective results. A coherent treatment of determinantal ideals from this point of view was given by Bruns and Vetter in their seminal book [28]. A new perspective to the study of the determinantal ideals was brought by Sturmfels’ paper [92], in which he applied the Knuth-Robinson-Schensted correspondence in order to determine Gr¨obner bases for determinantal ideals. Later on, this technique was extended by Herzog an Trung [68] to the study of 1-cogenerated ideals, ladder determinantal ideals and pfaﬃan ideals. Our book aims to a thorough treatment of all three types of determinantal rings in the light of the achievements of the last ﬁfteen years since the publication of Bruns and Vetter’s book [28]. We implicitly assume that the reader is familiar with the basics of commutative algebra. However, we include some of the main notions and results from [28], for the sake of completeness, but without proofs. We recommend the reader to ﬁrst look at the book of Bruns and Vetter in order to get a feel for the ﬂavor of this ﬁeld. The ﬁrst two iii

iv

Introduction

chapters of our book follow the line of investigation from Bruns and Vetter [28], while the rest of the book relies on the combinatorics of Gr¨obner bases, a method initiated by Sturmfels [92]. As often as possible, we discuss the case of 1-cogenerated ideals and their corresponding determinantal rings, though sometimes we restrict ourselves to the classical case of determinantal ideals generated by minors (pfaﬃans) of ﬁxed size, either due to lack of results or diﬃculties of exposition. An important and useful feature of the book is that every chapter contains a “Notes” section, where we present historical remarks, more bibliographical sources, open questions and further directions for research. Our book is meant to be a reference text for the current state of research in the theory of determinantal rings. It was structured in such a way that it can be used as textbook for a one semester graduate course in advanced topics in Commutative Algebra, at PhD level. We now include brief descriptions of the chapters of the book. In Chapter 1, we introduce the most important notions that will be used throughout the book. It is a mainly expository chapter, containing few proofs. We start with the deﬁnition of graded algebras with straightening law on a poset (doset) over an arbitrary commutative ring. In all sections of the book, the exposition is split into three parts that correspond to the three types of matrices we are interested in: in part (G) we study generic matrices, in part (S) generic symmetric matrices, and in part (A) generic alternating matrices. Chapter 2 deals with the computations of the divisor class group and of the canonical class of determinantal rings corresponding to 1-cogenerated ideals. We also determine which of the rings under consideration are Gorenstein rings. Chapter 3 is the core of the entire book. We ﬁrst describe the combinatorial algorithms INSERT and DELETE, and then present the Knuth-RobinsonSchensted correspondence, KRS for short, between standard (Young) bitableaux and two-line arrays of positive integers of a certain type. The theorems of Schensted and Greene are also reviewed. The theorem of Schensted [88] deals with the determination of the length of the longest increasing (decreasing) subsequence of a given sequence of integers. The length of the longest increasing (respectively decreasing) subsequence of a given sequence v is the length of the ﬁrst row (respectively column) in the tableau INSERT(v). An interpretation of the rest of the shape of INSERT(v) is given by Greene’s theorem [65]. We describe a KRS type correspondence between standard monomials and ordinary monomials of a set of indeterminates, and at the end of the chapter we use this correspondence to determine Gr¨obner bases for all three types of determinantal ideals. In Chapter 4, we ﬁrst recall some results on the primary decomposition of the powers (products) of determinantal ideals. It turns out that the primary decomposition depends on the characteristic of the ground ﬁeld. We then

Introduction

v

determine Gr¨obner bases for the powers of ideals of maximal minors (pfaﬃans) without reference to the characteristic. We conclude the chapter by exploiting the theorems of Schensted and Greene to determine Gr¨obner bases for the (symbolic) powers of determinantal ideals. Chapter 5 brings a new perspective to the study of determinantal ideals based on the principle of deriving properties of ideals and algebras from their initial counterparts. As a matter of fact, we associate to the initial ideal of any determinantal ideal a shellable simplicial complex in such a way that the corresponding residue class ring of the initial ideal is the Stanley-Reisner ring of the complex. Some of the combinatorial properties of simplicial complexes are then interpreted in terms of families of lattice paths of a certain type, and thus the determination of the multiplicity, the Hilbert series, or the a-invariant of determinantal rings becomes a matter of counting paths. All these techniques are brought together in Chapter 6 for the investigation of Rees algebras associated to determinantal (pfaﬃan) ideals and of algebras generated by minors (pfaﬃans) of ﬁxed size. We show that the Rees algebras of determinantal (pfaﬃan) ideals and the algebras generated by minors (pfaﬃans) are Cohen-Macaulay normal domains in non-exceptional characteristic. The rest of the chapter is devoted to the description of the divisor class group and the canonical class of Rees algebras of determinantal (pfaﬃan) ideals and of algebras generated by minors (pfaﬃans). Finally, we determine the Gorenstein rings among the rings under consideration. In chapter 7, we prove that the classical determinantal rings are F -rational, an extension of the well-known property that determinantal rings have rational singularities in characteristic 0. We also ﬁnd that the Rees algebras associated to determinantal (pfaﬃan) ideals and the algebras generated by minors (pfaﬃans) of ﬁxed size are F -rational (in non-exceptional characteristic). We want to point out that our book is concerned with 1-cogenerated determinantal ideals and determinantal rings associated with them. An important generalization of the classical determinantal rings is given by the ladder determinantal rings which are deﬁned by the minors of certain subregions, called ladders, of a generic (symmetric, respectively alternating) matrix. They were introduced by Abhyankar for the study of the singularities of Schubert varieties of ﬂag manifolds, and share many properties with classical determinantal rings, such as: ladder determinantal rings are Cohen-Macaulay normal domains, are F -rational, the Gorenstein property can be completely characterized in terms of the shape of the ladder, and there are determinantal formulas for the a-invariant and Hilbert series. As for the determinantal rings most of these results rely on the combinatorial structure of the Gr¨obner bases of their deﬁning ideals. For more details on these topics, we refer the reader to Conca [34], [35], Conca and Herzog [38], De Negri [45], Ghorpade [58], Herzog and Trung [68], Krattenthaler and Prohaska [80], Krattenthaler and Rubey [81],

vi

Introduction

and Narasimhan [86]. Recently, K. R. Goodearl and T. H. Lenagan [61], [62] initiated the study of quantum determinantal ideals that arise from quantum generic matrices. It is worth pointing out that a truly interesting question is to determine noncommutative Gr¨obner bases for these ideals.

Bucharest, January 2004

Cornel B˘aet¸ica

Chapter 1 Preliminaries The straightening law of Doubilet, Rota and Stein [47], in the approach of De Concini, Eisenbud and Procesi [41], is an indispensable tool in the study of determinantal rings. From the algebraic point of view, the determinantal rings are graded algebras with straightening law, and this yields some of their main properties. Deﬁnition 1.0.1 Let A be a B-algebra and ∆ ⊆ A a ﬁnite subset with a partial order , called a poset. Then A is a graded algebra with straightening law (for short ASL ) on ∆ over B if the following conditions hold: ⊕ (H0 ) A = i≥0 Ai is a positively graded B-algebra such that A0 = B, ∆ consists of homogeneous elements of positive degree and generates A as a B-algebra. (H1 ) The products δ1 · · · δu , u ∈ N, δi ∈ ∆, such that δ1 · · · δu are linearly independent over B. We call them standard monomials. (H2 ) (Straightening law) For all incomparable δ1 , δ2 ∈ ∆ the product δ1 δ2 has a representation ∑ δ1 δ2 = λµ µ, λµ ∈ B, λµ 6= 0, µ standard monomial , that satisﬁes the following condition: every standard monomial µ contains a factor ∑ ζ ∈ ∆ such that ζ δ1 , ζ δ2 (we allow that δ1 δ2 = 0, then the sum λµ µ being empty). In fact the standard monomials form a B-basis of A called the standard basis of A. The representation of an element of A as a linear combination of standard monomials is called its standard representation. The relations in (H2 ) will be referred to as the straightening relations. 1

2

1. Preliminaries

The most interesting examples of ASLs are rings related to matrices and determinants; see De Concini, Eisenbud and Procesi [42], Bruns and Vetter [28]. In order to apply the ASL theory to determinantal rings we have to know that these rings are indeed ASLs. This follows readily from the fact that their deﬁning ideals have a distinguished system of generators with respect to the underlying poset. To be precise, let us consider A an ASL on a poset ∆ (over B), and ∆′ ⊆ ∆ a poset ideal of ∆, i.e. ∆′ contains all elements x y if y ∈ ∆′ . Set I = ∆′ A the ideal of A generated by ∆′ . Proposition 1.0.2 The residue class ring A/I is a graded ASL on ∆ \ ∆′ over B. When ∆′ is the poset ideal cogenerated by a subset Ω of ∆, that is ∆′ = {ξ ∈ ∆ : ω 6 ξ for any ω ∈ Ω}, then the ideal I = ∆′ A is called the ideal cogenerated by Ω. In particular, when |Ω| = 1 the ideal cogenerated by Ω is said to be a 1-cogenerated ideal. An useful generalization of the notion of graded algebra with straightening law on a poset is that of graded algebra with straightening law on a doset. We deﬁne the underlying notions. Deﬁnition 1.0.3 Let H be a ﬁnite poset with order . A subset D of H × H is a doset if D satisﬁes the following conditions: (a) (α, α) ∈ D for all α ∈ H; (b) if (α, β) ∈ D, then α β; (c) if α β γ ∈ H, then (α, γ) ∈ D ⇐⇒ (α, β) ∈ D and (β, γ) ∈ D. ⊕ Deﬁnition 1.0.4 Let B be a ring, and let A = i≥0 Ai be a positively graded B-algebra such that A0 = B. Let D be a doset of a poset H, and suppose that D ⊆ A. Then A is a graded algebra with straightening law on doset D over B (for short DASL) if the following conditions are satisﬁed: (H0 ) D consists of homogeneous elements of positive degree in A. (H1 ) The products of the form (α1 , α2 ) · · · (α2k−1 , α2k ), k ≥ 1, (α2i−1 , α2i ) ∈ D, such that α1 α2 · · · α2k−1 α2k form a B-basis of A. We call them standard monomials.

1. Preliminaries

3

(H2 ) For (β2i−1 , β2i ) ∈ D, i = 1, . . . , l, let M = (β1 , β2 ) · · · (β2l−1 , β2l ). Moreover, let ∑ M= λN N , λN ∈ B, λN 6= 0, N standard monomial be the representation of M as a linear combination of standard monomials (standard representation). Let N = (γ1 , γ2 ) · · · (γ2h−1 , γ2h ) be any of the standard monomials appearing on the right side of the equation. Then for all permutations σ of the set {1, . . . , 2l} we have that the sequence (βσ(1) , . . . , βσ(2l) ) is lexicographically greater or equal than the sequence (γ1 , . . . , γ2h ). (H3 ) In the notation of (H2 ), suppose that there is a permutation σ such that βσ(1) · · · βσ(2l) . The standard monomial (βσ(1) , βσ(2) ) · · · (βσ(2l−1) , βσ(2l) ) must appear with coeﬃcient ±1 in the standard representation of M . Let us now consider the most important examples of ASLs related to matrices and determinants as they were described by De Concini, Eisenbud and Procesi [42]; see also Bruns and Vetter [28]. (G) Let X = (Xij ) be an m × n matrix of indeterminates over a commutative ring B, for short a generic matrix, ∆(X) the set of all minors of X, and B[X] = B[Xij : 1 ≤ i, j ≤ n] the polynomial ring over B. Consider the set ∆(X) ordered in the following way: [a1 , . . . , at | b1 , . . . , bt ] [c1 , . . . , cs | d1 , . . . , ds ] if and only if t ≥ s and ai ≤ ci , bi ≤ di for i = 1, . . . , s. (As usual, one denotes by [a1 , . . . , at | b1 , . . . , bt ] the minor det (Xai bj )1≤i,j≤t .) Let δ ∈ ∆(X), δ = [a1 , . . . , at | b1 , . . . , bt ]. We deﬁne I(X, δ) as being the ideal generated by all minors which are not greater or equal than δ. Thus the ideal I(X, δ) is the ideal of B[X] cogenerated by δ. Denote by R(X, δ) the residue class ring of B[X] with respect to the ideal I(X, δ) and by ∆(X, δ) the set of all minors which are greater or equal than δ. In particular, if δ = [1, . . . , t | 1, . . . , t] then I(X, δ) is the ideal It+1 (X) generated by all the (t + 1)-minors of X, and denote by Rt+1 (X) the analogous residue class ring. When B is a ﬁeld we call Rt+1 (X) the classical determinantal ring. In the investigation of the rings R(X, δ) we rely on the knowledge of their combinatorial structure enlightened by De Concini, Eisenbud and Procesi [41], [42], and on the methods developed by Bruns and Vetter in the fundamental

4

1. Preliminaries

book [28]. In this approach one considers all the minors of X as generators of the B-algebra B[X], and not only 1-minors Xij . So we may interpret the products of minors as “monomials”, but the price to be paid is that the computations may become tedious. Therefore we have to choose a proper subset of all these “monomials” as a B-basis: we will see that the standard monomials form a B-basis, and the straightening law will help us to express an arbitrary product of minors as a linear combination of standard monomials. Let µ = δ1 · · · δu be a monomial, i.e. a product of minors. We can always assume that the sizes of minors δi , denoted by |δi |, are in non-increasing order |δ1 | ≥ · · · ≥ |δu |. By convention, the value of the empty minor [ | ] is 1. The shape |µ| of µ is the sequence (|δ1 |, . . . , |δu |). Moreover, the monomial µ is said to be standard if and only if δ1 · · · δu . The combinatorial structure of B[X] with respect to the products of minors is clariﬁed by the following theorem; see De Concini, Eisenbud and Procesi [42, pg. 51], [41] or Bruns and Vetter [28, Chapter 4]. Theorem 1.0.5 The ring B[X] is a graded algebra with straightening law on ∆(X) over B, that is: (a) The standard monomials form a B-basis of B[X]. (b) The product of two minors δ1 , δ2 ∈ ∆(X) such that δ1 δ2 is not a standard monomial has a representation ∑ δ1 δ2 = λi ξi ηi , λi ∈ B, λi 6= 0, where ξi ηi is a standard monomial and ξi ≺ δ1 , δ2 ≺ ηi (we allow here that ηi is the empty minor). (c) The standard representation of an arbitrary monomial µ can be found by successive applications of the straightening relations in (b). Some remarks are in order here: Remark 1.0.6 Usually the proof of Theorem 1.0.5 goes by passing to the subalgebra B[Γ(X)] generated by the set Γ(X) of maximal minors of X. (When m ≤ n and B is a ﬁeld, the algebra B[Γ(X)] is the homogeneous coordinate ring of the Grassmann variety of the m-dimensional vector subspaces of B n .) The advantage of considering B[Γ(X)] instead of B[X] stands on the fact that the maximal minors satisfy the famous Pl¨ ucker relations; see, for instance, Bruns and Herzog [23, Lemma 7.2.3] or Bruns and Vetter [28, Lemma (4.4)]. Notice that the Pl¨ ucker relations are homogeneous of degree 2 (the maximal minors are considered in B[Γ(X)] as having degree 1). Therefore there are exactly two factors ξi and ηi in each term on the right side of the equation ∑ δ1 δ2 = λi ξi ηi ,

1. Preliminaries

5

with δ1 , δ2 maximal minors. But the straightening law in B[X] is a specialization of that in B[Γ(X ′ )], where X ′ is a generic matrix that contains X as a submatrix. This yields a justiﬁcation for the part (b) of Theorem 1.0.5. As a consequence of Theorem 1.0.5 and Proposition 1.0.2 we get: Corollary 1.0.7 The ring R(X, δ) is a graded algebra with straightening law on ∆(X, δ) over B. The sets ∆(X) and ∆(X, δ) are distributive lattices. By using [28, Theorem (5.14)], we can immediately conclude: Proposition 1.0.8 If B is a Cohen-Macaulay ring, then R(X, δ) is CohenMacaulay. In the following we give a formula for the dimension of the ring R(X, δ), where δ = [a1 , . . . , at | b1 , . . . , bt ] ∈ ∆(X). ∑ Proposition 1.0.9 We have dim R(X, δ) = dim B + (m + n + 1)t − ti=1 (ai + bi ). In particular, the classical determinantal ring Rt+1 (X) is Cohen-Macaulay of dimension (n + m − t)t. (S) Let X = (Xij ) be an n × n symmetric matrix of indeterminates over a commutative ring B (i.e. Xij = Xji for all i > j), for short a generic symmetric matrix, and B[X] = B[Xij : 1 ≤ i ≤ j ≤ n] the polynomial ring over B. Let H be the set of the non-empty subsets of {1, . . . , n} and α = {α1 , . . . , αt } ∈ H with α1 < · · · < αt . One deﬁnes I(X, α) as being the ideal of B[X] generated by all the i-minors of the ﬁrst αi − 1 rows of X, for i = 1, . . . , t, and by all (t + 1)-minors of X. Denote by R(X, α) the residue class ring of B[X] with respect to the ideal I(X, α). In particular, if α = {1, . . . , t} then I(X, α) is the ideal It+1 (X) generated by all the (t + 1)-minors of X, and denote by St+1 (X) the analogous residue class ring. When B is a ﬁeld we call St+1 (X) the classical ring of symmetric minors. There is also a standard monomial approach to the structure of R(X, α), in which DASLs replace ASLs. On the set H we deﬁne the following partial order: a = {a1 , . . . , at } b = {b1 , . . . , bs } ⇐⇒ t ≥ s and ai ≤ bi for i = 1, . . . , s. Since X is a symmetric matrix, it is obvious that we have [a1 , . . . , at | b1 , . . . , bt ] = [b1 , . . . , bt | a1 , . . . , at ]. Let M = M1 · · · Mu be a monomial, i.e. a product of minors. We can assume that the sizes of minors Mi , denoted by |Mi |, are in non-increasing order |M1 | ≥ · · · ≥ |Mu |. By convention, the value of the

6

1. Preliminaries

empty minor [ | ] is 1. The shape |M | of M is the sequence (|M1 |, . . . , |Mu |). A minor [a1 , . . . , at | b1 , . . . , bt ] of X is a doset minor if {a1 , . . . , at } {b1 , . . . , bt } in H. Let us denote by ∆(X) the set of all doset minors of the matrix X. Let a1 = {a11 , . . . , a1t1 }, b1 = {b11 , . . . , b1t1 }, . . . , as = {as1 , . . . , asts }, bs = {bs1 , . . . , bsts } be elements in H and suppose that [a1 |b1 ], . . . , [as |bs ] ∈ ∆(X). The product M = [a1 | b1 ] · · · [as | bs ] is a standard monomial if bi ≤ ai+1 in H for all i = 1, . . . , s − 1. For a standard monomial M , one denotes by min (M ) the minimum of its factors [a1 | b1 ]. The combinatorial structure of B[X] with respect to the products of minors is described by the following theorem of De Concini, Eisenbud and Procesi [42, pg. 82]: Theorem 1.0.10 The ring B[X] is a doset algebra on ∆(X) over B, that is: (a) The standard monomials form a B-basis of B[X]. (b) (Straightening law) Let Mi = [ai1 , . . . , airi | bi1 , . . . , biri ] ∈ ∆(X) with i = 1, . . . , s and N = [c11 , . . . , c1s1 | d11 , . . . , d1s1 ] · · · [cr1 , . . . , crsr | dr1 , . . . , drsr ] be one of the standard monomials that appear in the (unique) representation of the product M1 · · · Ms as a linear combination of standard monomials. If one considers ci = {ci1 , . . . , cisi }, di = {di1 , . . . , disi }, ai = {ai1 , . . . , airi } and bi = {bi1 , . . . , biri }, then in the lexicographic order on H the sequence c1 , d1 , . . . , cr , dr is less or equal than every sequence obtained by permuting the elements a1 , b1 , . . ., as , bs . The next result is a direct consequence of [40, Lemma 5.2]: Lemma 1.0.11 Every t-minor M = [a1 , . . . , at | b1 , . . . , bt ] can be written as a linear combination of doset t-minors. Furthermore, for any t-doset minor [c1 , . . . , ct | d1 , . . . , dt ] that appears in the representation of M we have ci ≤ ai for all i = 1, . . . , t. Set ∆(X, α) = {[a | b] ∈ ∆(X) : α a}. By Lemma 1.0.11 one deduces that the ideal I(X, α) is generated by the doset minors in Ω(X, α) = ∆(X) \ ∆(X, α), that is, the ideal I(X, α) is the ideal cogenerated by α. Note that the set Ω(X, α) is the set of all doset minors whose sequence of row indices is not greater or equal than α. Using the straightening law we get: Corollary 1.0.12 The set of all standard monomials M such that min (M ) ∈ Ω(X, α) is a B-basis of I(X, α), and the set of residue classes of all standard monomials M such that min (M ) ∈ ∆(X, α) forms a B-basis of the ring R(X, α). In [82] R. E. Kutz proved the following

1. Preliminaries

7

Proposition ∑t 1.0.13 (a) The dimension of the ring R(X, α) equals dim B + (n + 1)t − i=1 αi . (b) If B is a Cohen-Macaulay ring, then the ring R(X, α) is Cohen-Macaulay. In particular, the classical ring of symmetric minors St+1 (X) is a CohenMacaulay ring of dimension t(t + 1)/2 + (n − t)t. A proof of Proposition 1.0.13 will be given later in Chapter 5. (A) Let X = (Xij ) be an n × n alternating matrix of indeterminates over a commutative ring B, i.e. Xij = −Xji for all i > j and Xii = 0, for short a generic alternating matrix, and B[X] = B[Xij : 1 ≤ i < j ≤ n] the polynomial ring over B. It is well known that det (X) = 0 when n is odd, while for n even there exists an element pf (X) in B[X], called the pfaﬃan of X, such that det (X) = pf (X)2 . For more details about pfaﬃans we refer the reader to Bourbaki [13], Chapter IX, § 5, no. 2. Let Π(X) be the set of all pfaﬃans of X, and consider Π(X) ordered in the following way: [a1 , . . . , a2t ] [b1 , . . . , b2s ] if and only if t ≥ s and ai ≤ bi for i = 1, . . . , 2s. Set π ∈ Π(X), π = [a1 , . . . , a2t ]. One deﬁnes I(X, π) to be the ideal generated by all pfaﬃans which are not greater or equal than π. The ideal I(X, π) is called the ideal cogenerated by π. Denote by R(X, π) the residue class ring of B[X] with respect to the ideal I(X, π) and by Π(X, π) the set of all pfaﬃans which are greater or equal than π. In particular, if π = [1, . . . , 2t] then I(X, π) is the ideal It+1 (X) generated by all (2t + 2)-pfaﬃans of X, and denote by Pt+1 (X) the analogous residue class ring. The ideals It+1 (X) are called pfaﬃan ideals and the rings Pt+1 (X) the classical ring of pfaﬃans, provided B is a ﬁeld. Let ν = π1 · · · πu be a monomial, i.e. a product of pfaﬃans. We can always assume that the sizes of pfaﬃans πi , denoted by |πi |, are in non-increasing order |π1 | ≥ · · · ≥ |πu |. By convention, the value of the empty pfaﬃan [ ] is 1. The shape |ν| of ν is the sequence (|π1 |, . . . , |πu |). Moreover, the monomial ν is said to be standard if and only if π1 · · · πu . The combinatorial structure of B[X] with respect to products of pfaﬃans is settled by the following theorem; see De Concini and Procesi [40, Theorem 6.5] or De Concini, Eisenbud and Procesi [42, pg. 53]. Theorem 1.0.14 The ring B[X] is a graded algebra with straightening law on Π(X) over B, that is: (a) The standard monomials form a B-basis of B[X]. (b) The product of two pfaﬃans π1 , π2 ∈ Π(X) such that π1 π2 is not a standard monomial has a representation ∑ π1 π 2 = λi ξi ηi , λi ∈ B, λi 6= 0,

8

1. Preliminaries

where ξi ηi is a standard monomial and ξi ≺ π1 , π2 ≺ ηi (we allow here that ηi is the empty pfaﬃan). (c) The standard representation of an arbitrary monomial ν can be found by successive applications of the straightening relations in (b). The straighten of two pfaﬃans relies on the following result of De Concini and Procesi [40, Lemma 6.1]: Lemma 1.0.15 We have ∑ [a1 , . . . , a2t∑ ][b1 , . . . , b2s ] − 2t i=1 [a1 , . . . , ai−1 , b1 , ai+1 , . . . , a2t ][ai , b2 , . . . , b2s ] = 2s j−1 [b2 , . . . , bj−1 , bj+1 , . . . , b2s ][bj , b1 , a1 , . . . , a2t ]. j=2 (−1) Note that Lemma 1.0.15 yields an argument for part (b) of Theorem 1.0.14, as long as any straightening relation can be obtained by iterated use of Lemma 1.0.15. As a consequence of Theorem 1.0.14 and Proposition 1.0.2 we get: Corollary 1.0.16 The ring R(X, π) is a graded algebra with straightening law on Π(X, π) over B. The sets Π(X) and Π(X, π) are distributive lattices. By virtue of [28, Theorem (5.14)] we can immediately conclude: Proposition 1.0.17 If B is a Cohen-Macaulay ring, then R(X, π) is CohenMacaulay. Now we compute the dimension of the ring R(X, π) with π = [a1 , . . . , a2t ] ∈ Π(X). ∑ Proposition 1.0.18 We have dim R(X, π) = dim B + 2nt − 2t i=1 ai . In particular, the classical ring of pfaﬃans Pt+1 (X) is Cohen-Macaulay of dimension 2nt − (2t + 1)t. Proof. Since the poset Π(X, π) is a distributive lattice, it is a wonderful poset, so that any two maximal chains have the same length. The rank of Π(X, π) is the length of every maximal chain in Π(X) which starts from π. We can ﬁnd the following saturated chain [a1 , . . . , a2t−1 , a2t ] < [a1 , . . . , a2t−1 , a2t + 1] < · · · < [a1 , . . . , a2t−1 , n] < · · · < [a1 , . . . , n − 1, n] < [a1 , . . . , a2t−2 ]. By induction ∑2t−2 on t we get that rank Π(X, π) ∑2t= 2n − (a2t−1 + a2t ) + 2n(t − 1) − i=1 ai − 1, hence rank Π(X, π) = 2nt− i=1 ai −1. As R(X, π) is an ASL on Π(X, π) over B we can compute its dimension by using the general formula for computing the dimension of an ASL (see, for instance, Bruns and Vetter [28, Proposition (5.10)]), so that dim R(X, π) = dim B + rank Π(X, π) + 1.

Notes

9

Notes The straightening law of Doubilet, Rota and Stein [47] was used by De Concini and Procesi [40] in order to prove that the classical determinantal rings, classical rings of symmetric minors, and classical rings of pfaﬃans are ASLs. They also gave some insight on how one can straighten the product of two pfaﬃans. Later De Concini, Eisenbud, and Procesi [42] made a thorough investigation of the structure of ASLs, and proved that the ALSs are Cohen-Macaulay on wonderful posets; see also Bruns and Vetter [28, Chapter 4]. This is how the concept of ASL became a bridge between combinatorics and commutative algebra. The Cohen-Macaulay property of classical determinantal rings was ﬁrst proved by Eagon and Hochster [48] without reference to ASLs. (The proof from Eagon and Hochster [48] was done by means of principal radical systems.) The classical rings of symmetric minors were investigated by Kutz [82], who followed the same strategy as Eagon and Hochster [48] in order to show that these rings are Cohen-Macaulay domains. Later on, J´ozeﬁak [73] found a resolution of classical rings of symmetric minors in low dimensional cases. The classical rings of pfaﬃans were studied by J´ozeﬁak and Pragacz [74], [75]. They computed their dimension and pointed out a resolution of classical rings of pfaﬃans for small dimensions. Marinov [83], [84] and [85] proved the perfection of ideals generated by pﬀaﬁans. Subsequently, Kleppe and Laksov [76] generalized this class of rings, computed the dimension, the singular locus and proved that these rings are always Gorenstein. For a thorough study of the general case of determinantal rings determined by 1-cogenerated ideals, we refer the reader to Bruns and Vetter [28] for the generic case, Conca [30] for generic symmetric matrices, and Baetica [4] and De Negri [43] for generic alternating matrices.

Chapter 2 Divisor class group and canonical class of determinantal rings In this chapter we give a complete description of the divisor class group of determinantal rings in terms of the ”blocks” and ”gaps” of the cogenerator. At the end we determine which of these rings are Gorenstein. In order to solve this problem we ﬁrst determine the divisor class group of their canonical module, the so-called canonical class.

2.1

Integrity and normality of determinantal rings

In the following we show that the integrity and normality transfers from the ground rings to the determinantal rings. (G) For generic matrices we have the following result (see Bruns and Vetter [28, Theorem (6.3)]): Theorem 2.1.1 Let B be a (normal) domain. Then the ring R(X, δ) is a (normal) domain. Theorem 2.1.1 has an immediate consequence: Corollary 2.1.2 Let B be an integral domain, and Ω ⊂ ∆(X) a poset ideal. Then the minimal prime ideals of ΩR(X, δ) are the ideals I(x, ζ) = I(X, ζ)/I(X, δ), ζ running through the minimal elements of ∆(X) \ Ω, and ΩR(X, δ) is their intersection. 10

2.1. Integrity and normality

11

(S) We prove that R(X, α) is a (normal) domain as long as B is a (normal) domain. In order to do this, we need the following lemma which describes the localization of R(X, α) with respect to α within the total ring of quotients of R(X, α). The proof will follow closely the argument given by Bruns and Vetter [28, Lemma (6.4)]. Let α ∈ H, α = {α1 , . . . , αt }, and denote by f the residue class of the minor [α|α] in R(X, α). Consider Ψ = {[αi |αj ] : 1 ≤ i < j ≤ t}∪{[α|β] ∈ ∆(X, α) : β diﬀers from α in exactly one index} and denote by B[Ψ] the B-subalgebra of R(X, α) generated by the elements in set Ψ. Lemma 2.1.3 The set Ψ is algebraically independent over B and R(X, α)[f −1 ] = B[Ψ][f −1 ]. Thus R(X, α)[f −1 ] is isomorphic to B[T1 , . . . , Td ][ζ −1 ], where ζ ∈ B[T1 , . . . , Td ] and d = dim R(X, α) − dim B. If B is an integral domain, then ζ is a prime element. Proof. First we prove that R(X, α)[f −1 ] = B[Ψ][f −1 ]. It is clear that B[Ψ][f −1 ] is contained in R(X, α)[f −1 ]. The other inclusion will be done if we show that [u|v] ∈ B[Ψ][f −1 ] for all [u|v] ∈ ∆(X, α). If u (or v) is in {α1 , . . . , αt }, then we get [u, α1 , . . . , αt |v, α1 , . . . , αt ] = 0. Expanding the minor with respect to the ﬁrst row we get [u|v]f =

t ∑

(−1)j+1 [u|αj ][α1 , . . . , αt |v, α1 , . . . , α ˆ j , . . . , αt ],

j=1

where w ˆ means that the index w is missing. Because [u|αj ] ∈ Ψ and [α1 , . . . , αt |v, α1 , . . . , α ˆ j , . . . , αt ] ∈ Ψ (it diﬀers from α in exactly one index) or [α1 , . . . , αt |v, α1 , . . . , α ˆ j , . . . , αt ] = 0 in R(X, α), we conclude that [u|v] ∈ B[Ψ][f −1 ]. The general case is an immediate consequence of the relation above. Since f is the only minimal element of ∆(X, α), it is a non zero-divisor in R(X, α), therefore dim B[Ψ] = dim B[Ψ][f −1 ] = dim R(X, α)[f −1 ] = dim R(X, α). Note that an element ξ ∈ ∆(X, α) which diﬀers ∑t from α in exactly one index must ∑t be a t-minor, so that |Ψ| = t(t + 1) + i=1 (n − αi − (t − i)) = t(n + 1) − i=1 αi = dim R(X, α) − dim B. Consequently, when B is a ﬁeld the set Ψ is algebraically independent over B. The general case proceeds as in the proof of Lemma (6.1) from Bruns and Vetter [28].

12

2. Divisor class group and canonical class

When B is an integral domain, the element f is a prime element of the polynomial ring B[(Xai aj )1≤i≤j≤t ] because it is the determinant of the symmetric matrix (Xai aj )1≤i≤j≤t (if we use Theoem 2.1.9, it is immediately that the determinant of a symmetric matrix is a prime element). Therefore f is a prime element of B[Ψ], and the isomorphism above maps f in ζ. We get that ζ is a prime element of B[T1 , . . . , Td ]. Corollary 2.1.4 Let B be a domain. Then the ring R(X, α) is a domain. Proof. The localization R(X, α)[f −1 ] is a domain by Lemma 2.1.3, and f is a non zero-divisor. Therefore R(X, α) is a domain, too. In order to prove normality of the ring R(X, α) we shall use Serre’s normality criterion; see, for instance, Fossum [54, Theorem 4.1]. Since it is CohenMacaulay by Proposition 1.0.13, it is enough to show that R(X, α) satisﬁes the Serre’s condition (R1 ). By Lemma 2.1.3 any localization of R(X, α) to a prime which does not contain f is regular, as being a localization of a polynomial ring. Consequently we have to investigate the localizations of R(X, α) to minimal prime ideals of f . First let us describe the minimal prime ideals of f . Set U = {β ∈ H : β > α}, and denote by J the set of the minimal elements in U , that is J is the set of the upper neighbours of α in H. One considers J = ∅ if α = {n}, and I(X, ∅) = (Xij : 1 ≤ i ≤ j ≤ n). For β ∈ J , we have I(X, α) ⊂ I(X, β). Lemma 2.1.5 We have

∩ β∈J

I(X, β) = ([α|γ] : [α|γ] ∈ ∆(X, α)) + I(X, α)

Proof. It is obvious that the right side is contained in the left side. Let µ ∈ ([α|γ] : [α|γ] ∈ ∆(X, α)) + I(X, α). One may assume that µ is a standard monomial with min (µ) = [γ|δ]. Since γ 6≥ β for all β ∈ J , one deduces that γ = α or γ 6≥ α and the proof is done. Lemma 2.1.6 The ideals I(x, β) = I(X, β)/I(X, α) of R(X, α), with β ∈ J , are the minimal prime ideals of (f ). Proof. Note that the ideals I(x, β), β ∈ J , are distinct. Furthermore, we have ∩ ( β∈J I(x, β))2 ⊂ (f ). By Lemma 2.1.5, it is enough to show that for all γ, δ > α we have [α|γ][α|β] ∈ (f ). Let ν a standard monomial that appear in the standard representation of [α|γ][α|β], and set min (ν) = [ζ1 |ζ2 ]. The sequence ζ1 , ζ2 is less than or equal to the sequence α, α, hence ν = [α|α]ν1 or ν = 0 in R(X, α).

2.1. Integrity and normality

13

Now we want to describe the set J of all the upper neighbours of α. The standard way to do this is to break α into blocks of consecutive integers α = [β1 , . . . , βr ],

βi = (αki−1 +1 , . . . , αki ),

where k0 = 0 and kr = t. Each βi is followed by a gap χi = (αki + 1, . . . , αki +1 − 1), the sequence of integers properly between the last element of βi and the ﬁrst element of βi+1 , and χr = (αt + 1, . . . , n) or χr = ∅ if αt = n. For every i = 1, . . . , r we get an upper neighbour ζi of α by replacing αki with αki + 1, when i < r or i = r and αt < n. If αt = n, then ζr is obtained from α by deleting αt . The following lemma, borrowed from J´ozeﬁak and Pragacz [75, Lemma (1.1)], will allow us inductively proofs. ˜ be a symmetric matrix of size (n−1)×(n−1). Consider Lemma 2.1.7 Let X the K-algebras homomorphism −1 ˜ 11 , . . . , X1n ][X −1 ], ψ : K[X][X11 ] −→ K[X][X 11

deﬁned by the assignment ψ(X1j ) = X1j for all j = 1, . . . , n, and ψ(Xij ) = −1 ˜ i−1j−1 + X1i X1j X11 X for all 1 < i ≤ j ≤ n. Then ψ is an isomorphism. ˜ reLet us denote by [. . . | . . .]X , and by [. . . | . . .]X˜ the minors of X and X, −1 spectively. Let µ = [a1 , . . . , as |b1 , . . . , bs ]X ∈ ∆(X). Since (X1i X1j X11 )1≤i,j≤n is a rank 1 matrix, and using the linearity of the determinant with respect to the rows one obtains ψ(µ) = [a1 − 1, . . . , as − 1|b1 − 1, . . . , bs − 1]X˜ +

s ∑

−1 µuv , (−1)u+v X1bv X1au X11

u,v=1

(2.1) \ where µuv = [a1 − 1, . . . , a\ ˜. u − 1, . . . , as − 1|b1 − 1, . . . , bv − 1, . . . , bs − 1]X In particular, we get ψ([1, a2 , . . . , as |1, b2 , . . . , bs ]X ) = X11 [a2 −1, . . . , as −1|b2 −1, . . . , bs −1]X˜ . (2.2) If α1 = 1, then X11 6∈ I(X, α), and using (2.1) and (2.2) one shows ˜ α that the ideal ψ(I(X, α)) is the extension of the ideal I(X, ˜ ) to the ring −1 ˜ K[X][X11 , . . . , X1n ][X11 ], where α ˜ = {α2 − 1, . . . , αt − 1}. Thus we get an isomorphism −1 ˜ ˜ )[X11 , . . . , X1n ][X11 ], ψ˜ : R(X, α)[x−1 11 ] −→ R(X, α

(2.3)

14

2. Divisor class group and canonical class

where x11 denotes the residue class of X11 in R(X, α). If one denotes by f˜ the ˜ α residue class in R(X, ˜ ) of the minor [˜ α|˜ α], by ζ˜i the upper neighbours of α ˜, ˜ ˜ and by I(˜ x, ζi ) the corresponding minimal prime ideals of f , then by (2.2) we ˜ ) = X11 f˜. Moreover, if α2 = 2, then have ψ(f −1 ˜ ˜ α ψ(I(x, ζi )) = I(˜ x, ζ˜i )R(X, ˜ )[X11 , . . . , X1n ][X11 ],

(2.4)

for all i = 1, . . . , r, and, if α2 > 2, then −1 ˜ ˜ α ψ(I(x, ζi )) = I(˜ x, ζ˜i−1 )R(X, ˜ )[X11 , . . . , X1n ][X11 ],

(2.5)

−1 for all i = 1, . . . , r, while I(x, ζ1 )R(X, α)[x−1 11 ] = R(X, α)[x11 ].

Set R = R(X, α), and Pi = I(x, βi ). We are now ready to prove that the localizations RPi are discrete valuation rings (for short DVR). Lemma 2.1.8 (a) RPi is a regular ring for all i = 1, . . . , r. Hence RPi is a DVR, and let vi be the corresponding valuation. (b) If i < r, or if i = r and αt < n, then vi (f ) = 2. If αt = n, then vr (f ) = 1. Proof. The proof goes by induction on n. For n = 2, α is one of the following {1}, {2}, {1, 2}. 2 ), f = x11 , and P1 = (x11 , x12 ). If α = {1}, then I(X, α) = (X11 X22 − X12 −1 2 We get that P1 RP1 = (x12 )RP1 , and f = x22 x12 , therefore RP1 is regular and v1 (f ) = 2. If α = {2}, or α = {1, 2}, then I(X, α) = (X11 , X12 ), f = x22 , and 2 2 P1 = (x22 ), or I(X, α) = (0), f = (X11 X22 − X12 ), and P1 = (X11 X22 − X12 ). In this case all the assertions are trivial. Let us suppose now n > 2. If α1 > 1, Then R(X, α) ∼ = R(Y, γ), where Y is an (n + 1 − α1 ) × (n + 1 − α1 ) symmetric matrix of indeterminates and γ = {1, α2 + 1 − α1 , . . . , αt + 1 − α1 }. By induction, we may assume α1 = 1. If ˜ ˜ ˜ ), α2 = 2, then RPi is localization of R[x−1 11 ] for all i = 1, . . . , r. Set R = R(X, α ˜ ˜ and Pi = I(˜ x, βi ). From (2.3) and (2.4) we get that RPi is isomorphic to a ˜ ˜ [X11 , . . . , X1n ]. It is a regular ring by induction hypothesis, localization of R Pi hence RPi is regular. Since X11 is an unit, we have vi (f ) = v˜i (f˜), where v˜i is ˜ ˜ . Again by induction v˜r (f˜) = 1 if αt − 1 = n − 1, and the valuation on R Pi ˜ v˜i (f ) = 2 in the other cases. If α2 > 2 and i > 1, one proceeds as above using (2.3) and (2.5). It remains the case i = 1 and α2 > 2. By deﬁnition P1 = (x11 , . . . , x1n ). Since all the 2-minors of the ﬁrst two rows of X are 0 in R, we get x1i = x12 xi2 x−1 22 in RPi , and P1 RP1 = (x12 )RP1 . Therefore RP1 is regular. In RP1 −2 2 we have that f = det (xij )1≤i,j≤αt = [ζ1 |ζ1 ]x−2 22 x12 . As the element [ζ1 |ζ1 ]x22 is invertible in RP1 , one gets v1 (f ) = 2.

2.1. Integrity and normality

15

Now a simple application of Serre’s normality criterion gives the following Theorem 2.1.9 Let B be a normal domain. Then the ring R(X, α) is also a normal domain. Proof. Since R(X, α) is Cohen-Macaulay by Proposition 1.0.13, it is enough to show that R(X, α) satisﬁes the Serre’s condition (R1 ). By Lemma 2.1.3 any localization of R(X, α) to a prime which does not contain f is a localization of a polynomial ring, hence a regular ring. By Lemma 2.1.8(a) any localization of R(X, α) to a minimal prime ideal of f is regular. Consequently the ring R(X, α) satisﬁes the Serre’s condition (R1 ) and we are done. (A) In this part, as a normality criterion we shall use the following Lemma 2.1.10 Let S be a Noetherian ring, and x ∈ S a non zero-divisor. Assume S[x−1 ] is a normal domain, and S/(x) is reduced. Then S is normal. Proof. See Bruns and Herzog [23, Exercise 2.2.33] or Bruns and Vetter [28, Lemma (16.24)]. In order to prove that R(X, π) is a (normal) domain as long as B is a (normal) domain, we need the following lemma which describes the localization of R(X, π) with respect to π within the total ring of quotients of R(X, π). Our argument is a suitable modiﬁcation of the argument given in the proof of Lemma (6.4) from Bruns and Vetter [28]. Let π ∈ Π(X), π = [a1 , . . . , a2t ] ∈ Π(X), Ψ = {[ai aj ] : 1 ≤ i < j ≤ 2t} ∪ {ξ ∈ Π(X, π) : ξ diﬀers from π in exactly one index} and denote by B[Ψ] the B-subalgebra of R(X, π) generated by the elements in set Ψ. Lemma 2.1.11 The set Ψ is algebraically independent over B and R(X, π)[π −1 ] = B[Ψ][π −1 ]. Thus R(X, π)[π −1 ] is isomorphic to B[T1 , . . . , Td ][ζ −1 ], where ζ ∈ B[T1 , . . . , Td ] and d = dim R(X, π) − dim B. If B is an integral domain, then ζ is a prime element. Proof. At the beginning we prove that R(X, π)[π −1 ] = B[Ψ][π −1 ]. It is clear that B[Ψ][π −1 ] is contained in R(X, π)[π −1 ]. The other inclusion will be done if we show that [uv] ∈ B[Ψ][π −1 ] for all [uv] ∈ Π(X, π). If u (or v) is in {a1 , . . . , a2t }, then [u, v, a1 , . . . , a2t ] = 0. Expanding the pfaﬃan with respect to the ﬁrst two rows we get 2t ∑ [uv]π = (−1)j+1 [uaj ][v, a1 , . . . , a ˆj , . . . , a2t ], j=1

16

2. Divisor class group and canonical class

where w ˆ means that the index w is missing. Because [uaj ] ∈ Ψ and [v, a1 , . . . , a ˆj , . . . , a2t ] ∈ Ψ (it diﬀers from π in exactly one index) or [v, a1 , . . . , a ˆj , . . . , a2t ] = 0 in R(X, π), we conclude that [uv] ∈ B[Ψ][π −1 ]. The general case is an immediate consequence of the relation above. Since π is the only minimal element of Π(X, π), it is not a zero-divisor in R(X, π), therefore dim B[Ψ] = dim B[Ψ][π −1 ] = dim R(X, π)[π −1 ] = dim R(X, π). Note that an element ξ ∈ Π(X, π) which diﬀers from ∑2t π in exactly one index must be a 2t-pfaﬃan, so that |Ψ| = t(2t − 1) + i=1 (n − ai − (2t − i)) = ∑2t 2nt − i=1 ai = dim R(X, π) − dim B. Consequently, when B is a ﬁeld the set Ψ is algebraically independent over B. The general case proceeds as in the proof of Lemma (6.1) from Bruns and Vetter [28]. When B is an integral domain, the element π is a prime element of the ring B[(Xai aj )1≤i satisﬁes the condition Jh ∩ (Jh + h̸≥g Jg ) = Jh> for all h ∈ H, then we have the following ˜ gr F (R)) ' R(H, K[H])#H gr F (R). Lemma 7.2.2 gr F˜ (R(G, R)) ' R(G, Proof. The ﬁrst isomorphism is given by Lemma 7.2.1. The second follows ˜ gr F (R)) as a direct readily since we have the following decomposition of R(G, sum of its N × H-graded components ⊕⊕ ˜ gr F (R)) = (gr F (R))h T i . R(G, i∈N h∈Hi

Note that the algebra R(H, K[H])#H gr F (R) has the same decomposition. In particular, for K[X] we get the following Theorem 7.2.3 Let K be a ﬁeld with char K = p > 0, H = (Hi )i∈N a ﬁltration on the partial ordered∑ semigroup (Nm , ), and G = (Ii )i∈N a ﬁltration on R = K[X] given by Ii = w∈Hi Jw . Assume that R(H, K[Nm ]) is a ﬁnitely generated normal K-algebra. Then the Rees algebra R(G, R) is F -rational. Proof. By the same argument as Theorem 7.1.4 we obtain that the extensions of the ﬁltrations S and T to R(G, R) give rise to isomorphic K-algebras

7.2. F -rationality of Rees algebras

131

gr S (R(G, R)) and gr T (R(G, R)). By (TC5) it is enough to show the F -rationality of gr T (R(G, R)). Since gr T (R(G, R)) is isomorphic to gr S (R(G, R)), it suﬃces to show that gr S (R(G, R)) is F -rational. From Lemma 7.2.2 this is isomorphic to the Segre product of T = R(H, K[Nm ]) with F (X). By hypothesis, T is a normal aﬃne semigroup ring and therefore a direct summand of a polynomial ring over K. This shows that T ⊗K F (X) is a direct summand of a polynomial extension of F (X), and then so is the Segre product T #Nm F (X). By (TC6), Theorem 7.1.6, and (TC2) we obtain that T #Nm F (X) is F -regular, and the proof is ready. Corollary 7.2.4 (a) The symbolic Rees algebra Rs (It ) is F -rational. (b) If the characteristic of K is non-exceptional, that is, char K > min (t, m − t), then the Rees algebra R(It ) is F -rational. Proof. To begin with, we deﬁne a ﬁltration on Nm that satisﬁes the hypothesis of Theorem 7.2.3. Let l1 , . . . , lv be linear forms with nonnegative coeﬃcients on Qm , and set Hi = {a ∈ Nm : lj (a) ≥ i for all j = 1, . . . , v}. Then H = (Hi )i∈N is the desired ﬁltration. Actually, the Rees semigroup R(H, Nm ) is the subsemigroup of Nm+1 consisting of all a′ ∈ Nm+1 with the property that li (a′1 , . . . , a′m ) ≥ a′m+1 for all i = 1, . . . , v. It is ﬁnitely generated and normal since it is the intersection of Zm+1 with a ﬁnite number of rational halfspaces; see Bruns and Herzog [23, Proposition 6.1.2]. (a) Set v = 1 and l1 = γt . The result follows by Theorem 4.1.1. (b) Set v = t and li = 1/(t − i + 1)γi for i = 1, . . . , t. Now we use Corollary 4.1.4. Remark 7.2.5 (a) We can also deduce the F -rationality of Rs (It ) and R(It ) from the structure of the initial algebras that we studied in Chapter 6 by using Conca, Herzog and Valla [39, Corollary 2.3]. (b) It follows readily from Corollary 7.2.4 that the associated graded ring gr It (X) (K[X]) is Cohen-Macaulay if char K > min (t, m − t). We have a similar result for the algebra At generated by the t-minors. Proposition 7.2.6 Suppose that the characteristic of K is non-exceptional. Then At is F -rational. Proof. Let S ′ and T ′ stand for the restrictions of the ﬁltrations S and T to At . It is easily seen that gr T ′ (At ) ' gr S ′ (At ) ' K[H]#F (X), where H is the subsemigroup of Nm consisting of the weights w with the property that a bitableau of weight w belongs to At . Then H is a normal semigroup (it is ﬁnitely generated and can be written as an intersection of a full subsemigroup and a radical subsemigroup).

132

3. F -rationality of determinantal rings

(A) If K is an algebraically closed ﬁeld of characteristic 0, then the Rees algebra R(It ) has rational singularities; see Bruns [17, Remarks (3.4)(b)]. By using a result of Boutot [14] we get that the K-subalgebra At of K[X] generated by all 2t-pfaﬃans has also rational singularities. Let us suppose that char K = p > 0. Again the non-exceptional characteristic it proves to be the right setting; see Example 6.1.8. Proposition 7.2.7 (a) If (p, n!) = 1, then the symbolic Rees algebra Rs (It ) is F -rational. (b) Suppose in addition that char K > min (2t, n − 2t). Then the Rees algebra R(It ) is also F -rational. Proof. Similar to the proof of Corollary 7.2.4.

A similar result holds for the algebra At generated by 2t-pfaﬃans and we single it out. Proposition 7.2.8 If (p, n!) = 1 and char K > min (2t, n − 2t), then At is F -rational. Remark 7.2.9 The condition (p, n!) = 1, required by our method of proof, is not really necessary. We can also deduce the results of this section from the structure of the initial algebras that we studied in Chapter 6 by using Conca, Herzog and Valla [39, Corollary 2.3].

Notes The fact that the classical determinantal rings are F -regular was ﬁrst proved by Hochster and Huneke [70] by using their F -regularity criterion. The proof given here belongs to Bruns and Conca [19]. For the classical rings of pfaﬃans, the results of this chapter are borrowed from Baetica [7]. We have no doubt that the classical rings of symmetric minors St (X) are also F -regular, but this is not yet established. Bruns, R¨omer and Wiebe [26] have recently proved that Rt (X), with respect to its classical generic point, has a normal semigroup algebra as its initial algebra. Using general results about ﬂat deformations (see Conca, Herzog and Valla [39]) they obtained that Rt (X) has rational singularities in characteristic 0, and is F -rational in positive characteristic. In the same manner, they get that the determinantal rings R(X, δ) share the same properties. With respect to the case of generic symmetric matrices one knows that for an algebraically closed ﬁeld of characteristic 0 the Rees algebra R(It ) has rational singularities; see Bruns [17, Remarks (3.4)(b)]. By Boutot’s result [14] it follows that the K-subalgebra At of K[X] generated by all t-minors of X has also rational singularities. The positive characteristic case is still open.

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