Étale Cohomology (PMS-33), Volume 33 9781400883981

Table of contents :
Contents
Preface
Terminology and Conventions
Chapter I. Etale Morphisms
§1. Finite and Quasi-finite Morphisms
§2. Flat Morphisms
§3. Etale Morphisms
§4. Henselian Rings
§5. The Fundamental Group:Galois Coverings
Chapter II Sheaf Theory
§1. Presheaves and Sheaves
§2. The Category of Sheaves
§3. Direct and Inverse Images of Sheaves
Chapter III. Cohomology
§1. Cohomology
§2. Cech Cohomology
§3. Comparison of Topologies
§4. Principal Homogeneous Spaces
Chapter IV. The Brauer Group
§1. The Brauer Group of a Local Ring
§2. The Brauer Group of a Scheme
Chapter V. The Cohomology of Curves and Surfaces
§1. Constructible Sheaves: Pairings
§2. The Cohomology of Curves
§3. The Cohomology of Surfaces
Chapter VI. The Fundamental Theorems
§1. Cohomological Dimension
§2. The Proper Base Change and Finiteness Theorems
§3. Higher Direct Images with Compact Support
§4. The Smooth Base Change Theorem
§5. Purity
§6. The Fundamental Class
§7. The Weak Lefschetz Theorem
§8. The Kiinneth Formula
§9. The Cycle Map
§10. Chern Classes
§11. The Poincare Duality Theorem
§12. The Rationality of the Zeta Function
§13. The Rationality of L-Series
Appendix A. Limits
Appendix B. Spectral Sequences
Appendix C. Hypercohomology
Bibliography
Index

Citation preview

Etale Cohomology

Princeton M athem atical Series Editors:

Luis

A.

C ak earelli, John

N.

M athf.r,

and

Elias

M.

Stein

1. T he C lassical G roups by H ermann Weyl 3. A n In tro d u ctio n to D ifferential G eom etry by Luther P fahler E isenhart 4. D im en sio n T h eory by W. H u rew icz an d H. Wallman 8. T h eo ry o f L ie G roups: I by C. C h evalley 9. M ath em atical M ethods o f S tatistics by H arald C ram er 10. Several C om p lex V ariables by S. B ochner an d W. T. M artin 11. Intro d u ctio n to T op o lo g y by S. L efschetz 12. A lg eb raic G eo m etry and T opology e d ite d by R. H. Fox, D. C. Spencer, a n d A. IV. Tucker 14. T he T o p o lo g y o f Fibre B undles by Norm an S teen rod 15. F o u n d atio n s o f A lgebraic T opology by Sam uel E ilen berg an d N orm an S teen rod 16. F u n ctio n als o f F inite R iem ann S urfaces by M enahem Schijfer an d D on ald C. S p en cer 17. In tro d u ctio n to M athem atical L ogic, V ol. I by A lonzo Church 19. H om o lo g ical A lgebra by H. Cartan an d S. E ilen berg

20. The Convolution Transform by 1.1. H irschm an a n d D. V. W idder 21. G eo m etric In tegration T heory by H. W hitney 22. Q u alitativ e T h eory o f D ifferential E quations by V. V. N em ytskii a n d V. V. Stepan ov 23. T o p o lo g ical A nalysis by Gordon T. Whyburn ( revised 1964) 24. A naly tic F un ctions by Ahlfors, Behnke, Bers, G rauert e ta l. 25. C o n tin u o u s G eo m etry by John von Neumann 26. R iem ann S u rfaces by L. Ahlfors an d L. Sario 27. D ifferential and C om binatorial T opology ed ited by S. S. Cairns 28. C o n v ex A n aly sis by R. T. R ockafellar 29. G lobal A n aly sis ed ite d by D. C. S pen cer an d S. Iyanaga 30. S in g u lar In tegrals and D ifferentiability P roperties o f F unctions by E. M. Stein 3 1. P roblem s in A nalysis ed ited by R C. Gunning 32. Intro d u ctio n to F ourier A nalysis on E uclidean S paces by E. M. Stein an d G. Weiss 33. E tale C o h o m o lo g y by J. S. M ilne 34. P seu d o d ifferen tial O perators by M ichael E. Taylor 35.

T h ree-D im en sio n al G eo m etry and T opology, V olum e 1 by William P. Thurston, E d ited b y S ilvio L evy

36. R ep resen tatio n T heory o f S em isim ple G roups: A n O v erview B ased on E xam ples by Anthony W. K n app 37. F o u n d atio n s o f A lgebraic A nalysis by M asaki K ashiw ara, Takahiro Kaxvai, an d Tatsuo Kimura. T ran slated by Got o K ato 38. Spin G eo m etry by //. B laine Lawson, Jr., a n d M a rie-L o u ise M ichelsohn 39. T o p o lo g y o f 4 -M an ifo ld s by M ichael H. Freedm an a n d Frank Quinn 40. H y p o -A n aly tic S tructures: Local T heory by F rancois Treves 41. T he G lobal N o n lin ear S tability o f the M inkow ski S pace by D em etrios C hristodoulou a n d Sergiu K lainerm an 42. E ssays on F o u rier A nalysis in H onor o f Elias M. Stein ed ited by C. Fefferm an, R. Fejferm an, a n d S. W ainger 43. H arm onic A nalysis: R e al-V ariable M ethods, O rthogonality, and O scilla to ry Integrals b y E lias M. Stein 44. T opics in E rgodic T heory b y Ya. G. Sinai 45. C o h o m o lo g ical Induction and U nitary R epresentations by Anthony W. K n app an d D a vid A. Vogan, Jr.

s

Etale Cohomology

J. S. Milne

P R I N C E T O N U N I V E RS I T Y PRESS PRIN CETO N , NEW JERSEY

Copyright © 1980 by Princeton University Press Published by Princeton University Press, Princeton, New Jersey In the United Kingdom: Princeton University Press, Chichester, W est Sussex All Rights Reserved L ib ra ry o f C o n g r e ss C a ta lo g in g -in -P u b lic a tio n D ata

Milne, James S 1942Etale cohomology. (Princeton mathematical series : 33) Bibliography: p. Includes index. 1. Geometry, Algebraic. 2. Homology theory. 3. Sheaves, Theory of. I. Title. II. Series. QA564.M 52 514*23 79-84003 This book has been com posed in Monophoto Times Roman Princeton University Press books are printed on acid-free paper and meet the guidelines for perm anence and durability o f the Com mittee on Production G uidelines for Book Longevity o f the Council on Library Resources Printed in the United States o f America 5 7 9

10 8 6

ISB N -13: 9 7 8 -0 -6 9 1 -0 8 2 3 8 -7 (cloth) ISB N -10: 0 -6 9 1 -0 8 2 3 8 -3 (cloth)

La m er eta it eta le, m ais le reflux c o m m e n g a it a se faire sentir. H u g o , L es tra v a illeu rs de la m er.

Contents Preface Term inology and C onventions C hapter

C h apter

C hapter

C hapter

C hapter

C h a p te r

E tale M orphism s

ix xiii

I. §1. §2. §3. §4. §5.

Finite and Quasi-finite Morphisms Flat Morphisms Etale Morphisms Henselian Rings The Fundamental Group:G alois Coverings

IF

Sheaf Theory

46

§1. §2. §3.

Presheaves and Sheaves The Category of Sheaves Direct and Inverse Images o f Sheaves

46 56 68

III. §1. §2. §3. §4.

IV.

C ohom ology

3

4 7 20 32 39

82

Cohomology Cech Cohomology Comparison o f Topologies Principal Homogeneous Spaces

82 95 110 120

The B rauer G ro u p

§1. §2.

The Brauer Group o f a Local Ring The Brauer Group o f a Scheme

136 136 140

V.

The C ohom ology o f Curves and Surfaces

155

§1. §2. §3.

Constructible Sheaves: Pairings The Cohomology o f Curves The Cohomology o f Surfaces

155 175 197

The F und am en tal T heorem s

220

Cohomological Dimension The Proper Base Change and Finiteness Theorems Higher Direct Images with Compact Support The Smooth Base Change Theorem

220 222 227 230

VI. §1. §2. §3. §4.

§5. §6. §7. §8. §9. §10. §11. §12. §13. A p p e n d ix A . A p p e n d ix B. A p p e n d ix C.

Purity The Fundamental Class The Weak Lefschetz Theorem The Kiinneth Formula The Cycle Map Chern Classes The Poincare Duality Theorem The Rationality o f the ZetaFunction The Rationality o f L-Series

241 247 253 256 268 271 276 286 289

Lim its Spectral Sequences H ypercohom ology

304 307 310

B ibliography

313

Index

321

Preface

The purpose o f this book is to provide a com prehensive introduction to the etale topology, sheaf theory, and cohom ology. W hen a variety is defined over the com plex num bers, the complex topology may be used to define cohom ology groups th at reflect the structure of the variety m uch m ore strongly th an do those defined, for example, by the Zariski topology. F o r an arb itrary scheme the complex topology is not available, but the etale topology, whose definition is purely algebraic, may be regarded as a replacem ent. It gives a sheaf theory and cohom ology theory with properties very close to those arising from the complex topology. W hen both are defined for a variety over the com plex num bers, the etale and com plex cohom ology groups are closely related. On the other hand, when the scheme is the spectrum of a field and hence has only one point, the etale topology need not be trivial, and in fact the etale cohom ology of the scheme is precisely equivalent to the G alois cohom ology of the field. Etale cohom ology has achieved an im portance for the study of schem es com parable to th at of complex cohom ology for the study of the geom etry of com plex m anifolds or of G alois cohom ology for the study of the arithm etic of fields. The etale topology was initially defined by A. G rothendieck and developed by him with the aid of M. A rtin and J.-L. V erdier in order to explain W eil’s insight (Weil [1 ]) that, for polynom ial equations with integer coefficients, the com plex topology of the set of com plex solutions of the equations should profoundly influence the num ber of solutions of the equations m odulo a prim e num ber. In this, the etale topology has been brilliantly successful. We give a sketch of the explanation it provides. It m ust be assum ed the equations define a scheme proper and sm ooth over some ring of integers. The complex topology on the complex points of the scheme determ ines the complex cohom ology groups. The com ­ parison theorem says th at these groups are essentially the sam e as the etale cohom ology groups of the scheme regarded as a variety over the complex num bers. T he pro p er and sm ooth base change theorem s now show th at these last groups are canonically isom orphic to the etale co­ hom ology groups of the scheme regarded as a variety over the algebraic

closure of a finite residue field. But the points of the schem e with co­ o rdinates in a finite field are the fixed points of the F robenius operato r acting on the set of points of the schem e with coordinates in the algebraic closure of the finite field. T he Lefschetz trace form ula now shows th at the n um ber of points in the finite field m ay be com puted from the trace of the Frobenius o p e ra to r acting on etale cohom ology groups th at are essentially equal to the original com plex cohom ology groups. A large p art o f this book m ay be regarded as a justification of this sketch. T o give the reader som e idea o f the sim ilarities and differences to be expected between the etale and com plex theories, we consider the case of a projective n onsinguiar curve X o f genus g over an algebraically closed field k. If k is the com plex num bers, then X may be regarded as a one­ dim ensional com pact com plex m anifold 2f(C), and its fundam ental group 7ii(X, x) has 2g generators and a single, w ell-known relation. The m ost in­ teresting cohom ology group is H \ and H l(X (C), A) = Horn {nx(X, x), A) for a con stan t abelian sheaf A; for exam ple, H l(X(C), Z) = I 29. If k is arb itrary , then it is possible to define in a purely algebraic way a funda­ m ental g roup 7iil8(X, x) that, when k is o f characteristic zero, is the profinite com pletion o f n x(X, x). T he etale cohom ology group H l( X et, A) = H orn (ri\x*(X, x), A) for any co n stan t abelian sheaf A, b u t now Horn refers to continuous hom om orphism s. T h u s H l( X e,, A) = A 29, if A is finite or is the /-adic integers Z,. But H l( X el,Z ) = 0, for Z m ust be given the discrete topology, an d the im age of any co ntinuous m ap n\lg(X, x) -> Z is finite. Therefore the etale cohom ology is as expected in the first two cases but is an om olous in the last. It may seem th at the etale topology should be superfluous when k is the com plex num bers, but this is n o t so; the etale groups have one im p o rtan t advantage over the com plex groups, namely, th at if X is defined over a subfield k0 of /c, then any au tom orphism of k/k0 acts on «'(* « « , A). T he b o o k ’s first ch ap ter is concerned with the properties of etale m o r­ phism s, H enselian rings, an d the algebraic fundam ental group. It had been my original intention to state these w ithout proof, but this would have been unsatisfactory since one of the essential differences between etale sheaf theory an d the usual sheaf theory is th at trivial facts from point set topology m ust frequently be replaced by subtle facts from algebraic geom etry. O n the o th er h and to give a com plete treatm ent of these topics w ould require a book in itself. T hus C h ap ter I is a com ­ prom ise; alm ost everything ab o u t etale m orphism s and H enselian rings is proved and alm ost n o th in g a b o u t the fundam ental group. The pre­ requisites for this ch ap ter are a solid know ledge of basic com m utative algebra, for exam ple, the contents o f A tiyah and M acdonald [1], plus a reasonable understan d in g of the language o f schemes.

The next two chapters are concerned with the basic theory of etale sheaves and with elem entary etale cohom ology. The prerequisite for these chapters is som e know ledge of hom ological algebra and G alois cohom ology. The fourth chapter treats A zum aya algebras over schemes and the B rauer groups o f schemes. H ere it is assum ed th at the reader is fam iliar with the corresponding objects over fields. This chapter may be skipped. The fifth chap ter contains a detailed analysis of the cohom ology of curves and o f surfaces. The section on curves assum es a knowledge of the representation theory of finite groups and th a t on surfaces assum es a m ore detailed know ledge of algebraic geom etry than required earlier in the book. The sixth chapter proves the fundam ental theorem s in etale cohom ology and applies them to show the rationality of som e very general classes of zeta functions and L-series. The appendixes list definitions and results concerning limits, spectral sequences, and hypercohom ology that the reader may find useful. The m ost striking application of etale cohom ology, th at of Deligne to proving the W eil-R iem ann hypothesis, is not included, but anyone who reads this book will find little difficulty with D eligne’s original paper. Essentially the only results he uses th at are not included here concern Lefschetz pencils of odd fiber dim ension. However, we do treat Lefschetz pencils of fiber dim ension one, and the general case is very sim ilar and only slightly m ore difficult. I have tried to keep things as concrete as possible. O nly enough founda­ tional m aterial is included to treat the etale site and sim ilar sites, such as the flat and Zariski sites. In particular, the word topos does not occur. Derived categories are not used although their spirit pervades the last part of C hapter VI. F o r an account of the origins of etale cohom ology and its results up to the m id 1960s, I recom m end A rtin’s talk at the International C ongress in M oscow, 1966 [3 ]; for a “p o p u lar” account of the history of the Weil conjectures (which is intim ately related to the history of etale cohom ology) and of D eligne’s solution, I recom m end K atz’s article [2], and for a survey of the m ain ideas and results in etale cohom ology and their relations to their classical analogues, I recom m end D eligne’s A reata lectures [SG A . 4 |, A reata]. The best introduction to the m aterial from algebraic geom etry required for reading this book is provided by H artsh o rn e [2]. It is a pleasure to thank M. A rtin for explaining a num ber of points to me, R. H oobler for his com m ents on C h ap ter IV, and the Institute for Advanced Study and l’ln stitu t des H autes Etudes Scientifiques where parts o f the book were written.

Terminology and Conventions

All rings are N oetherian and all schemes are locally N oetherian. A variety is a geom etrically reduced and irreducible scheme of finite-type over a field, and a curve or surface is a variety of dim ension one or two. F o r a field k, ks o r fcsep is the separable algebraic closure of k and kal the algebraic closure. If K is G alois over k, then G al (K/k) or G(K/k) is the corresponding G alois g roup; Gk denotes G al (kjk). F o r a ring A, A* denotes the group of units of A and k(p) the field of fractions of A/ p , where p is a prim e ideal in A. F o r a scheme X , R(X) is the ring o f ratio nal functions on X , X t the set of points x of codim ension i (that is, such that dim Ox x = i), and X ( the set of points x of dim ension i (that is, such th at {x} has dim ension i). A geom etric point of A" is a m ap z - + X where z is the spectrum of a separably closed field. Sets is the category of sets, Ab the category of abelian groups, Gp the category o f groups, G-sets the category o f finite sets, on which G acts (continuously on the left), G-mod the category o f (discrete) G-modules, Sch/ X the category o f schemes over X, FEt / X the category o f schemes finite and etale over X , L F T /T the category o f schemes locally o f finitetype over X , and Fun(C, A) the category of functors from C to A. The symbols f^J, Z, Q, !R, € , f q denote respectively, the natu ral num bers, the ring of integers, the field of ratio n al num bers, the field of real n u m ­ bers, the field of com plex num bers, and the finite field of q elements. The symbols oap, fjuni G w, G a denote certain group schemes (II.2.18). An injection is denoted by a surjection by an isom orphism by %, a quasi-isom orphism (or hom otopy) by and a canonical iso­ m orphism by = . The sym bol X U Y m eans X is defined to be 7, or that X equals Y by definition. The kernel and cokernel of m ultiplication by n, M A M , are denoted respectively by M n and M [n). The em pty set and em pty scheme are both denoted by 0 . The symbol b » a m eans b is sufficiently greater than a.

Etale Cohomology

CHAPTER I /

Etale Morphisms

A flat m orphism is the algebraic analogue of a m ap whose fibers form a continuously varying family. F o r exam ple, a surjective m orphism of sm ooth varieties is flat if and only if all fibers have the sam e dim ension. A finite m orphism to a reduced schem e is flat if and only if, over any connected com ponent, all fibers have the sam e num ber of points (counting multiplicities). The image of a flat m orphism of finite-type is open, and flat m orphism s th at are surjective on the underlying spaces are epim orphism s in a very strong sense. An etale m orphism is a flat quasi-finite m orphism Y X with no ram ification (that is, branch) points. Locally Y is then defined by an equation T m 4- a l T m~ 1 -f • • • 4- am = 0, where a i 9. . . , am are func­ tions on an open subset U of X and all roots of the equation over a point of U are simple. An etale m orphism induces isom orphism s on the tangent spaces and so m ight be expected to be a local isom orphism . This is true over the com plex num bers if local is m eant in the sense of the complex topology, but the Zariski topology is too coarse for this to hold algebraically. However, an etale m orphism induces an isom orphism on the com pletions o f the local rings at a point where there is no residue field extension. M oreover, it has all the uniqueness properties of a local isom orphism . A local scheme is H enselian if, for any scheme etale over it, any section of the closed fiber extends to a section of the scheme. It is strictly H enselian, or strictly local, if any scheme etale and faithfully flat over it has a sec­ tion. T he strictly local rings play the sam e role for the etale topology as local rings play for the Z ariski topology. The fundam ental g roup of a scheme classifies finite etale coverings of it. F or a sm ooth variety over the com plex num bers, the algebraic funda­ m ental group is simply the profinite com pletion of the topological fundam ental group. T here are algebraic analogues for m any of the results on the topological fundam ental group.

§1. Finite and Quasi-Finite Morphisms Recall th at a m orphism of schem es f : Y -> X is affine if the inverse im age of any open affine subset U of X is an open affine subset of Y. If, m oreover, T { f ~ l(U), &Y) is a finite F (l/, 0 x)-algebra for every such U , then / is said to be finite. These conditions need only be checked for all V in som e open affine covering of X (M um ford [3, III.l, Prop. 5]). Exam ples o f finite m orphism s abound. Let X be an integral scheme with field of rational functions R(X), and let L be a finite field extension o f R(X). T he normalization o f X in L is a pair (X'9f ) where X' is an integral scheme with R(X') = L and f : X ' - > X is an affine m orphism such that, for all open affines U o f X , T( f ~ 1(U ),O x>) is the integral closure of T(U, (9X) in L. P ro po sitio n 1.1. I f X is normal and f \ X ' X is the normalization of X in some finite separable extension o f R(X), then f is finite. Proof. O ne has only to show th at T ( / _1((7), 0 X) is a finite F(U, (9X)algebra for U an open affine in X, but this is done in A tiyah-M acdonald [1, 5.17]. Remark 1.2. T he above pro p o sitio n holds for m any schemes X w ith­ out the separability assum ption, for exam ple, for reduced excellent schemes and so for varieties ([E G A . IV.7.8] and B ourbaki [2, V.3.2]). (A field is excellent and a D edekind do m ain A is excellent if the com ple­ tion K of its field of fractions K at any m axim al ideal of A is separable over K; any scheme of finite type over an excellent scheme is excellent.) P ro po sitio n 1.3. (a) A closed immersion is finite. (b) The composite of two finite morphisms is finite. (c) Any base change of a finite morphism is finite, that is, if f :Y -► X is finite, then so also is f {}r) : *(*') X' for any morphism X' -* X. Proof. These reduce easily to statem ents ab o u t rings, all of which are obvious. The “going up” theorem of C ohen-Seidenberg has the following geom etric interpretation. P ro po sitio n 1.4. Any finite morphism f : Y ^ X is proper, that is, it is separated, of finite-type, and universally closed. Proof. F o r any open affine covering (L/() of X , / restricted to f ~ l(Ui) -> U i is separated for all i, and so / is separated. (H artshorne [2, II.4.6]). T o show th at finite m orphism s are universally closed it suffices, according to (1.3c), to show th at they are closed, and for this it suffices, according to (1.3a,b), to show th at they m ap the whole space on to a closed set. T hus we m ust show th at f ( Y ) is closed. This re-

duces easily to the affine case with, for exam ple, / = ag where g: A -> B is finite. Let 3 = ker (g). Then / factors into spec B -> spec A /3 -► spec A. The first m ap is surjective (A tiyah-M acdonald [1 , 5 .1 0 ]), and the second is a closed im m ersion. F o r m orphism s X -► spec k, with k a field, there is a topological characterization o f finiteness. P ro po sitio n 1.5. Let f \ X - > sp ec k be a morphism of finite-type with k a field. The following are equivalent: (a) X is affine and T(Ar, Ox) is an Artin ring; (b) X is finite and discrete {as a topological space): (c) X is discrete; (d) / is finite. Proof. See A tiyah-M acdonald [1, C h ap ter V III, especially exercises 2, 3, 4]. A m orphism f : Y - * X is quasi-finite if it is of finite-type and has finite fibers, th at is, / ' l(x) is discrete (and hence finite) for all x e X. Similarly an /4-algebra B is quasi-finite if it is of finite-type and if B X be quasi-finite and X' -> X arbitrary. If x' x under X' -> X, then the fiber P r o p o s it io n 1.7.

/(*') = and hence is discrete.

® k{x) k(x')

If / : y -> X is finite and U is an open subschem e of 7, then it follows from the above p ro p o sitio n th at U -> X is quasi-finite. T he rem arkable thing is th at essentially every quasi-finite m orphism comes in this way. T heorem

1.8.

(Z ariski’s M ain Theorem ). I f X is a quasi-compact, then

any separated, quasi-finite morphism f : Y - + X factors as Y 7' X where / ' is an open immersion and g is finite. Proof. T he m ost elem entary p ro o f m ay be found in R aynaud [3, C h ap ter IV]. W e sketch the deduction o f (1.8) from the following affine form of it, proved in R aynaud [3, p. 42]: let B be an /4-algebra th at is quasi-finite, and let A be the integral closure of A in B\ then the m ap spec B -> spec A is an open im m ersion. C onsider a schem e X. A ssociated with any quasi-coherent sheaf A of ^ - a lg e b r a s , there is a pair ( X \ g) where X ' is a scheme and g \ X ’ -> X is an affine m orphism such th at g*(9x> = A (H artshorne [2, II. Ex. 5.17] and [E G A . 1.9.1.4]). O ne writes X' = spec A. F o r any A'-scheme Y ^ X, to give an Ar-m orphism Y -► X' is the sam e as to give a hom om orphism A -►f^(9y o f fi^-algebras. N ow let f : Y - > X be separated and of finite-type. T hen f^OY is a quasi-coherent tfV algebra [E G A . 1.6.7.1], and the ^ - a lg e b r a A such th at r (U , A ) is the integral closure o f T(L/, 0 X) in T(U, / * 0 y) for all open affines U cz X is also quasi-coherent [EG A . II.6.3.4]. T he associated A'-scheme X ' = spec A is called the n o rm alization of X in Y. A ssume further th at / is quasi-finite. It follows easily from the affine result q uoted above, th at the m orphism Y -+ X' induced by the inclusion A c= f J 9 y is an open im m ersion. N ow let (A,) be the family of all coherent tfV subalgebras of A . O ne checks easily th at the m orphism Y -* spec A h induced by the inclusion A { c= f^(9Y, is an open im m ersion for all suffi­ ciently large A { (using the fact th at A = t )A com pare the p ro o f of R aynaud [3, p. 42, C or. 2(2)]). Since spec A xis finite over X, this proves (1.8). Remark 1.9. Z arisk i’s m ain theorem is, m ore correctly, the m ain theo­ rem of Z ariski [2]. T here he was interested in the behavior of a singularity on a norm al variety under a birational m ap. The original statem ent is essentially th at if f : Y -> X is a birational m orphism of varieties and (9XtX is integrally closed, then either f ~ l(x) consists of one point and the inverse m orphism f ~ 1 is defined in a neighborhood of x or all com po­ nents of / - 1 (x) have dim ension > 1 . T o relate this to G rothendieck’s version, note th at if in (1.8) A" and Y are varieties, / is birational and X is norm al, then g is an isom orphism . F o r a m ore com plete discussion of the theorem , see M um ford [3, III.9]. C orollary

1.10.

Any proper, quasi-finite morphism f : Y - > X i s finite.

Proof. Let / = gf' be the factorization as in (1.8). As g is separated and / is proper, / ' is proper. (Use the factorization f

= f r ) o V r : Y Y x x Y ' ^ Y ’.)

Thus / ' is an im m ersion with closed image, th at is, a closed im m ersion. N ow both / ' and g are finite. Remark l . l I. T he separatedness is necessary in both of the above results; for if X is the affine line with the “origin doubled” (H artshorne [2, 11.2.3.6]), and f : X A 1 is the natural m ap, then / is universally closed and quasi-finite, but is not finite. (It is even fiat and etale; see the next two sections.) Exercise 1.12. Let f : Y -► X be separated and of finite-type with X irreducible. Show th at if the fiber over the generic point rj is finite, then there is an open neighborhood U of rj in X such th at / _1(l/) -* U is finite. §2. Flat Morphisms A hom om orphism f : A -► B of rings is flat if B is flat when regarded as an ,4-m odule via / . Thus, / is flat if and only if the functor - ® aB from /1-modules to B-m odules is exact. In particular, if 3 is any ideal of A and / is flat, then 3 ® A B -> A ® A B = B is injective. The converse to this statem ent is also true. P ro po sitio n 2.1. A homomorphism f :A -*> B is flat if (a ® b\-> f(a)b): 3 ® A B -> B is injective for all ideals 3 in A. Proof. Let g:M ' -► M be an injective m ap of /1-modules where, fol­ lowing A tiyah-M acdonald [1, 2, 19], we m ay assum e M to be finitely generated. Case (a) M is free. We prove this case by induction on the rank r of Af. If r = 1, then we m ay identify M with A and M ' with an ideal in A; then the statem ent to be proved is the statem ent given. If r > 1, then M = © A /2 with M x and M 2 free of rank < r . C onsider the exact com m utative diagram :

0 ------------- ► g, 0 -------►

►M

>M2

9

92

►0

►M ' --------►p g { M ') -------►0

W hen tensored w ith B, the to p row rem ains exact, and p , an d g 2 rem ain injective. This implies th at g N —

> M* i

-h

►M --------►0 0

0 -------> N ------- ►h ~ l g(M') -------> M' ------- > 0. By case (a) i ® 1 is injective, and it follows th at g ® 1 is injective. P ro po sitio n 2.2. / / f : A -> B is flat, then so also is S ' 1A -► T ~ l B for all multiplicative subsets S cz A and T c= B such that f ( S) cz T. Con­ versely, if A f - 1(„) -> Bn is flat for all maximal ideals n o f B, t/ien A -> B is flat. Proof. S~ l A -> S ' l B is flat according to A tiyah-M acdonald [1,2.20], and S ~ l B -> T ~ l B is flat according to A tiyah-M acdonald [1, 3.6]. F o r the converse, let M' -> M be an injective m ap of ^-m odules. To show th at B ® A M ’ -+ B ® A M is injective, it suffices to show th at

Bn ® B (B ® A M')

Bn ® B (B ® A M)

is injective for all n, but this follows from the flatness of A v -> Bn with P = f ~ Hn) and the isom orphism Bn ® B (B ® A N) % Bn ® Ap (Av ® A JV), which exists for any ,4-m odule N\ Remark 2.3. If a e A is not a zero-divisor and f : A - + B is flat, then f(a) is not a zero-divisor in B because the injectivity of (x ax): A -► A implies th at of (x h-* f(a)x):B -► B = A ® A B. Thus, if A is an integral dom ain and B ^ 0, then / is injective. Conversely, any injective h om o­ m orphism f : A -+ B, w here A is a D edekind dom ain and B is an integral dom ain, is flat, In proving this, we may localize and hence assum e that A is principal. A ccording to (2.1), it suffices to prove that for any ideal 3 ^ 0 of A, 3 ® A B -» B is injective, but 3 ® A B is a free B-m odule of rank one, and we know th at the g enerator o f 3 is not m apped to zero in B. A m orphism / : Y -* X of schemes is flat if, for all points y of Y, the induced m ap (9x,ny) -* @y,y is flat. E quivalently / is flat if for any pair V and U of open affines of Y and X such th at f ( V ) c= U, the m ap T((7, C0X) -> T(K, @Y) is flat. F ro m (2.2) it follows that the first condition needs only to be checked for closed points y o f Y. P ro po sitio n 2.4. (a) An open immersion is flat. (b) The composite of two flat morphisms is flat. (c) Any base extension o f a flat morphism is flat. Proof, (a) and (b) are obvious. (c) If f:A-^> B is flat and A -> A' is arbitrary, then to see th at A' -► B ® A A' is flat, one m ay use the canonical isom orphism (B ® A A ) ® A> M & B ® A M, which exists for any ^ '-m o d u le M.

In o rder to get less trivial exam ples of flat m orphism s we shall need the following criterion. P ro po sitio n 2.5. Let B be a flat A-algebra, and consider h e B. I f the image o f b in B/mB is not a zero-divisor for any maximal ideal m of A , then B/(b) is a flat A-algebra. Proof. After applying (2.2), we may assum e th at A -► B is a local h om om orphism of local rings. By assum ption, if c e B and be = 0, then c e m B. W e shall show by induction that in fact c e m rB for all r, and hence c e f ] m rB cz Q t f = (0), where n is the m axim al ideal of B. Assume that c e mrB, and w rite C =

X «A

where the ax form a m inim al generating set for m r. Then 0 = be = Y, aflib, i

and so, by one o f the stan d ard flatness criteria (proved in (2.10b') below), there are equations btb = X aUb'i j

w ith the b'j e B, a e A such th at X afiij = 0 i for all j. F rom the choice of the ah all a e m. Thus btb e ntB, and since b is n o t a zero-divisor in B/mB, this implies th a t bt e mB. T hus c e mr+ lB, which com pletes the induction. We have show n that b is not a zerodivisor in B, and the sam e argum ent, with A replaced by ,4 /3 and B by B /3B , shows th at b is not a zero-divisor in B /3 B for any ideal 3 of A. Fix such an ideal, and consider the exact com m utative diagram : 0

0

3® B

B

B /3 B

0

B

£ —» B / 3 B

0

B/ b)

♦ (B/(b) )/3(B/(6)) -------►0

0

in which b m eans m ultiplication by b. An ap plication of the snake lem m a shows that 3 M" o f A-modules is exact whenever B ® A Af' B ® A M -> B ® A M" is exact; (c) af : spec B -► spec A is surjective; (d) for every maximal ideal m of /l,/(m )B ^ B. In particular, a flat local homomorphism of local rings is automatically faithfully flat. Proof, (a) => (b). Suppose th at Af' ^ Af tensoring with B. T hen im (g2g i) = 0 because

M" becom es exact after

B ® A im (9 i9 i) = im ((1 0 g 2)(\ ® g x)) = 0, and im (gt) = ker (g 2) because B ® (ker 02/im g {) = ker (1 (x) gf2)/im (1 0 gf) = 0. (b) => (a). Af Af 0 is exact if and only if Af = 0. (a) => (c). F o r any prim e ideal p of A, B ® A k(p) # 0, and so af ~ l(v) — spec {B ® A k(p)) is nonem pty. (c) => (d). This is trivial. (d) => (a). Let x e Af, x ^ 0. Because / is flat, it suffices to show that B ® A N ^ 0, where N = Ax cz Af. But N » A /3 for som e ideal 3 of A, and hence B ® N % B /3B. If in is a m axim al ideal of A containing 3 , then 3 B c /(tn )B ^ B, and so B /3B ^ 0. 2.8. Let f : Y X he flat; let y e Y , und /e£ x' be such that x = f ( y ) is in the closure {x'} o f |x '} . Then there is a / such that __ y e { / } and f ( y ’) = x'. Proof. T he x' such th at x e {x'} are exactly the points in the image of the canonical m ap spec (9X -> X. The corollary therefore follows from the fact th at the m ap spec Oy -► spec (9X induced by / is surjective. A m orphism f \ Y - * X is faithfully flat if it is flat and surjective. Ac­ cording to (2.7c), this agrees with the previous definition for rings. We now consider the question of flatness for finite m orphism s. The next theorem shows that, for such a m orphism f :Y -* X, flatness has a very explicit interp retatio n in term s of the properties of / * 0 y as an tP^-module. Corollary

T h e o r e m 2.9. Let Af he a finitely generated A-module. The following are equivalent: (a) M i s flat; (b) Afm is a free A m-module for all maximal ideals m of A; (c) Af is a locally free sheaf on spec A; (d) Af is a projective A-module. Moreover, if A is an integral domain, they are equivalent to: (e) dim fc(p) (Af 0 ^ /c(p)) is the same for all prime ideals p o f A.

Proof, (d) => (a). This im plication does not use the finite generation of M. As tensor p roducts com m ute with direct sums, any free m odule is flat, and any direct sum m and of a flat m odule is flat. (b) => (c). Let m be a m axim al ideal of A, and let x l 9. . . 9xr be ele­ m ents of Af whose im ages in M m form a basis for M m over A m. Then the hom om orphism g( at, . . . , ar) = X aix h

g :A r -* M,

induces an isom orphism A ar -> M a for som e a e A, a $ tn,because the kernel and cokernel of g are zero at m and, being finitely generated, have closed su p p o rt in spec A. (c) => (a). Let a l9. . . 9ar be elem ents of A such th at the ideal (al9. . . , ar) = A and M a. is a free / ^ -m odule for all i. Let # = H A ar T hen B is faithfully flat over A, and B ® A M = . is clearly a flat fl-m odule. It follows that Af is a flat /4-m odule (apply (2.7b)). T o prove the rem aining im plications, (a) => (d), (a) => (b), we shall need the following lemm a. L e m m a 2.10. Let 0 - > N - > F ^ > M - + 0 b e a n exact sequence o f A modules with N a submodule of F. (a) I f M and F are flat over A , then N n 3 F = 3 N for all ideals 3 of A. (b) Let M be flat and F free, with basis ( y t) over A. If

n = Z «i-Vi g N, then there exist nt e N such that n = X aini(b') Let M be any flat A-module. If

I

=

i

0,

a{ e A, xi e M , then there are equations Xi = X aiJX'j j with x) e M, axj e A, such that X aiau = 0 i for all j. (c) Let M be flat and F free. For any finite set [nl9. . . , nr} o f elements o f N, there exists an A~linear map f : F - > N with f(nj) = nj9 j — 1, . . ., r.

§2.

Proof,

13

F L A T M O R P H IS M S

a. From the given exact sequence, we obtain exact sequences,

0 -------> N n 3 F ---------- ►3 F ------------ ►3 M ---------- ►0, 3 ® N -------- > 3 ® F

►3 ® M

►0.

As M and F are flat, 3 ® F and 3 ® M may be identified with 3 F and 3 M , and then the image of 3 ® N in 3 ® F = 3 F becom es identi­ fied with 3 N. But from the first sequence, this is also N n 3 F. (b) Let 3 be the ideal generated by the n{ occurring in

= X

n

Then « g J V o 3 F = 3 N , and so there are n, e N such that

« = X ai"i(b') W rite M as a quotient of a free m odule, as in (b), and let a tx j + • • • + a rx,. = 0. It is possible to choose F so th at it has a basis (^ f) with g(Vi) = A'„ / = 1 Then w = X a^ t 6 j v > and so it may be w ritten

« = X tii e N. W rite

«i = >’i - X au)0> some a^. Then n

= X

=

X X

» -

j

i a ia u ) y j ’

i

and so X i

= 0

each j. A lso

=

^aijg(yj),

and so x'y may be taken to be g ( y j ) . (c) We use induction on r. Assume first th at r = 1, and write s

«i = X

j= 1

°jyij

where (y,) is a basis for F. Then »1 = X ain'j j= i for som e «'■ e N, and / m ay be taken to be the m ap such that /(y,v) = n), j = 1 , . . . , s, and f ( y t) = 0 otherwise. N ow suppose th at r > 1, and

there are m aps f u f 2:F

N such th at M n t) - n{ and

M " i ~ M " i)) = ni -

/ = 2 ,. . . , r.

T hen f:F -

jv,

/(>-) = M y ) +

My)

-

hMy)

has the required property. W e now com plete the p ro o f of (2.9). (a) => (d). Em bed M into an exact sequence 0 -y v -* F -> M -0 in which F is free and N an d F are both finitely generated. A ccording to (2.10c), this sequence splits, and so M is projective. (a) => (b). W e may assum e th a t M is a finitely generated flat m odule over a local ring A. Let x 1?. . . , x r e M be such th at their images in M /m M form a basis for this over the field ,4/m. E m bed M in a sequence 0 -> J V - > F ^ M - > 0 where F is free with basis { y l 5. . . , y r} and g(yi) = x f. As iV cz m F, miV = N n (m F) = JV, and N is zero according to N ak ay am a’s lemma, (c) => (e). This is obvious. (e) => (c). Fix a prim e ideal p o f A , and choose elem ents x l 9. . . , xr of M a, som e a 4 P, whose images in M k(p) form a basis. According to N a k ay am a’s lem m a the m ap g :A '~ * M a,

g(a u . . . , ar) = XI ^ x «

defines a surjection -> M p. O n changing a, we may assum e th at # itself is surjective. F o r any prim e ideal q o f A a, the m ap k(q)r -* M k(q) is surjective, and hence is an isom orphism because dim (M k(q)) = r. T hus ker (g) cz qAra for any q, which implies that it is zero as Aa is re­ duced. T hus M a is free. Remark 2.11. Let / : Y -> X be finite and flat. I claim th at / is open. Follow ing (2.9), we may assum e th a t X = spec /I, Y — spec B, and £ % as an ,4-module. Let T r -h a { T r~ l -f • • • + ar be the characteristic poly­ nom ial over A of an elem ent b e B. A prim e ideal p of A is in the image of spec (Bb) -> spec (A) exactly when Bb/pB b is nonzero. But Bh/p B b % (B/pB)b and so this ring is nonzero exactly when 5 is not nilpotent in B/ pB or, equivalently, when som e coefficient of T r + a 1T r ~ 1 -f • • • 4- ar is nonzero in A/p. T hus the image of spec Bb in spec A is (J spec Aa., which is open. A m uch m ore general statem ent holds. T heorem 2.12.

Any flat morphism that is locally o f finite-type is open.

L emma 2.13. Let f : Y X be of finite-type. For all pairs (Z , U) where Z is a closed irreducible subset of Y and U is an open subset such that

U n Z # 0 , there is an open subset V o f X such that f ( U n Z ) 3 /( Z ) ^ 0 . (Here, / ( Z ) denotes the closure of the set f ( Z) ).

Vn

Proof. First note the following statem ents. (a) The lem m a is true for closed im m ersions. (b) The lemma is true for / if it is true for fred' ^red ~* ^red* (c) The lemm a is true for g f if it is true for / and g. For, if V' satisfies the conclusion of the lem m a for the pair (/(Z ), V) and the m ap g, then it also satisfies the conclusion for the pair (Z, U) and the m ap gf. (d) It suffices to check the lem m a locally on Y and X. (e) In checking the lem m a for a given Z, we note th at X may be replaced by /( Z ) , and hence may be assum ed to be irreducible. Using (a), (c), and (d), we may reduce the p ro o f to the case th at / is the projection A" x X X where X is affine. U sing (b) and (e), we reduce the p ro o f further to the case th at X — spec A, A an integral dom ain. Finally, using (c) again, we reduce the proof to the case th at / is the projection A 1 x X -+ X. Let Z be a closed irreducible subset of A*, say Z = spec B where B = A\T]/c\. We m ay assum e th at q ^ 0, for otherw ise the lem m a is easy. We may also assum e, according to (e), that q n A = (0), th at is, th at /( Z ) = X. Let K be the field of fractions of A , and let f = T (mod q). Since q contains a nonco n stan t polynom ial, t is algebraic over K , and so there is an a e A, a ^ 0, such th at at is integral over A. T hen Ba is finite over Aa, and so spec Ba spec Aa is surjective (A tiyah-M acdonald [1, 5.10]). Thus we are reduced to show ing th at the image of a n o n ­ em pty open subset U of spec Ba contains a nonem pty open subset of spec A. But if U contains (spec Ba)b, and b satisfies the polynom ial T m + a, T m~ 1 + • • • + a„, = 0, «, e Aa, then f ( U ) = (J (spec A a)at. Proof of (2.12). (C om pare H artsh o rn e [2, III. Ex. 9.1].) Let f \ Y -> X be as in the proposition. It suffices to show th at f ( Y ) is open. We may assum e that X is quasi-com pact. Let W = X - j \ Y ) and let Z Zn be the irreducible com ponents of W. Let z; be die generic point of Zj. If zj e f ( Y) , say zj = f (y), then (2.13) applied to ({y}, Y) shows th a t there exists an open U in X such th at f ( Y ) U n Zj 3 {zj}. But then f(Y)^ U n ( x -

\ J Z ^

{zj}.

and, as U and (X - U ; * j z this implies th at z} $ W, which is a contradiction. T hus z} e W, and, according to (2.8), all specializations o f Zj belong to W. T hus W => Z ; , W => [ j Z j = W, and f ( Y ) is open.

Remark 2.14. If / : Y -> X is finite and flat, then it is both open and closed. Thus, if X is connected, then / is surjective and hence faithfully flat (provided Y # 0 ) . Exercise 2.15. G ive an exam ple to show th at (2.12) is false w ithout the finiteness condition, even if / is surjective. (Start with the exam ple in ( 1.6 b)). If f : Y X is finite, and for som e y e F, (9y is free as an ®/(y)-m odule, then clearly r ( / _1(t/), &Y) is free over T (l/, (9X) for som e open affine U in X containing f(y). (See the p ro o f of (b) => (c) in (2.9.) Thus the set of points y e Y such th at (0y is flat over (9X is open in Y and is even n o n ­ em pty if X is integral and f ( Y ) — X. Again this holds m ore generally. T h e o r e m 2.16. Let f : Y -* X be locally of finite-type. The set o f points y e Y such that (Py is flat over &fiy) is open in Y; it is nonempty if X is integral. Proof. A reasonably self-contained p ro o f of this may be found in M atsu m u ra [1, C h ap ter V III]. See also [E G A , IV .11.1.1]. Recall th at, in any category with fiber products, a m orphism Y -> X is a strict epimorphism if the sequence

Y xx Y = t Y -

X

P2

is exact, that is, if the sequence o f sets H orn (X, Z ) -» H orn ( Y, Z) = 3 Horn ( Y x Y, Z) P2

is exact for all Z, th at is, if the first arrow m aps H orn (X , Z ) bijectively o n to the subset o f H orn (F, Z ) on which p* and p f agree. C learly the condition th at a m orphism o f schemes be surjective is not sufficient to imply th at it is a (strict) epim orphism (consider spec k -+ spec A, where A is a local A rtin ring with residue field k), but for flat m orphism s it is (almost). T h e o r e m 2.17. Any faithfully flat morphism f : Y -> X of finite-type is a strict epimorphism. It is convenient to prove the following result first. P r o p o s it io n

2.18.

I f f : A -> B is faithfully flat, then the sequence

0 _► A £ B ^ B®2 - » • • • - ► B®r

U B®r+ 1 -> • • •

is exact, where B®r = B ® a B ® • • • ® A B

(r times)

d' - 1 = £ ( - 1 ) % ei(b0 ® ' ' ' ® 6r - i ) = b0 ® * ’ ’ ® fej-i ® 1 ®

® • • * ® 6r _i.

Proof. The usual argum ent shows that drd r~ 1 = 0. We assum e first that / adm its a section, th at is, th at there exists a hom om orphism g:B -> A such th at g f = 1, and we construct a contracting hom otopy kr:B®r* 2 -> B®r +i . Define kr(b0 ® ■■■® br +l ) = g(b0)bl ® b2 ® ■■■® br +l ,

r > - 1.

It is easily checked th a t kr +l dr+l + drkr = 1, r > - 1 , and this shows th at the sequence is exact. N ow let A' be an A-algebra, let B' - A' ®,, B, and let / ' = 1 ® / : A' -* B'. The sequence corresponding to / ' is obtained from the sequence for / by tensoring with A' (because B®r ®,, A' % B'®r). Thus, if A' is a faithfully flat A -algebra, it suffices to prove the theorem for / ' . Take A' = B, and then / ' = (b>~* b ® \ ):B -> B ® A B has a section, namely, g(b ® b') = bb', and so the sequence is exact. Remark 2.19. A sim ilar argum ent to the above shows th at i f/ : A -» B is faithfully flat and M is an A-m odule, then the sequence 0 - M -* M ® A B

m

® A B®2

■

-> M ® B®r - ®d- ‘> M ® B®r+i is exact. Indeed, one may assum e again th at / has a section and con­ struct a contracting hom otopy as before. Proof of 2.17. W e have to show th at for any scheme Z and any m orphism h:Y -* Z such th at hp{ = hp2, there exists a unique m orphism g :X Z such th at g f = h. Case (a) X = spec A, Y — spec B, and Z = spec C are all affine. In this case the theorem follows from the exactness of 0

A -* B — — > B ® A B

(since % = Pi* °e \ = Pi)Case (b) X = spec A and Y = spec B affine, Z arbitrary. We first show the uniqueness o f g. I f g i , g 2- X Z are such th at g vf = g 2f , then g Y and g 2 m ust agree on the underlying topological space of X because / is surjective. Let x e X \ let U be an open affine neighborhood of g^x) ( = g 2(x)) in Z, and let a e A be such th at x e X a and g i ( Xa) = g2{ Xa) c U. T hen Bb, where b is the im age of a in B is faithfully flat over Aa, and it therefore follows from case (a) th at g t \ X a = g 2 \XaN ow let h: Y -> Z have hpj = hp2. Because of the uniqueness ju st proved, it suffices to define g locally. Let x e X , y e f ~ l{ x\ and let U be an open affine neighborhood of h(y) in Z. Then f ( h ~ l(U)) is open in X (2.12), and so it is possible to find a n a e A such th at x e X a cz f ( h ~ l(U)). I claim / “ T O is contained in h ~ \ U ) . Indeed, if f ( y {) = f ( y 2), there

is a y' e 7 x 7 such th at p x(y') = y x an d p 2(v') = 7 2 ^ if Yi-G h ~ l(U), then /i(>’i) = hpx{ y f) = hp2(y') = My 2) e I/, which proves the claim. If now b is the im age of a in B, then /z(7b) = h ( f ~ l( Xa)) cz U, and Bb is faithfully flat over T fl. T hus the problem is reduced to case (a). C use(c) G eneral case. It is easy to reduce to the case where X is affine. Since / is quasi-com pact, Y is a finite union, 7 = 7Xu • • • u 7„, of open affines. Let 7* be the disjoint union Yx 1L • • • 1L Then 7* is affine and the obvious m ap 7* -> X is faithfully flat. In the com m utative diagram , Horn (X, Z)

►H orn (7 , Z)

H orn (X, Z )

> Horn (7 * , Z ) = 4

Horn (7 x 7, Z)

H orn (7 * x 7*, Z),

the low er row is exact by case (b) and the m iddle vertical arrow is obviously injective. An easy diagram chase now shows th at the top row is exact. Exercise 2.20. Show th a t s p e c / c [ T ] s p e c fc[7 3, T 5] is an epi­ m orphism , but is not a strict epim orphism . R e m a r k 2 2 \. Let f : A -> B be a faithfully flat hom om orphism , and let M be an /1-module. W rite A f for the B-m odule /*A / = B ® AM. The m odule e0*Af= (B ® A B) ® B A f may be identified with B ® 4 A f where B ® A B acts by (bx ® b 2)(b ® m) = b xb ® b2m, and e x*M' may be identified with Af' ® A B where B ® A B acts by (bx ® b2)(m ® b) = b xm ® b2b. T here is a canonical isom orphism 0 : e x*M' -*> e0*Af arising from

ei*M' = (eJ)tM = (e0/)*A7 = *0,Af; explicitly it is the m ap M' ® A B -> B ® A A f,

(b ® m) ® b' h-> b ® (b' ® m),

m e Af.

M oreover, A/ can be recovered from the pair (A/', 0) because Af = {m e Af' 11 ® m = 0(w ® 1)} according to (2.19). C onversely, every pair (Af', 0) satisfying certain conditions does arise in this way from an T-m odule. G iven (p:Mf ® A B -+ B ® A M' define 0 i : B ® A M' ® A B -> B ® A B ® A Af', 02 •A f (X)^ B ® A B -> B ® A B ® A A f, 03: A f ® A B ® A B -+ B ® A M' ® A B

by tensoring 0 with idB in the first, second, and third positions respec­ tively. Then a pair (Af\ 0) arises from an /1-module Af as above if and only if 0 2 = 0 ! 0 3. T he necessity is easy to check. F o r the sufficiency, define Af = (m e Af' Jl (x) m = 0(m ® 1)}. There is a canonical m ap (b (x) mi—> bm):B ® A Af -> Af', and it suffices to show th at this is an isom orphism (and th a t the m ap arising from Af is 0). C onsider the diagram M’ ®AB = = } P®I

B ®A M’ ®A B

:p*Z' -> p*Z' satisfying

P* i(0) = P h W P u W ) Remark 2.24. The above is a sketch of p art of descent theory. A nother part describes which properties of m orphism s descend. C onsider a

20

C artesian square Y X be a faithfully flat morphism that is locally of finite-type. Then there exists an affine scheme X \ a faithfully flat quasi-finite morphism h : X f -> X, and an X-morphism q \X ' -* Y. Proof. O ne has to show th at, locally, there exist sequences satisfying the conditions of (2.6d) and of length equal to the relative dim ension of Y/X. (See [E G A . IV.17.16.2] for the details.)

§3. Etale Morphisms Let k be a field and Tc its algebraic closure. A k-algebra A is separable if A = A ® k k has zero Jaco b so n radical, th at is, if the m axim al ideals of A have intersection zero. P r o p o s i t i o n 3.1. Let A be a finite algebra over a field k. The following are equivalent: (a) A is separable over k; (b) A is isomorphic to a finite product o f copies o f k ; (c) A is isomorphic to a finite product o f separable field extensions of k; (d) the discriminant o f any basis o f A over k is nonzero (that is, the trace pairing A x A -* k is nondegenerate).

Proof, (a) => (b). F ro m (1.5) we know th at A has only finitely m any prim e ideals and th at they are all m axim al. N ow (a) implies th at their intersection is zero and (b) follows from the Chinese rem ainder theorem (A tiyah-M acdonald [1, 1.10]). (b) => (c). The Chinese rem ainder theorem implies th a t A/ I r, where Ir is the Jacobson radical of A , is isom orphic to a finite product kt of finite field extensions o f k. W rite [ K : k ]s for the separable degree of a field extension K/k. Then H o m k_alg (A, Tc) has

It W l elements. But Hom*_alg (A, Tc) % Hom*_alg (A, Tc\ and this set has [ A : k ] elem ents by (b). Thus [ A: k ] = £ [ * < :* ] . < l [ k t:k] = [A/Tr:k] < [ A : k ]. Since [A :£ ] — [ A : k f equality m ust hold th roughout and we have (c). (c) => (d). If A = Y[ ki> where the k,- are separable field extensions of k, then disc (A) = f ] disc (kf), and this is nonzero by one of the standard criteria for a field extension to be separable, (d) => (a). The discrim inants of A and A are the same. If x is in the radical of A, then xa is nilpotent for any a e A, and so T r ^ ( x a ) = 0 all a. Thus x = 0. A m orphism f : Y - > X th at is locally of finite-type is said to be un­ ramified at y e Y if &Y,y/ mx&Y,yls a finite separable field extension of k(x), where x = f(y). In term s of rings, this indicates that a hom om orphism f : A -> B of finite-type is unram ified at q e spec B if and only if p = / - l (q) generates the m axim al ideal in Bq and k(q) is a finite separable field extension of k(p). T hus this term inology agrees with th at in num ber theory. A m orphism / : Y -> X is unramified if it is unram ified at all y e Y. P ro po sitio n 3.2. Let f : Y - > X be locally of finite-type. The following are equivalent: (a) / is unramified; (b) for all x e X , the fiber Yx -► spec k(x) over x is unramified; (c) all geometric fibers of f are unramified (that is, for all morphisms spec k -> X, with k separably closed\ Y x x spec k -> spec k is unramified); (d) for all x e X, Yx has an open covering by spectra of finite separable k(x)-algebras; (e) for all x e X, Yx is a sum J J spec kh where the k,- are finite separable field extensions of fc(x);

(If f is of finite-type, then Yx itself is the spectrum of a finite separable k(x)-algebra in (d), and Yx is a finite sum in (e); in particular f is quasi-finite). Proof, (a) (b). This follows from the isom orphism (9y y % ©Vx.r (b) => (d). Let U be an open affine subset of Yx, and let q be a prim e ideal in B = T((7, &yJ- A ccording to (b), Bq is a finite separable field extension of k(x). Also k(x)

cz

B/q

cz

BJqBq

=

Bq,

and so B/q is also a field. T hus q is m axim al, B is an A rtin ring (AtiyahM acdonald [1, 8.5]), and B = f ] Bq, where q runs through the finite set spec B. This proves (d). A sim ilar argum ent show s th at (c) => (d), and (d) => (e) => (c) and (d) => (b) are trivial consequences o f (3.1). N otice th a t according to the above definition, any closed im m ersion Z c* X is unram ified. Since this does not agree with our intuitive idea of an unram ified covering, for exam ple, o f Riem ann surfaces, we need a m ore restricted notion. A m orphism of schem es (or rings) is defined to be etale if it is flat and unram ified (hence also locally of finite-type). P r o po sitio n 3.3. (a) Any open immersion is etale. (b) The composite of two etale morphisms is etale. (c) Any base change of an etale morphism is etale. Proof. After applying (2.4), we only have to check th at the three statem ents hold for unram ified m orphism s. Both (a) and (b) are obvious (any im m ersion is unram ified). Also, (c) is obviously true according to (3.1) if the base change is of the form k -> k \ where k and k' are fields, but, according to (3.2), this is all th at has to be checked. Example 3.4. Let k be a field an d P( T) a m onic polynom ial over k. T hen the m onogenic extension /e[T ]/(P ) is separable (or unram ified or etale) if and only if P is separable, th at is, has no m ultiple roots in Tc. This generalizes to rings. A m onic polynom ial P( T) e A [ T ] is separable if (P, P ) = A [ T ], th at is, if P '(T ) is a unit in 4 [ T ] /( P ) where P '(T ) is the formal derivative of P(T). It is easy to see that P is separable if and only if its image in /c(p)[T ] is separable for all prim e ideals p in A. Let B = 4 [ T ] /( P ) , where P is any m onic polynom ial in A[T~\. As an 4-m o d u le, B is free of finite rank equal to the degree of P. M oreover, B ® A k(p) = k (p )[T ]/(P ) where P is the im age of P in /c(p)[T]. It follows from (3.2b) th at B is unram ified and so etale over A if and only if P is separable. M ore generally, for any b e B, Bb is etale over A if and only if P' is a unit in Bb.

F o r example, B = A [ T ] / ( T r — a) is etale over A if and only if ra is invertible in A (for ra e A* ra e /c(p)*, all p T r — a is separable in k( p) [ T] all p). F o r algebras generated by m ore than one element, there is the following Jacobian criterion: let C = A \ T x, . . . , Tw], let P u . . . , P n e C, and let B = C /(P 1?. . . , P n); then B is etale over A if and only if the im age of det (dPi/dTj) in B is a unit. T hat B is unram ified over A if and only if the condition holds follows directly from (3.5b) below. (The P -m odule Qg/A has generators d T u . . . , dT n and relations £ “(dPJdTfi' dT} = 0.) T h at B is flat over A m ay be proved by repeated applications of (2.5). (See M um ford [3, III. §10. Thm . 3 '] for the details.) N o te th at if Y = spec B and X = spec A were analytic manifolds, then this criterion w ould indicate th at the induced m aps on the tangent spaces were all isom orphism s, and hence Y -> X w ould be a local isom orphism at every point o f Y by the inverse function theorem . It is clearly not true in the geom etric case th at spec B -> spec A is a local isom orphism (unless local is m eant in the sense o f the etale topology: see later). F o r example, consider s p e c Z [ T ] /( T 2 - 2) -► sp e c Z , which is etale on the com ple­ m ent o f {(2)}. P r o p o s i t i o n 3.5. Let f :Y - > X be locally o f finite-type. The following are equivalent: (a) / is unramified; (b) the sheaf is zero; (c) the diagonal morphism Ayjx'-Y -> Y x x Y is an open immersion. Proof, (a) => (b). Since £l\jX behaves well with respect to base change, one can reduce the p ro o f to the case th at Y = spec B and X = spec A are affine, then to the case that A -> B is a local hom om orphism of local rings and using N ak ay am a’s lem m a to the case where A and B are fields. Then B is a separable field extension of A , and it is a stan d ard fact th a t this implies th at Q^/A = 0. (b) => (c). Since the diagonal is always at least locally closed, we may choose an open subschem e U of Y x x Y such that A y/x: Y U is a closed im m ersion and regard Y as a subschem e of U. Let / be the sheaf of ideals on U defining Y. T hen 7 //2, regarded as a sheaf on T, is isom orphic to Qy,x and hence is zero. Using N ak ay am a’s lemma, one sees this implies th at Iy = 0 for all y e T, and it follows th at / = 0 on some open subset V of U containing Y. Then (F, &Y) = (K, &v) is an open subschem e of Y x Y. (c) => (a). By passing to the geom etric fiber over a point of X , we m ay reduce the problem to the case of a m orphism / : Y spec k where k

is an algebraically closed field. Let y be a closed point of Y. Because k is algebraically closed, there exists a section g: spec k -* Y whose image is {y}. T he following square is C artesian: y — ij

{ y}

> Y xx Y ujf,

1)

y.

Since A is an open im m ersion, this implies th at {y} is open in Y. M oreover, spec &y = { y } -> spec k still has the property th at spec (Sy ^ spec ( 0 y ® k Gy) is an open im m ersion. But (9y is a local A rtin ring with residue field /c, and so spec (9y ® k (9y has only one point, and Oy ® k (9y -> (Vy m ust be an isom orphism . By counting dim ensions over /c, one sees then th a t (9y = k. T hus, by applying (3.1) and (3.2), we have (a). C o r o l l a r y 3.6. Consider morphisms f : X - > S , g : Y X. If fg is etale and f is unramified, then g is etale. Proof. W rite g == p2Tg where T3: Y -+ Y x s X is the graph of g and p 2: Y x s X -> X is the projection on the second factor. Tg is the pull-back of the open im m ersion Ax/s:X -* X x s X by g x 1 : Y x s X -+ X x s X , and p 2 is the pull-back of the etale m ap fg: Y -+ S by f : X -► S. Thus, by using (3.3), we see th a t g is etale. Remark 3.7. Let J’: Y - > X be locally of finite-type. The an n ih ilato r of Q.\iX (an ideal in C°Y) is called the different br/x of Y over X. T hat this definition agrees with the one in num ber theory is proved in Serre [7, III.7]. T he closed subschem e o f Y defined by b Y/x is called the branch locus o f y over X. T he open com plem ent of the branch locus is precisely the set on which Q Y/x = 0, th at is, on which f : Y - + X is unram ified. The theorem of the purity of branch locus states th at the branch locus (if nonem pty) has pure codim ension one in Y in each of the two cases: (a) when / is faithfully flat and finite over X ; or (b) when / is quasi-finite and dom inating, Y is regular and X is norm al. (See A ltm an and K leim an [1, VI.6.8], [SG A . 1, X.3.1], and [SG A . 2, X.3.4].) P r o p o s i t i o n 3.8. If f : Y -> X is locally of finite-type, then the set o f points y of Y, such that (9Y y is flat over (9XJ{y) and = 0, is open in Y. Thus there is a unique largest open set U in Y on which f is etale. Proof. This follows im m ediately from (2.16). Exercise 3.9. Let / : Y -+ X be finite and flat, and assum e th at X is connected. T hen f ^ 6 Y is locally free, of con stant rank r say. Show that there is a sheaf of ideals T>\ix 011 called the discriminant of Y over

X , with the property th at if U is an open affine in X such th at B = n / - 1 (^ )i is free basis {h1?. . . , br} over yt = r(l7 , (9X\ then T(U, T)yix) is the principal ideal generated by det (TrB/A(bibj)). Show th at / is unram ified, hence etale, at all y e f ~ l(x) if and only if (Dy/*)* = GXtX (use (3.Id)). Use this to show th at if / is unram ified at all y e f ~ l (x) for som e x e X, then there exists an open subset U c X containing x such th at f : f ~ l( U) -* V is etale. Show th at if B — A\_T]/(P(T)) with P monic, then the discrim inant T>B/A = (D(P)), where D(P) is the discrim inant of P, th at is, the resultant, res (P, P'), of P and P'. Show also th at the different bfl/,4 = where t — T (mod P). (See Serre [7, III.§6].) The next propo sitio n and its corollaries show th at etale m orphism s have the uniqueness properties of local isom orphism s. P r o p o s i t i o n 3.10. Any closed immersion f :Y X that is flat (hence etale) is an open immersion. Proof. A ccording to (2.12), f ( Y ) is open in X and so, after replacing X by f ( Y) , we may a ssu m e / to be surjective. A s / is finite, f^.(9Y is locally free as an Cx-m odule (2.9). Since / is a closed im m ersion, this implies th at (9X % /* 0 y , th at is, that / is an isom orphism . Remark 3.11. By using Z ariski’s m ain theorem , we may prove a stronger result, nam ely, th at any etale, universally injective, separated m orphism / : Y -► X is an open im m ersion. (Universally injective is equiv­ alent to injective and all maps k( f ( y) ) -> k(y) radicial [EG A . 1.3.7.1].) In fact, by proceeding as above, one can assum e th at / is universally bijective, hence a hom eom orphism (2.12), hence proper, and hence finite (1.10). N ow / being etale and radicial implies th at /* 0 y m ust be free of rank one.

C o r o l l a r y 3 .12. I f X is c o n n e c t e d a n d f :Y -> X is e t a l e ( r e s p e c t i v e l y etale and separated), then any section s o f f is an open immersion (respec­ tively an isomorphism onto an open connected component). Thus there is a one-to-one correspondence between the set o f such sections and the set of those open (respectively open and closed) subschemes Yt o f Y such that f induces an isomorphism Y( -► X. In particular, a section is known when its value at one point is known if f is separated. Proof. O nly the first assertion requires proof. Assume first that / is separated. Then s is a closed im m ersion because fs - 1 is a closed im ­ m ersion, and / is separated (com pare with the p roof of (3.6)). A ccording to (3.6) s is etale, and hence it is an open im m ersion. Thus 5 is an isom or­ phism onto its image, which is both open and closed in Y. If / is only assum ed to be etale, then it is separated in a neighborhood of y and x = f(y), and hence the above argum ent shows that s is a local isom or­ phism at x.

C o r o lla r y 3.13. Let f , g \ Y ' - > Y be X-morphisms where Y f is a con­ nected X-scheme and Y is etale and separated over X. I f there exists a point y' e Y' such that f(y') = g(y') = y and the maps k(y) -+ k(y') induced by f and g coincide, then f = g. Proof. The graphs o f / and g, Y f , Yg: Y' -> Y' x x Y, are sections to P i -Y' x x Y -+ Y' The conditions imply th at T f and Yg agree at a point, and so Yf and Yg are equal (3.12). T hus f = p 2Yf = p 2Yg = g. W e saw in (3.4) above th at given a m onic polynom ial P( T) over A, it is possible to construct an etale m orphism spec C -> spec A by taking C’ = Bb where B = A [T ]/(P ) and b is such th at P'(7") is a unit in £*,. W e shall call such an etale m orphism standard. The interesting fact is th at locally every etale m orphism Y -> X is standard. G eom etrically this m eans th at in a neighborhood of any point x of X, there are functions fl j , . . . , a,, such th at Y is locally described by the equation T r + a xT r~ l + • • • + ar = 0, and the roots o f the eq u atio n are all simple (at any geo­ m etric point). T heorem 3.14. Assume that f \ Y - + X is etale in some open neighbor­ hood of y e Y. Then there are open affine neighborhoods V and U o f y and f(y), respectively, such that f \ V : V -* U is a standard etale morphism. Proof. Clearly, we may assum e th at Y = spec C and X = spec A are affine. Also, by Z ariski’s m ain theorem (1.8), we may assum e th at C is a finite A -algebra. Let q be the prim e ideal of C corresponding to y. We have to show th at there is a stan d ard etale A -algebra Bb such th at Bb « Cc for som e c $ q. It is easy to see (because everything is finite over A) th at it suffices to do this with A replaced by Ap, where p = / -1 (q)> th at is, th at we m ay assum e th at A is local and th at q lies over the m axim al ideal p of A. C hoose an elem ent t e C whose image 1 in C /pC generates /c(q) over /c(p), that is,7 is such th at /c(p)[7] = k(c\) c C/pC. Such an elem ent exists because C /pC is a pro d u ct /c(q) x C , and /c(q)/k(p) is separable. Let q' = q o A [r]. I claim th a t A [ t \ Cq is an isom orphism . N o te first th at q is the only prim e ideal of C lying over q' (in checking this, one may tensor with k(p)). T hus the sem ilocal ring C (x)^] A [ t \ is actually local and so equals Cp. As A [t] -> C is injective and finite, it follows th at

A [ t \ -» C

= Cq

is injective and finite. It is surjective because /c(q') -» k(q) is surjective, and N ak ay am a’s lem m a may be applied. A [r] is finite over A (it is a subm odule of a N oetherian A-module), and the isom orphism A[ t ] q>-» Cq extends to an isom orphism A [f]c> A Cc for some c $ q, c $ q'. T hus C may be replaced by A[t], th at is, we may assum e th at t generates C over A.

Let n = [fc(q):fc(p)]» so th at 1 , 7, . . . ,7 M-1 generate k(q) as a vector space over k(p). T hen 1, t , . . . , tn~ 1 generate C = A[ t ] over A (according to N ak ay am a’s lemma), and so there is a m onic polynom ial P( T) of degree n and a surjection h:B = A[ T ] / ( P ) -> C. Clearly P( T) is the characteristic polynom ial ofT in /c(q) over k(p) and so is separable. Thus Bb is a stan d ard etale /1-algebra for som e b U c* X where V -+ V' is etale and V' is affine n-space over U; (c) for any y e Y, there exist open affine neighborhoods V = spec C o f y and U = spec A o f x = f ( y ) such that

c = 4 [ T „ . . . , T „ ] /( P j,. . . , P J ,

m < n,

and the ideal generated by the m x m minors o f ( d PJ d T f is C; (d) / is flat and for any algebraically closed geometric point x of X, the fiber Y* -> x is smooth; (e) / is fiat and for any algebraically closed geometric point x o f X, Yf is regular; (f) f is flat and Qy/X is locally free of rank equal to the relative dimension of YjX. Proof. See [SG A . 1, II] or D em azu re-G ab riel: [ 1 , 1. §4.4]. Remark 3.25. (a) In the case th at / is of finite-type, conditions (d) and (e) may be parap h rased by saying th a t Y is a flat family of nonsingular varieties over X. (b) C ondition (b) shows that for a m orphism of finite-type etale is equivalent to smooth and quasi-finite. Finally we note th at (2.25) has an analogue for sm ooth m orphism s. 3.26. Let f :Y -> X be smooth and surjective, and assume that X is quasi-compact. Then there exists an affine scheme X ', a surjective etale morphism h\X' X, and an X-morphism g:X' -> Y. Proof. See [E G A . IV .17.16.3]. Exercise 3.27. (H ochster). Let A be the ring k [ T 2, T 3] localized at its m axim al ideal ( T 2, T 3) (that is, A is the local ring at a cusp on a curve); let B = A [S ]/(S 3Y2 + S 4- T 2), and let C be the integral closure of A in B. Show that B is etale over A , but th at C is not flat over A. (H int: show P r o p o s it io n

th at TS and T 2S are in C; hence TS e ( T 2: T 3)c . If C were flat over A , then ( T 2: T \ = ( T 2: T \ C = ( T 2, 7 3); b ut TS e ( T 2, T 3) w ould imply S e C.) Exercise 3.28. Let 7 and X be sm ooth varieties over a field k ; show th at a m orphism 7 -» X is etale if an d only if it induces an isom orphism on tangent spaces for any closed point o f 7. Exercise 3.29. D o H artsh o rn e [2, III. Ex. 10.6].

§4. Henselian Rings T h ro u g h o u t this section, A will be a local ring with m axim al ideal m and residue field k. T he hom om orphism s A -> k and ^4[T] -> k \ T ] will be w ritten as (a a) and ( / J). Tw o polynom ials / ( T ) , g( T) w ith coefficients in a ring B are strictly coprime if the ideals ( / ) and (g) are coprim e in £ [ T ] , th at is, if ( / , gf) = B [T ]. F o r exam ple, f ( T ) and T — a are coprim e if and only if /(a ) # 0 and are strictly coprim e if and only if f(a) is a unit in B. If A is a com plete discrete valu atio n ring, then HensePs lem m a (in n um ber theory) states the following: if / is a m onic polynom ial with coefficients in A such th at J factors as / = g0h0 with g 0 and h0 m onic and coprim e, then / itself factors as / = gh with g and h m onic and such that g — g0, h = h0. In general, any local ring A for which the co n ­ clusion of HensePs lem m a holds is said to be Henselian. Remark 4.1. (a) The g and h in the above factorization are strictly coprim e. M ore generally, if / , g e >4[T] are such th at J, g are coprim e in /c[T ] and / is m onic, then / and g are strictly coprim e in A [ T \ Indeed, let M — / 4 [ T ] / ( /, g). As / is m onic, this is a finitely generated ,4-m odule; as ( / , g) — /c[T ], ( / , g) -f m v4[T] = A [ T ] and m M = Af, and so N ak ay am a’s lem m a im plies th at M = 0. (b) T he factorization / = gh is unique, for let / = gh = g'h' with gt h, g \ h all m onic, g = g',H = W, and g and h coprim e. Then g and b! are strictly coprim e in A [T ], and so there exist r , s e A [ T ] such th at gr + h's = 1. N ow g' = g'gr T g'h's = gf'gfr + ghs, and so gf divides gf'. As b oth are m onic and have the sam e degree, they m ust be equal. T h e o r e m 4.2. Let x be the closed point o f X are equivalent: (a) A is Henselian;

—

spec A. The following

(b) any finite A-algebra B is a direct product o f local rings B = n * « (the B( are then necessarily isomorphic to the rings Bm., where the m, are the maximal ideals of B); (c) if f : Y - + X is quasi-finite and separated, then Y = Y0 i i y, 1 L • • • Yn where f ( Y 0) does not contain x and Y( is finite over X and is the spec o f a local ring, i > 1 ; (d) if f : Y - * X is etale and there is a point y e Y such that f ( y ) = x and k(y) — /c(x), then f has a section s: X -> Y; (d') let / i , . . . , f , e A [ T U . . . , Tn]; if there exists an a = (au . . . , « „ ) 6 kn such that f ( a) = 0, i = 1, . . . , w, and det ( ( d f / c T })(a) ) ^ 0, then there exists a b e An such that 5 = a and f ( b) = 0, i = 1 , . . . , n; (e) /ct / ( T ) e A [ T ] ; * / / factors as J = g0h0 with g0 monic and g0 and h0 coprime, t/icn f factors as f = #/i with g monic and g — g0,Ti = h0. Proof, (a) => (b). A ccording to the going-up theorem , any m axim al ideal of B lies over m. T hus B is local if and only if B/tnB is local. Assum e first that B is of the form 13 = y 4 [T ]/(/) with f ( T ) monic. If J is a pow er of an irreducible polynom ial, then B/mB = /c [ T ] /( /) is local and B is local. If not, then (a) implies th at f — gh where g and h are monic, strictly coprim e, and of degree > 1 . T hen B « A\ T ] K g ) x A[T~\/(h) (A tiyah-M acdonald [1 , 1 .10]), and this process may be con­ tinued to get the required splitting. N ow let B be an arb itrary finite A-algebra. If B is not local, then there is a h e B such th a t 5 is a nontrivial idem potent in B/mB. Let / be a m onic polynom ial such th at f(b) = 0; let C = A [ T ] /( /) , and let 0 :C -» B be the m ap th at sends T to b. Since C is m onogenic over A, the first part implies th at there is an idem potent c e C such that cj)(c) — 5. N ow 0(c) = e is a nontrivial idem potent in B\ B — Be x B(1 — e) is a n o n ­ trivial sp littin g, an d th e p r o cess m ay be co n tin u e d . (b) => (c). A ccording to (1.8), / factors into Y ^ Y' X with / ' an open im m ersion and g finite. T hen (b) implies th at Y' = {J spec (Or y) where the y run thro u g h the (finitely m any) closed points of Y'. Let Y* = U spec ( 0 Y',y)> where the y runs thro u gh the closed points of Y' th at are in Y. Then Y* is contained in Y and is both open and closed in Y because it is so in Y\ Let Y = Y* J J Y0. T hen clearly /(Y 0) does not contain x. (c) => (d). U sing (c), we m ay reduce the question to the case of a finite etale local hom om orphism A -> B such th at k(rn) = k(n) where n is the m axim al ideal of B. A ccording to (2.9b), B is a free A-m odule, and since k(n) = B ® A k(nt) = k(m) it m ust have rank 1, that is, A « B. (d) =* (d'). Let B = A [T „

= A[ t t , t „ ] ,

and let J ( T U . . . , 7„) = det (dfi/dTj). The conditions imply th at there exists a prim e ideal q in B lying over ni such that J(t l f . . . , t„) is a unit in It follows th at j \ t v, . . . , t„) is a unit in Bb for som e b e B, b $ q and thus th at Bb is etale over A (com pare (3.4); to convert Bb to an algebra of the form considered there, use the trick of the p ro o f of (3.16). N ow apply (d) to lift the solution in k" to one in A". (d') => (e). W rite f ( T ) = a j " 4- an„ J n' { + • • • 4- a 0, and consider the equations, * 0^0 = «0*

X qYx +

=

fll,

X o Y i + X 1Y 1 + X 2 Yq = a2, X r- \ Ys + Ys_ j Ys

!, an

where r = deg (g 0) and s = n — r. Clearly (b0 i . . . , br solution to this system o f equ atio n s if and only if f ( T ) = ( T 4- b ' ^ T ' - ' + ••• 4- b0)(csT s + The Jacobian of the eq uations is / y° \ Y 1 Y0 det

; ^ y, ;

*0 * ,X 0

\

X 2X ,X 0 1

= res (g, h i the resultant of g and /t,

/

where g = T r + X r _ 1 r ~ 1 4- • • • + * 0 an d h = FST S 4- * • • + T0. To prove (e) we have only to show th a t res (g0i h0) 7* 0. But res (0O>^0) can be zero only if both deg (0O) < v and deg (h0) < s, or g0 and /t0 have a com m on factor, and neither of these occurs. (e) => (a). This is trivial. C o r o l l a r y 4.3. If A is Henselian, then so also is any finite local A-algebra B and any quotient ring A/3. Proof. O bviously B satisfies co ndition (b) of the theorem , and A /3 satisfies the definition of a H enselian ring. P r o p o s i t i o n 4.4. If A is Henselian, then the functor B \-+ B ® A k induces an equivalence between the category o f finite etale A-algebras and the category o f finite etale k-algebras.

Proof. After applying (4.2b), we need only consider local 4-algebras B. The canonical m ap H om ^ (B, B ) -> H o m k (B ® fc, B' ® fc) is injective according to (3.13). T o see the surjectivity, note th at a khom om orphism B ® k -+ B' ® k induces an 4-h o m o m o rp h ism g:B -+ B' ® A k by com position w ith B B ® k and hence an 4-h o m o m o rp h ism (b' ® b\-+ b'g(b)):B' ® A B - + B ' ® A k. N ow apply (4.2d) to the m ap spec (Bf ® B) -* spec (B') to get an 4 hom om orphism B' ® A B -* B' th at induces the required m ap B -» B'. Thus the functor is fully faithful. T o com plete the proof, one only has to observe that any local etale /c-algebra k' can be w ritten in the form fc [r]/(/o (T )), w here f 0{T) is m onic and irreducible and then th at B = 4 [ T ] / ( / ( T ) ) , w here J( T) = f 0(T) and / is m onic, has the property that B ® A k = k'. So far, we have had no exam ples o f H enselian rings. The following is a generalization o f HenseFs lem m a in num ber theory. P ro po sitio n 4.5. Any complete local ring A is Henselian. Proof. Let B be an etale 4-alg eb ra, and suppose th at there exists a section s0:B -» k. We have to show (4.2d) th at this lifts to a sec­ tion s:B -+ A. W rite Ar = 4 /m r+ 1 ; if we can prove th at there exist com patible sections sr:B -> Ar, then these m aps will induce a section s:B -> lim 4 r = A. F o r n = 0 the existence of sr is given. F o r r > 0 the existence of sr follows from th at of sr _ x because of the property of the functor defined by an etale m orphism (3.22). Remark 4.6. (a) The last two propositions show th at the functor B\-> B ® A A gives an equivalence between the categories of finite etale algebras over A and over its com pletion when A is H enselian. U nder certain circum stances, notably when X is p roper over a H enselian ring 4 , this result extends to the categories of schemes finite and etale over X and over X = X ® A A. See Artin [2] and [5]. (b) T he result in (4.4) has the following generalization. Let X be a scheme p ro p er over a H enselian local ring 4 , and let X 0 be the closed fiber o f X. The functor Y h* Y x x X 0 induces an equivalence between the category of schem es Y finite and etale over X and those over X 0. W hen 4 is com plete, proofs of this m ay be found in A rtin [9, VII.11.7] and M u rre [1, 8.1.3]. T he H enselian case is deduced from the com plete case by m eans o f the approxim ation theorem (Artin [9, II]; see also Artin [5, T heorem 3.1]).

(c) P art (c) of (4.2) also has a generalization. Let / : Y -> X be sepa­ rated and of finite-type, w here X = spec A with A H enselian local. If y is an isolated point in the closed fiber Y0 of / , so th at y0 = {)’} YL Y'0 (as schemes), then Y = Y" i i Y ’ w ith Y" finite over X and Y" and Y ’ having closed fibers { y } and Y'0 respectively. F o r a proof, see A rtin [9,1.1.10]. (d) If X is an analytic m anifold over C, then the local ring at a point x of X is H enselian. (F o r this, and sim ilar exam ples, see R aynaud [3, VII.4].) Remark 4.7. Let f \ Y - + X be etale, and suppose k(x) = k(y) for som e y e Y, x = f(y). T hen the m ap on the com pletions G)x,x ®Y,y is etale and, according to (4.5) and (4.2), has a section, which implies (3.12) it is an isom orphism (9X x A C Y y. (See H artsh o rn e [2, III. Ex. 10.4] for a converse statem ent.) This may be used to give an exam ple o f an injective unram ified m ap of rings th at is not etale. Let A be a curve over a field, which has a node at x 0, and let / : Y X be the n orm alization of X in k(X). It is obvious this m ap is unram ified, and 0 XJ[y) c* 0 Y y is injective for all y, but if y lies over x 0, then (PXtXo -> &Yy is n o t an isom orphism because ®XtXQ is not an integral dom ain. (It has tw o m inim al prim e ideals; see H artsh o rn e [ 2 , 1.5.6.3].) A is a subring o f its com pletion A (since we require rings to be N oetherian). T hus any local ring A is a subring of a H enselian ring, and the sm allest such ring will be called the H enselization of A. M ore pre­ cisely, let i:A -+ A h be a local hom om orphism of local rings; A h is the Henselization of A if it is H enselian and if any other local hom om orphism from A into a H enselian local ring factors uniquely through i. Clearly (Ah, i) is unique, up to a unique isom orphism , if it exists. Before proving the existence of A h, we introduce the notion of an etale neighborhood of a local ring A. It is a pair (£, q) w here B is an etale T -algebra and q is a prim e ideal of B lying over m such th at the induced m ap k -> fc(q) is an isom orphism . Lemma 4.8. (a) If (£, q) and (B\ q') are etale neighborhoods of A such that spec B' is connected, then there is at most one A-homomorphism f : B -> B’ such that / _ 1(q') = q. (b) Let (£, q) and (B\ q') be etale neighborhoods of A; there is an etale neighborhood (B'\ q") of A with sp e c B" connected and A-homomorphisms f : B -> B \ f ':B ' -> B" such that f ~ l(q") = q, / / _ 1(q") = q'. Proof, (a) This is an im m ediate consequence of (3.13). (b) Let C = B ® 4 B \ The m aps B -* k(q) = k, B' -» fc(q;) = k induce a m ap C -> k. Let q" be the kernel of this m ap. Then (ZT, q"B"\ where B” = Cc with som e c $ q" such th at spec B" is connected, is the required etale neighborhood.

It follows from the lem m a th at the etale neighborhoods of 4 with connected spectra form a filtered direct system. Define ( 4 h, m h) to be its direct limit, (A h, m h) = lim (£, q). It is easy to check th at A h is a local 4 -alg eb ra with m axim al ideal m h, /4h/m h = k and A h is the H enselization of A. Also 4 h is obviously flat over A. (Slightly less trivial is the fact th at 4 h is N oetherian, th at is, we have n o t passed outside our category. This m ay be found in A rtin [1, III.4.2].) Exercise 4.9. Instead of building A h from below, it is possible to descend on it from above. Let A be the intersection of all local H enselian subrings H of A, containing A, which have the property th at rn n H = m H. Show th a t (4 , /), where i:A A is the inclusion m ap, is a H enselization of A. (H int: to show th a t A satisfies the definition of H enselian ring, note th at the factorization J = g 0h0 lifts to a factorization / - gh in any H ; use the uniqueness of the factorization to show th at g and h are in 0 " = *■) Examples 4.10. (a) Let A be norm al; let K be the field of fractions of A, and let K s be a separable closure of K. The G alois group G of K s over K acts on the integral closure B of A in K s. Let n be a m axim al ideal of B lying over m, and let D c G be the decom position group of it, th at is, D = {a e G|er(n) = n}. Let 4 h be the localization at n D of the integral closure BD of A in /Cf. (Here BD = { b s B \ a ( b ) = h all a e D] etc.) I claim th at 4 h is the H enselization of A. Indeed, if A h were not H enselian, there would exist a m onic poly­ nom ial f ( T ) th at is irreducible over A h but whose reduction J ( T ) factors into relatively prim e factors. But from such an / one can construct a finite G alois extension L of K f such th at the integral closure A' of A h in L is not local. This is a contrad ictio n since the G alois group of L over K f perm utes the prim e ideals of A lying over nD and hence cannot be a q uotient of D. T o see th at A h is the H enselization, one only has to show th a t it is a union of etale neighborhoods of A, but this is easy using (3.21). (b) Let k be a field, and let A be the localization of k [ T u .. ., T„] at ( T j , . . . , Tn). The H enselization of A is the set of pow er series P e k[ [ T u . . . , T„] ] th at are algebraic over A. (F or a good discussion of why this should be so, see A rtin [ 8]; for a proof, see A rtin [9, II.2.9].) (c) The H enselization of 4 / 3 is 4 h/ 3 4 h. This is im m ediate from the definition of the H enselization and (4.3). Every ring is a quo tien t of a norm al ring, and so it w ould have sufficed to construct 4 h for 4 norm al. This is the approach adopted by N ag ata [ !] ■

Remark 4.11. W e have seen th a t if A is norm al, then so also is A h. It is also true th at if A is reduced o r regular, then A h is reduced or regular and dim A b = dim A. These statem ents follow from (3.18) or (3.17). Let X be a schem e and let x e X. An etale neighborhood of x is a pair (F, y) where Y is an etale X -schem e an d y is a point o f Y m apping to x such th at k(x) = k(y). The connected etale neighborhoods of x form a filtered system and clearly the lim it lim T(F, 0 Y) — &x,xBy definition, A being H enselian m eans th at it has no finite etale extensions w ith trivial residue field extension (4.2d), except those of the form A -► A x. T hus if the residue field o f A is separably algebraically closed, then A has no finite etale extensions at all. Such a H enselian ring is called strictly Henselian o r strictly local. M ost of the above theory can be rew ritten for strictly H enselian rings. In particular, the strict Henselization of A is a pair ( 4 sh, i) w here / l sh is a strictly H enselian ring and i:A -+ A sh is a local hom o m o rp h ism such th at any other local h o m o ­ m orphism f : A - > H w ith H strictly H enselian extends to a local h om o­ m orphism f ' : A sh -* / / ; m oreover, / ' is to be uniquely determ ined once the induced m ap ,4sh/n tsh -+ H / m H on residue fields is given. Fix a separable closure ks of k. T hen ,4sh = lim B, where the limit runs over all com m utative diagram s B -------►ks

A in which A B is etale. If A = k is a field, then Ash is any separable closure o f k ; if A is norm al, then ,4sh can be constructed the sam e way as A h except th at the decom position gro u p m ust be replaced by the inertia g ro u p ; if A is norm al and H enselian, then A sh is the m axim al unram ified extension o f A in the sense of the num ber theorists. Finally, ( A/ 3 ) sh = A sh/ 3 A sh. Let X be a schem e and x -+ X a geom etric point of X. An etale neigh­ borhood of x is a com m utative d iagram : x -------►U

X with U -> X etale. Clearly (9sx tX — lim T (I/, Ov) where the limit is taken over all etale n eighborhoods of x. W e write or simply 0 XtX for this

limit. As we shall see, 0 XtX is the analogue for the etale topology of the local ring for the Z ariski topology, th at is, it is the local ring relative to a stronger notion o f localization. N ote th at its definition is form ally the same, for (9Xx = lim T( U, (9V) where the limit is taken over all Zariski (open) neighborhoods of x. Exercise 4.12. Study the properties of the ring Aqh = lim B, where the lim it is taken over all diagram s B

A -------> k in which spec B is connected and spec B -> spec A is finite and etale or an open im m ersion o r a com posite of such m orphism s. Exercise 4.13. Let X be a sm ooth scheme over spec A, where A is H enselian with residue field k. Show th a t X(A) -► X(k) is surjective. (Use 3.24b).

§5. The Fundamental Group: Galois Coverings In this section we sum m arize som e of the basic properties of the funda­ m ental group of a scheme. Proofs may be found in M urre [1]. The fundam ental g roup 7c1(^T, x0) of an arcwise connected, locally connected, and locally simply connected topological space with base point x 0 may be defined in tw o ways: either as the group of closed paths through x 0 m odulo hom otopy equivalence or as the autom orphism group of the universal covering space of X. T he first definition does not gen­ eralize well to schemes— there are simply too few algebraically defined closed paths— but the second does. T hus the im portant, defining property of the fundam ental group of a schem e is that it classifies in a natu ral way the etale coverings of X, etale being the m ost natural analogue of local hom eom orphism . T hus let X be a connected scheme, and let x X be a geom etric point of X. Define a functor F: ¥ Et / X -> Sets, where FEt/A" is the category of X-schem es finite and etale over X , by setting F(Y) = Horn* (x, Y). Thus to give an elem ent o f F(Y) is to give a point y e Y lying over x and a /c(x)-hom om orphism k(y) -► k(x). It may be show n th at this functor is strictly prorepresentable, th at is, th at there exists a directed set /, a p ro ­ jective system ( X h < ^)i6/ in F E t/ X in which the transition m orphism s (frij'-Xj -* Xi (i < j) are epim orphism s and elem ents / e F(Arl) such that (a) f t = 4>ij ° f p and

(b) the n atu ral m ap ljm H orn (X h Z ) -> F(Z) induced by the f is an isom orphism for any Z in F E t/X . T he projective system X = (X,, 0) will play the role of the universal covering space o f a topological space, and we want to define n x to be its auto m o rp h ism group. F o r an X -schem e Y, we write A utx ( Y ) for the group of X -autom orphism s o f Y acting on the right. F o r any Y e F E t/X , A u t* (7 ) acts on F(Y) (on the right), and if Y is connected, then this action is faithful, that is, for any g e F(Y), a h-> cr o g : Aut x (Y) -> F(Y) is injective (this follows from (3.13)). If Y is connected and A u tx (T ) acts transitively on F(Y) so th at the above m ap A u tx (T ) F(Y) is bijective, then Y is said to be Galois over X. F o r any Y e F E t/X there is a Y' e F E t/X th at is G alois and an A'-m orphism Y' Y (see 5.4 below). It follows th at the objects X t in X m ay be assum ed to be G alois over X. Now, given j > i, we can define a m ap A u tx (Xj) -+ A utx (X t) by requiring th at ^ ( a ) / j = f a o o o fj. W e define n x(X, x) to be the profinite group lim A utx (X t). Remark 5.1. (a) If x ' is any o th er geom etric point of X, then n x(X, x') is isom orphic to n x(X, x) and the isom orphism is canonically determ ined up to an inner au to m o rp h ism of n x(X, x). (b) If the above process is carried through for an arcwise connected, locally connected, and locally sim ply connected topological space X, a point x on it, and the category o f covering spaces of X, then one finds th at F is representable, th a t is, not merely prorepresentable, by the universal covering space X o f X. T hus 7i,(X , x) can be directly defined as the auto m o rp h ism g ro u p of X over X. (c) Let X be a sm ooth projective variety over C, and let X an be the associated analytic m anifold. The R iem ann existence theorem states that the functor, which associates with any finite etale m ap Y -► X the local isom orphism of analytic m anifolds Tan -► X an, is an equivalence of cate­ gories. T hus the etale fundam ental gro u p n x{X) and the analytic funda­ m ental g roup 7r!(X an) have the sam e finite quotients. It follows that their com pletions with respect to the topology defined by the subgroups of finite index are equal. But n x(X) by definition, is already com plete, and hence n x(X) ^ n^X™). The reason th at no algebraically defined fundam ental group can equal 7Ti(Xan) is th at the covering space does n o t exist algebraically (at least, not as an elem ent o f F E t/X ). (d) A theorem of G ra u e rt and R em m ert implies th at (c) also holds for nonprojective varieties. This fact is the basis of the p ro o f of the theorem com paring etale and com plex cohom ology. (See III.3.) Examples 5.2. (a) Let X = spec k, k a field. T he X t may be taken to be the spectra of all finite G alois extensions X, of k contained in k(x).

Thus 7ii(X, x ) is the G alois group over k of the separable closure of k(x) in k(x). C hanging x corresponds therefore to choosing a different sepa­ rable algebraic closure. (b) Let X be a norm al scheme. Let x = spec k(x)sep where x is the generic point of X. Then the X t- may be taken to be the norm alizations o f X in K {, where the K { run through the finite G alois extensions of k{x) contained in k(x) such th at the norm alization of X in K, is unramified. Thus n {{X, x) is the G alois group of /c(x)un over /c(x), where /c(x)un = u * , . (c) Let X = spec A, where A is a strictly H enselian local ring. Then 7i j(X, x) = {1} since F E t/X consists only of direct sums of copies of X. (If X is a scheme and x a geom etric point of X, then spec 0 X 5 is the algebraic analogue of a sufficiently small ball about a point x on a m ani­ fold; thus 7r ! = {1 } agrees with the ball being contractible.) (d) Let X = spec A, where A is H enselian. Let x be a geom etric point over the closed point x of X. The equivalence of categories F E t/X has a double pole at infinity and no o th er poles o r zeros. T hus / * ( co) has 2n poles, where n is the degree of / , an d no zeros. But - 2n = 2g - 2 > —2, where g is the genus of 7, and so n = 1 and / is an isom orphism . (The sam e argum ent show s th a t there is no nontrivial m ap Y -> P 1 th a t is etale over A 1 c= P l and tam ely ramified at infinity. In this case / * ( co) has < 2 n — (n — 1 ) = n -f 1 poles and no zeros, and —(n 4- 1 ) > 2g — 2 > —2 show s again th at n = 1 .) (g) Let X be a p ro p er schem e over a H enselian local ring A such that the closed fiber X 0 of X is geom etrically connected. Then it follows from (4.6b) that n {( X, x ) is canonically isom orphic to n x( X 0, x 0 ) if x is the com posite x 0 -> X 0 c> X. (h) Let U be an open subschem e of a regular scheme X, where the closed com plem ent Z = X — V has codim ension > 2. It follows im m e­ diately from the theorem of the purity of branch locus (3.7) and the description of the fundam ental g roup of a norm al scheme given in (b) above, th at n {(U, x) « nfi X, x) for any geom etric point x of U. It follows th at the fundam ental g roup is a birational invariant for varieties com plete and regular over a field k (because any dom inating rational m ap of such varieties is defined on the com plem ent of a closed subset of codim ension > 2 ; see H artsh o rn e [2, V.5.1]). (i) Let A" be a schem e p ro p er and sm ooth over spec A where A is a com plete discrete valuation ring w ith algebraically closed residue field of characteristic p. Assum e th a t Xg is connected, where K is the algebraic closure o f the field of fractions o f A, and th a t the special fiber of X is connected. Then the kernel of the surjective h om om orphism n ^ Xg , x) -> n {(X, x), for any geom etric poin t x of Xg, is contained in the kernel of any h om om orphism of nfi Xg, x) into a finite group of o rder prim e to p (see [SG A . 1, X.] or M urre [1]).

(j) Let X 0 be a sm ooth projective curve of genus g over an algebraically closed field k of characteristic p. T hen there is a com plete discrete valua­ tion ring A with residue field k and a sm ooth projective curve X over A such that X 0 = X ® A k. (The obstructions to lifting a sm ooth projective variety lie in second (Zariski) cohom ology groups and hence vanish for a curve ([SG A . 1, III.7])). It follows from (g) above that 7r1(A"0, 3c) % n^ X, x) and from (i) th at there is a surjection n ^ X g , x) n ^ X , x) with small kernel. T he com parison theorem (5.1c) shows that n ^X ^, x) is the profinite com pletion of the topological fundam ental group of a curve of genus g over C and hence is well-known. O n putting these facts together, one finds th at n f i X 0, x 0 ) (p) % G(p) where G is the free group on 2g gen­ erators uh (i = 1 , . . . , g) with the single relation (t/|

Wf ‘ t>f ') ' ' ' {UgVgUg l Vg l ) =

1

and where the superscript (p) on a gro u p H m eans replace H by lim H t with the Hi running thro u g h all finite q u otients of H of order prim e to p. This com pu tatio n of the prim e-to-p part of the fundam ental group of a curve in characteristic p was one of the first m ajor successes of G ro th en d ieck ’s approach to algebraic geom etry. Once nfiX, x) has been constructed, the im portant result is that it really does classify finite etale m aps Y -» X. T heorem 5.3. Let x be a geometric point of the connected scheme X. The functor F defines an equivalence between the category F E t/X and the category n {( X , x)-sets of finite sets on which n x( X , x) acts continuously (on the left). Proof. See M u rre [1]. Remark 5.4. Recall th at if Y -> X is finite and etale and both Y and X are connected, then we defined Y to be G alois over X if Aut* ( Y) has as m any elem ents as the degree of Y over X. W e wish to extend this notion slightly. If G is a finite group, then Gx (for any scheme X) denotes the scheme LLec, where X a = X for each a. N o te th at we may define an action of G on Gx (on the right) by requiring o \ X xto be the identity m ap X x -> X xa. Let G act on Y over X ; then Y -* X is Galois, with Galois group G, if it is faithfully flat and the m ap

Y X y ,^ |y o = ( y ^ ( y , y a ) ) is an isom orphism . Equivalently, Y -> X is G alois with G alois group G if there is a faithfully flat m orphism U -> X, locally of finite-type, such th a t Yv is isom orphic (with its G-action) to Gv . In term s of rings, a ring B 3 A on which G acts by /1-autom orphism s (on the left) is a G alois extension of

A with G alois group G if B is a finite flat /1-algebra and E n d ^ B ) has a basis {ex | a e G} as a left B-m odule with m ultiplication table (bo)(cr) = bo(c)oT,

b, c e B,

o, x e G.

We leave it to the reader to check th at these are all equivalent and th at if Y -> X is G alois with G alois g ro u p G, then so also is Y{X ) -> X ' for any X' X. Also the reader may check th a t any finite etale m orphism Y -> X can be em bedded in a G alois extension, th at is, th at there exists a G alois m orphism I" -► X th at factors th ro u g h Y -► X. (F or X norm al, this is obvious from the G alois theory of fields; the general case is not m uch m ore difficult; see, for exam ple, M u rre [1, 4.4.1.8].) Let X be connected, with geom etric p oint x. A ccording to (5.3), to give an x-pointed G alois m orphism Y X with a given action of G on Y over X is the sam e as to give a co ntinuous m orphism n x(X, x) G. We shall som etim es write n l(X, x; G) = H o m conls (rc^X, x), G) % set of xpointed G alois coverings of X w ith G alois g roup G (m odulo isom orphism ). As an application of etale cohom ology we shall show how to com pute n l( X 9 x; G) for som e schemes X w hen G is com m utative. N ote th a t in this case it is not necessary to give the G alois coverings x-points. Remark 5.5. Let A be a ring w ith no idem potents other than 0 and 1, th at is, such th at X = spec A is connected, and Jet X = (A"*, *;) be the universal covering schem e (as above) relative to some geom etric point x of X. Each X, is an affine schem e Spec and A = lim A i has the following properties: spec A = lim X ,; there is no nontrivial finite etale m ap A -►B; u i(X, x) = A uG (,4). T hus A may reasonably be called the etale closure of A. If A has only a finite num ber of idem potents, then A = f ] Aj (finite product), X = J J Xj (Xj = spec Aj), and F E t/X « ]~| F E t / X I f A has an infinite num ber o f idem potents, then A = \J Aj where each A} has only a finite num ber of idem potents, X = lim X j9 and F E t/X = fc‘fin}’’ FEt/X j [E G A . IV. 17]. T hus the study of F E t/X , X affine, essentially can be reduced to the case with X connected. Remark 5.6. We have seen th at n x(X, x) prorepresents the functor th at takes a finite group N to the set of isom orphism classes of x-pointed G alois coverings of X with a given action o f N. Clearly this property determ ines 7rx(X, x), and, since the category of finite groups is A rtinian, a standard theorem (G rothendieck [3 ]) shows th at the existence of n ^ X , x) is equivalent to the left exactness of the functor. It is n atu ral to ask w hether there exists a larger fundam ental group that, in addition, classifies coverings whose stru cture groups are finite group schemes. M ore precisely, consider a variety X over a field A:; fix a spec (kal)point x on X, and consider the functor th at takes a finite group scheme N over k to the set of isom orphism classes of pairs ( Y, y) where Y is a principal

hom ogeneous space for N over X (III.4) and y is a spec (kaZ)-point of Y lying over x. Unless X is com plete and k is algebraically closed, this functor will not be left exact, as one may see by looking at the cohom ology sequence of a short exact sequence of com m utative finite group schemes. However, when these conditions hold, then the functor is left exact and is represented by a profinite g roup scheme 7c1(X n , x), (N ori [1]). The fundam ental group 7Cj(X, x) is the m axim al proetale quotient of this true fundam ental group n i ( Xfu x). See also [SG A . 1, p. 271, 289, 309]. Comments on the Literature The first source for m uch of the m aterial in this chapter is G rothendieck’s B ourbaki talks [3 ] and [SG A . 1], and the ultim ate source is C hapter IV of [E G A .]. Some of the same m aterial can be found in R aynaud [3] and Iverson [2] in the affine case and in the notes of A ltm an-K leim an [1]. The first chapter of A rtin [9] also contains an elegant, if brief, treatm ent of etale m aps and H enselian rings. See also K urke-Pfister-R oczen [ I ] . The m ost useful intro d u ctio n to the fundam ental group is M urre [1]. The theory of the higher etale hom otopy groups is developed in A rtin-M azur

[!]•

C H A P T E R II

Sheaf Theory

In order to o b tain a sufficiently fine topology on a schem e X, it is necessary to relax the requirem ent th at a covering consists o f subsets of X. T hus a covering for the etale topology is a surjective family of etale m orphism s ([/, -► X). M ost of the definitions and results from sheaf theory generalize quite easily to the new situation. F o r exam ple, it is possible to define the sheaf associated w ith a presheaf, the stalk of a sheaf a t a geom etric point, an d the direct and inverse images of sheaves relative to a m orphism of schemes. As a first indication that the theory we obtain is interesting, we show th at the category of sheaves over the spectrum of a field is equivalent to the category o f discrete m odules over the G alois group o f the field. Any g ro u p schem e or quasi-coherent m odule defines a sheaf for the etale topology. T here are m any exam ples of sequences of sheaves th at are not exact relative to the Z ariski topology, but becom e exact when considered relative to the finer etale topology. Sometimes, particularly when considering p-torsion sheaves in characteristic p, it is necessary to use an even finer topology than the etale topology: the flat topology.

§1. Presheaves and Sheaves We shall be concerned with classes E of m orphism s o f schemes satisfying the following conditions: (ex) all isom orphism s are in E\ (e2) the com posite of tw o m orphism s in E is in E; (e3) any base change o f a m orphism in E is in E. A m orphism in such a class E will be referred to as an E-morphism. The full subcategory of Sch/A" of X -schem es whose structure m orphism is an £-m o rp h ism will be w ritten E/X. The following exam ples of such classes will be particularly im portant: (a) the class E = (Zar) of all open im m ersions; (b) the class E = (et) of all etale m orphism s of finite-type; (c) the class E = (fl) of all flat m orphism s th at are locally of finite-type.

In each of these exam ples, the £ -m orphism s are open (1.2.12) and any open im m ersion is an £-m orphism . This will be so in alm ost all examples. The £ -m orphism s are to play the role of the open subsets in an £-topology. N ow fix a base schem e X , a class £ as above, and a full subcategory C / X of Sch/X th at is closed under fiber products and is such that, for any Y -> X in C / X and any £-m o rp h ism U -> Y, the com posite U -> X is in C/X. An E-covering o f an object Y of C / X is a family (£/,• ^ Y)ie[ of £-m orphism s such th at Y — (Jg,(Xi). T he class of all such coverings of all such objects is the E-topology on C/ X. T he category C / X together with the £ -topology is the E-site over X , and will be w ritten (C/X)E or simply as X E. T he small E-site on X is (E/X) e and, in the case th at all £-m orphism s are locally of finite-type, the big E-site on X is (LFT/Ar)E where L FT /X is the full subcategory of Sch/X of X -schem es whose structure m orphism is locally of finite-type. By the Zariski site X Zar on X we shall always m ean the sm all (Zar)-site ((Z ar)/X )Zar; by the etale site X et we shall always m ean the sm all (et)-site ((et)/X )et, and by the flat site X fl, we shall always m ean the big (fl)-site (L F T /X )fl. N ote th at in the first two of these exam ­ ples, but not in the third, any m orphism in C /X is an £-m orphism (see (1.3.6) for the etale site). A small site is to be tho u g h t of as being analogous to a topological space in the usual sense and a big site to the category of all topological spaces and continuous m aps over a given topological space. N ote that the (Zar)-topology on a scheme in the above sense is the sam e as the Zariski topology in the usual sense, up to the identification of any open im m ersion with its image. Remark 1.1. The category C /X , together with the family of £-coverings, satisfies the following conditions: 1. if : U -> U is an isom orphism in C /X , then it is a covering; 2. if (Ui -► U)i is a covering, and, for each i, (Vy Uf j is a covering, then (Vy -> U)itj is a covering; 3. if (Ui -> U) is a covering, then, for any m orphism V -► U in C /X , (Ui x v V -> V) is a covering. T hus C /X , together w ith the family of £-coverings, is a Grothendieck topology in the sense of A rtin [1]. A presheaf P on a site (C /X )E is a co n trav arian t functor (C/X)° -> Ab. T hus P associates with each U in C /X an abelian group P ( U \ which we shall som etim es w rite as T(L/, P), and whose elem ents we shall som etim es refer to as sections o f P over U. W ith each m orphism f : U ' - + U of C/X , P associates a m orphism P ( f ) : P ( U ) -► P(U'). Since P ( f ) corresponds to the usual notion o f restricting a function to an open subset, we shall som etim es write it as resf o r simply as vcsVtV or (s s | U'). N ote, however,

the confusion in these last n o tatio n s; unlike the case of topological spaces, there will in general be m any m aps V -» U and their restriction m o r­ phism s need not agree. A morphism (f)\P -> P' of presheaves on (C/X)E is sim ply a m orphism of functors P -* P'. T hus cj) associates w ith each U in C / X a hom om orphism cf)(U):P(U) -* P f(U) and these h o m om orphism s com m ute with restriction maps. The presheaves an d presheaf-m orphism s over (C /X )£ form a category P (X £) = P( (C /X )£), which inherits m ost of the properties of Ab. If P and P' are presheaves on X E, then the presheaf Q with Q(U) = P{U) © P { U ) and Q ( f ) = P ( f ) © P( f' ) is the direct sum of P and P' in P ( X E). Thus P (X £) is an additive category. The kernel (respectively cokernel) of a m orphism :P -> P ' is the presheaf Q w ith Q(U) = ker ((/>(£/)) (respectively coker ((£/))) for all U and the obvious restriction maps. T he direct sum (respectively product) of a family of presheaves (Pi)ie/ is the presheaf Q with Q{U) = ffiP ^ L ) (respectively f ] Pi(U)). A sim ilar statem ent holds for direct and inverse limits of presheaves; a sequence P ^ P ^ P" o f presheaves is exact if and only if all of the sequences P'(U) - U)-+P(U) ~ {~U P \ U ) are exact. If one puts all of these obvious rem arks together, then one sees th at P (X £) is an abelian category satisfying AB5 and AB4* (A ppendix A). Examples 1.2. (a) F o r any abelian g roup M, the constant presheaf PM on X E is defined to have P M(U) = Af, all U, and P M( f ) = l M, all / , except th at we require P m ( 0 ) = 0 if 0 is the em pty scheme. (b) The presheaf G a has G a(U) = r ( U , (9V) regarded as an additive g roup for all U ; for any m orphism f : U - > U \ G a( f ) is the m ap V(U', (9V ) -> T((7, (9V) induced by / . (c) T he presheaf G m has G m{U) = T(L/, CVv)* for all V, and the obvious restriction maps. (d) Let F be a sheaf of ^ - m o d u le s in the usual sense of the Zariski topology. Define W(F) as the presheaf with W{F)(U) = T((7, F (9V) for all U and with the obvious restriction maps. F o r example, W(&x) = G a. If X is an S-schem e and F = Q |/s, then W(0.lXjS) should not be confused with U i-» T((7, Qt//S). T here is a canonical m ap from IT (Q i/s) to this second presheaf, but it is an isom orphism only for the small etale (or coarser) site. P r o p o s i t i o n 1.3. For any U / X etale, H/(Q^/S)((/) = F(U, £ll/s). Proof. F o r any sequence o f m orphism s U -> X 5, there is a canonical m ap (g ® df\->gdf):C% ® Q |/s -► Q///s. In the case that S = spec K, X — spec A, U = spec C, and the m ap U -> X is a standard etale m orphism (1.3.14), this m ap is an isom orphism because any Rderivation D:A -» M, where M is a C-m odule, extends uniquely to an

^-d eriv atio n D ' : C - * M. As any etale m orphism is locally of this form (1.3.14), we see th at 0 V 0 % Q 1UIS as sheaves (in the usual sense) on U and that H i / , o v 0 a lxis) a n i ; , a b /s)The usual criterion for a presheaf to be a sheaf has an obvious tran sla­ tion in term s o f the present set-up once it is realized th at the intersection of two open sets should be replaced by the fiber product of tw o objects of Sch/X . Indeed, the fiber product, in the category of topological spaces, of two open subsets of a space is their intersection. A presheaf P on X F is a sheaf if it satisfies: (S^) if s e P ( U ) and there is a covering (C/t U)ieI of U such that VQSVi lj(s) = 0 for all i, then s = 0; (S2) if (Ui -» U)ieI is a covering and the family (si)ieI, s, e P ( Ui ) is such that r e S (/i x i ; U j , U , ( S i) “

r e S (7, x v V j , U j ( Sj )

for all i a n d j, then there exists an s e P(U) such th at resUitU(s) — st for all i. In other words, P is a sheaf if a section is determ ined by its restriction to a covering, and a com patible family of sections on a covering always arises from a global section. In o th er sym bols, P is a sheaf if the sequence

(S)

p(u) - n nut n p{Vi x„ Uj)

is exact for all coverings ( f /t U). A presheaf P th at satisfies is called a separated presheaf N ote that the em pty scheme 0 has an em pty covering, that is, the em pty set of m orphism s is a covering of 0 . Since a pro d u ct of abelian groups over an em pty indexing set is zero, P ( 0 ) = 0 for any separated presheaf. (Those w ho find this reasoning too bizarre may prefer to take this as part of the definition of a separated presheaf and sheaf). In the case th at E contains all open im m ersions, any open covering U = \ JU t in the usual sense of the Zariski topology is an open covering in the sense of the E -topology, and so a sheaf F on X Edefines by restriction sheaves in the usual sense on all schemes U in C /X . T he condition (S) for coverings by open subsets is ju st the usual sheaf condition. T o get some idea of w hat (S) m eans for oth er coverings, we look at a G alois covering. P r o p o s i t i o n 1.4. Let Y be Galois over X with Galois group G, and let P be a presheaf for the etale (or finer) topology on X that takes disjoint sums of schemes to direct products. Then G acts on P(Y) (on the left), and the sequence (S) for the covering (T -» X) may be canonically identified with

the sequence, P( X) -+ P { Y ) = = = b P(Y)", where G =

. . . , P( Y) identifies P( X) with the subset P( Y) G = {a e P(Y)\ o{a) = a, a// c t g G} of P(Y). Proof. F ro m the definition of a G alois covering (1.5.4) we know th at Y x x Y ^ G y = ] J Ya., and once 7 x 7 has been identified via this m ap with J_J Ya., the diagram Y xx YH Y P2

becom es identified with T X

(1 ,1 ,.. .,1)

11 Y.t = = 3 Y . ( f f lt . . . , a „ )

O n applying P, we o b tain (i

P(Y ) =

i)

=

( &1 * • • •

t

,Gn)

P(Y)",

and this gives the required statem ent. We m ake the sheaves on ( C/ X) E into a category S ((C /X )£) = S (X £) by defining a m orphism of sheaves to be the sam e as a m orphism of pre­ sheaves. T hus S (X £) is a full subcategory of P(A"£). We shall prove in the next section th a t S ( X E) inherits m ost, but not all, of the good properties o f P ( * £). Before giving som e exam ples of sheaves, we prove a result th at m akes it easier to check th at a presheaf is a sheaf. P r o p o s i t i o n 1.5. Let P be a presheaf for the etale or flat site on X. Then P is a sheaf if and only if it satisfies the following two conditions: (a) for any U in C/A", the restriction of P to the usual Zariski topology on V is a sheaf; (b) for any covering (U' -> U) with U and U' both affine, P ( U ) -> P(U') i t P(U' x v U') is exact. Proof. T he necessity of the tw o conditions is obvious. W e prove the sufficiency in the case of the flat site and leave the reader to m ake the m inor changes required for the etale site. C ondition (a) im plies th at if a schem e V is a sum V = [ J V{ of sub­ schemes Vh then P( V) = [~| P(VJ). F ro m this it follows th at the sequence (S) for a covering ( l / f -► C )t is isom orphic to the sequence (S) arising from the single m orphism (U' I/), U' = ]J 17, because

(Li u,) xv (U ut) = LJ (u, *u Uj)

[EG A . 1.3.2.4]. T hus (b) implies th at (S') is exact for coverings ( U t -v U)ie/ in which the indexing set / is finite and each U { is affine, for then ] J U t is affine. Let f : U ' U be the m orphism [ J U) -► I/, where (Uj U)j is a covering, and w rite U as a union of open affines U — (JC,-. T hen is a union of open affines = \JU'ik. Each f(U'ik) is open in U t (1.2.12), and is quasi-com pact, and therefore there is a finite set K t of fc’s such th at {U'ik -► (/,•)* 6K|. is a covering. By proceeding in this way and possibly introducing infinite redundances into the union U = [ j U h one may write U = and U ’ = \ J U i’k as unions of open affines in such a way th at for any i, (U'ik -* Ui)keK. is a finite covering of U t. C onsider the diagram : P ( U ) ------------------------ ►P(U')

ni nvd

i

.z:..::. > P ( U’ x v U ’)

nkn pw*) = * ni n kj

^ t/y

n p(Ui n Uj) — ► n n w *n ij i>j k,l F rom (a) the two colum ns are exact, and from the extension of (b) the m iddle row is a p ro d u ct of exact sequences and so is exact. It follows th at P(U) -> P{U ) is injective and th a t P is a separated presheaf. Hence the bottom arrow is injective and an easy diagram chase now shows that the to p row is exact. C o r o l l a r y 1.6. For any quasi-coherent (9x-module F , W(F) is a sheaf on X n and, a fortiori, on X et. Proof. O bviously W(F) satisfies condition (a), and (1.2.19) shows that it satisfies condition (b). In particular, the presheaf G a is a sheaf for the flat and etale topologies. We give a second p ro o f of this fact. Let G a%x be the scheme X x specZ spec Z [T ]. Then, for any scheme U over X,

Horn* (I/, G„,x) = H o m spec(Z) (U, spec Z [ T ] ) = H orn

(z[r], n u ,

*). N ow (1.2.17) implies that condition (b) of (1.5) is satisfied. Similarly, G m(l/) = H o m x (t/, G m* = X ® z Z [T , T - 1 ], and this description of G m(t7) shows, as for G fl, that both conditions of

(1.5) are satisfied. T hus G,„ is a sheaf for the flat or etale sites on any scheme X. Both G fl and G„, are exam ples of sheaves defined by com m utative g roup schemes. Recall th at an X -schem e G is a com m utative group scheme if there is a given factorization S c h/ X -------> Ab

Sets in which / is the forgetful functor. This sim ply says that G( Y) is an abelian gro u p for all Y over X and th a t all the m aps G( Y) -► G(Y') corresponding to X -m orphism s Y' -> Y are hom om orphism s. Alternatively one can say th at G is a com m utative group in the category Sch/X . F rom this last description it is easy to see how to m ake Gx = I I X a, X a = X a eG

into a group scheme, where G is any com m utative group (namely, Gx x Gx = L K * * x x Xi), and X a x x X x % X (canonically), and so Gx x Gx -> Gx may be defined to be the m ap th at, on x x X T, is simply the identity m ap X a x x X x ^> X ax). Any com m utative group scheme o b ­ viously defines a presheaf for any site on X. C o r o l l a r y 1 .7 . The presheaf defined by a commutative group scheme on X is a sheaf for the flat, etale, and Zariski sites on X. Proof. (1.5b) follows from (1.2.17), and (1.5a) is easy to check. If G is a com m utative g roup schem e over X (or spec Z), then we shall w rite G for both the g roup schem e and the sheaf defined by it (or by GiX)). T he next exam ple is very im p o rtan t. It shows that giving an etale sheaf over the spectrum o f a field is the sam e as giving a discrete m odule over the G alois gro u p of the separable closure o f the field. Fix a field K and consider the etale site on X = spec K. Recall (1.3) th at any schem e U th a t is etale and of finite-type over X is finite over X and is a disjoint union C7 = J J C/f- of spectra of finite separable field extensions of K. Fix a separable closure K s of K and let G = G al ( K J K \ th at is, choose a geom etric point x -> X of X and write G = n ^ X , x). N ote th at G acts on K s on the left and on spec (K J = x on the right. Let P be a presheaf on X et. If K' is a finite separable field extension of K, then we write P(K') instead of P(spec K'). Define M P = lim P(K') where the limit is taken over all subfields K ' of K s th at are finite over K. Then G acts on P(K ) on the left through its action on K' whenever K ’/ K

is Galois, and it follows that G acts on the limit M P. Clearly M P = \ J M P, where H runs through the open subgroups of G, and so M P is a discrete G-m odule (in the sense of Serre [8,1.2]). Conversely, given a discrete G-m odule M , we can define a presheaf FM such th at: (a) F m (K') = M ", if H — G al ( K J K ’)\ (b) F M( f ] K t) = U FM( K ty O ne only has to take F m {U) = H o m c (F (C ),M ) where Fis the functor FEt/X G-sets defined by F(U) = H orn* (x, U) (com pare 1.5.3) and note th a t (a) F(K') - G/H\ (b) F ( [ ] K () = J~]F (K f). L emma 1.8. FM is a sheaf. Proof. It suffices to check conditions (a) and (b) of (1.5). P a rt (b) of the definition of FM implies condition (a) o f (1.5). It is clear from the p ro o f o f (1.5) that in checking (b) we may take U' and U to be “arbitrarily Z a risk i-sm a ir affines and, in this case, to be spec L and spec L respec­ tively where L => L are finite separable field extensions of K. Let L" be a finite G alois extension of L containing L', and consider the diagram , F m (L )

---------- ► F A f ( L ' ) —

F m(L) -------> Fm(L") —

> 14m { F '

® l F )

F m(L" ® l L")

The bottom row is exact according to (1.4) since, by definition, F m (L) = FM(L")G*liL"lL). Also F m (L) -+ F m ( L) and FM(L') -+ FM(L") are injective. An easy diagram chase now shows th at the top row is exact. T heorem 1.9. The correspondences F «-►M f , M FM induce an equiv­ alence between the category S (X et) and the category G-mod of discrete G-moduIes. Proof. Clearly a G -hom om orphism M M ’ induces a m orphism F m -> Fm .. Conversely, let H be an open norm al subgroup of G, and let K' = Ks - If 0 : F F* is a m orphism of sheaves, then (K'):F(K') F\K') com m utes with the action of G because of the functoriality of (j). Thus lim (K') is a G -hom om orphism M f M F. It is trivial to check that H o m G (M, M') -> Horn (F M, FM ) is an iso­ m orphism and th at the canonical m ap F -+ F Mf is an isom orphism , and this suffices to show th at the tw o categories are equivalent. Remark 1.10. W ith obvious m odifications, the above theory (and some of w hat follows) is also valid for presheaves and sheaves of sets,

groups, rings, and m odules over a sheaf of rings. There is a canonical presheaf of rings on the site ( C/ X) E for any E, X and C, nam ely, U T((7, V' J

X of schemes defines a morphism of sites (C'/X')E' -* (C/'X)E if: (a) for any Y in C / X , Y{X1 is in C / X ' ; (b) for any £-m o rp h ism U -> Y in C / X , -> Y ^ is an £'-m orphism . Since the base change of a surjective family of m orphism s is again surjective, we see th at n defines a functor n ^ (Y ^ Y ^ y.C /X

C/X'

th at takes coverings to coverings. F requently we shall sim ply refer to n as a continuous morphism n \ X ’E‘ -> X E. Examples 2.1. (a) The identity m ap o f X defines a m orphism of sites (C/X)E> -> {C/X)E if and only if the £ '-to p o lo g y on X is as fine as the E-topology. (b) The identity m ap on X defines continuous m orphism s, X f l -+ X - * Zar. (c) Any m orphism n : X ’ -> X defines a continuous m orphism n:X'E -> X E provided it respects the underlying categories. Let n:X'E' -> X E be continuous. W ith any presheaf P' on X'E>we may associate the presheaf np{P') — P’ o 7r'on X E. Explicitly, np(Pf) is the pre­ sheaf on X E such th at r ( U , np(P')) = V{UiX1, F). T he presheaf np(P') is called the direct image of P \ Clearly np defines a functor P(X'E) PfX^r), and we define the inverse image functor np: P ( X E) -> P ( X ’E) to be the left adjoint of np, th at is, np is such that H o m P(^-) (tipP, F ) * H o m m ) (P, npF). The existence of np is given by a sta n d a rd result in category theory. P r o p o s i t i o n 2.2. Let C and C; be small categories, and let p be a functor C -* C . Let A b e a category with direct limits, and write Fun (C, A) and Fun (C', A) for the categories of functors C -> A and C' -» A. The functor (f f ° p ) : Fun (C , A) Fun (C, A)

has a left adjoint. Proof. See, for exam ple, H ilton-S tam m bach [1, p. 321].

We apply this to C = C /X , C' = C / X ' , A = Ab and p = n : The smallness condition on C and C is no p articular problem if each is a subcategory of a category of schemes of finite-type over som e schem e X, for such a category has a small subcategory containing at least one object in every isom orphism class. (Cover X by open affines U{ — spec A h a n d consider all schemes th at can be built up of schemes of the form spec B, where B is a q u o tien t of some A([ X l9 X 2, . . . ] . ) In any specific situation such set-theoretic questions will cause no special difficulty, but to be both rigorous and general one should use universes [SG A. 4 , 1. A ppendix] or the following m ethod of W aterhouse. In tu ition suggests that the only presheaves and sheaves we need consider are determ ined by a small am o u n t of inform ation, and this prevents difficulties even though we may be considering functors on Sch/X . In W aterhouse [2] a presheaf P is defined to be basically bounded if there is an infinite cardinal m such th at P(spec A) = lim P(spec B) for all affine schemes spec A/ X, where the limit is over subrings B c A of cardinality X in C /X . A m orphism from one such square {g, U) to a second (gu U x) is an X -m orphism h:U -► U l such that hg = g x. I f h!\V' -* U' is a m orphism in C'/X', then there is an obvious restric­ tion m ap r(U', npP) -> T(V', npP) because the first group is the direct limit over a sm aller category than the second. Thus npP is a presheaf. T o give a m orphism P -► npP' is the sam e as to give m aps P(U) -+ P'(U{X')) for all U in C /X , which are com patible with the restriction

maps. T o give a m orphism n pP -► P' is the same as to give m aps P(U) -> P(U' ) for all com m utative squares >U

U1

which are com patible with the restriction maps. But a m orphism V -► U factors uniquely as U' -> U{X1 -> I/, and so if we are given a m ap P(U) -> P'(U(X )), then we get m aps P ( U) - + P'{U') autom atically. Thus a m orphism P -» npP' defines a m orphism npP P' and conversely. P r o p o s i t i o n 2.3. 77ie limit defining T ( U \ n pP) is cofiltered if finite inverse limits exist in C/X. Proof. O ne has only to construct the ap p ro p riate fibered product or difference kernel to show th at the category satisfies the duals of ( /j) and ( / 2) and is connected (A ppendix A). N ote th at (2.3) holds for the Zariski, etale, and flat sites (1.13). W hen the limits are cofiltered, it is possible to give a slightly m ore explicit description o f T (U \ n pP). A section of n pP over U’ is represented by a pair (s, g) w here g:U' -> U is a m orphism over n and s e P ( l/); tw o such pairs (su g {) and (s2, g 2) represent the sam e elem ent in T (U \ n pP) if there is a com m utative diagram

w

X' such that rest/(/l

(^ i) =

rQsUU2

( s 2).

The elem ent

g)” in T { \ J \ n pP)

restricts to the elem ent %s, gh)” in Y( V\ npP) under V' X U \ Examples 2.4. (a) If P is a constant presheaf, then npP is a constant presheaf and is defined by the sam e group. (b) If n:X' X is in C / X and C'/X' = ( C / X ) / X \ then there is an initial object in the category over which the lim it is being taken, namely, U '

X'

— iiU

JJ'

X

T hus T{U\ n pP) = V(U\ P f and so n p simply restricts the functor P to the category C / X ' . In this case we often write P \ X ' for n pP. (c) If n : X -> X is the identity map, then npnp(P) = P for all presheaves P on X E (we assum e C / X 3 C/X). Exercise 2.5. C onsider n p when n is the identity m ap X n -> X ei or ^et Xzar l f° r exam ple, does npG a = G a? P r o p o s i t i o n 2.6. The functor tcp is exact and np is right exact. More­ over, np is left exact in each of the following two cases: (a) finite inverse limits exist in C / X ( for example, when C / X ~ LFT /X , (€t)/x\ or (Zar/X ), (1.13)); (b) 7i is in C / X and C / X ' = ( C / X ) / X\ Proof. The exactness of np is obvious from its definition, and np is right exact because a rb itrary direct limits are right exact in Ab. The left exactness of np follows from (2.3) and the fact that filtered direct limits are exact in Ab (Appendix A3) in case (a), and is obvious from (2.4b) in case (b).

2.7. I f F is a sheaf then npF is a sheaf. Proof. Let n be a m orphism (C'/X')E -> (C/X)E. F or any U in C / X we write W = U x x X'. If ((7, U) is a covering, then so also is (U'i -► U ' \ and so P r o p o s it io n

f(v)-n w,) ^ n ni/; i

ij

u'j)

is exact. But U- x w U) % (C, x v Uf)\ and so this sequence is isom orphic to the sequence (7XpF)(u) -

ni *pF(ut) =1n U

which proves npF is a sheaf. Remark 2.8. It is not true in general th at n pF is a sheaf when F is. However, it is obviously true in the exam ple described in (2.4b). It is possible to define stalks o f sheaves and presheaves at a point of a site once one has decided w hat a point is. A lthough there is a theory o f points and stalks for alm ost any site, we shall only study it in the case of the etale site because there the points may be described very explicitly. The im p o rtan t property of a point of a topological space for the usual theory of sheaves is that to give a sheaf on a one-point space is simply to give a set (or abelian group). This is not true for etale sheaves on a one-point schem e unless the scheme is the spectrum of a separably closed field. Thus a point for the etale topology on X should be a geom etric point.

Let x be a point (of the underlying topological space) of X. We shall always use x to denote the spectrum of som e separably closed field k{x) containing k(x) and ux:x -> X to denote the m ap induced by the inclu­ sion o f k(x) in k(x). A ccording to (1.9), the functor F F(x) gives an equivalence of categories betw een S (xel) and Ab. Let P be a presheaf on X et. T he stalk of P at x, P x, is the abelian group (upP)(x). M ore explicitly, P x = lim P(U) where the lim it runs over all com m utative triangles

with U etale over X , th a t is, over all etale neighborhoods of x in X. Clearly Px is independent of the field k(x) chosen. Remarks 2.9. (a) T he functor (P i-> P x): P(Aret) -+ Ab is exact, because up is exact (2.6). (b) The abelian g ro u p Px is acted on by G al (fc(x)sep/fc(x)). Indeed the m orphism ux factors into x x X and Px is the underlying abelian g ro u p of the G al (/c(x)sep//c(x) )-m odule associated w ith the presheaf ip(P) on spec k(x). (c) Let U -+ X be an etale m orphism whose image contains x. In general U can be given an x-point, th a t is, the structure of an etale neighborhood, in m any different ways, and so there is no canonical m ap P(U) -> Px. O f course, once U has been given an x-point, there will be such a canonical m ap P{U) -> Px, which we often write s i—> sx. (d) Let Y be a schem e locally of finite-type over X. If (Ud is a filtered inverse system of A -schem es with each (7, affine, then it is easy to see th at the canonical m ap Y(lim U^) lim Y(Ui) is an isom orphism ((3.3) below). T hus if P is the sheaf defined by a g roup scheme G th at is locally of finite-type over X , then Px - lim G(U) = G(lim U) = G((9X x)

(1.4).

F o r exam ple, ( G J S = (9xrx and (G m)s = & l x. P r o p o s i t i o n 2.10. Let F be a sheaf on X et. I f s e F(U) is nonzero, then there is an x e X and an x-point of U such that sx is nonzero. Proof. Suppose th at s m aps to zero under all such m aps F(U) -> Fx. C learly any u e U is the im age o f an x-point of U, where x is the image of u in X. Thus, from the definition of Fx, for any u e U, there is a scheme Vu, etale over L, whose im age in U contains u and which is such th at s \ Vu = 0. This family (Vu -* U)uel, is a covering of 17, and so, by (5 J , 5 - 0. The next theorem shows th at every presheaf has an associated sheaf.

2.11. For any presheaf P on X E there is a sheaf aP on X E and a morphism :P -> aP such that any other morphism from P into a sheaf F factors uniquely as T heorem

P — *—+ aP

F Proof. W e give one p ro o f for the etale and Zariski sites and sketch a second th at w orks in general. Lemma 2.12. Let X E be a site. (a) A product of sheaves on X F is again a sheaf. (b) I f (F f),6/ is a family of subsheaves of a sheaf F, then f ] F f (where (f]Fi)(U) = n ( F f ( l / ) ) is again a sheaf. (c) If :F -* F' is a morphism of sheaves and F0 is a subsheqf of F', then 0 _1F O (where (~ 1F0)(U) = 0 " I(F 0( L ) ) is again a sheaf; in particular, ker ((f) is a sheaf. Proof. It is easy to check these directly. A lternatively they follow from the description of the sheaf condition in (1.14) and the fact th at inverse lim its com m ute with inverse limits. We now prove (2.11) for the etale site (the p roof for the Z ariski site is similar). N ote first th a t if X is the spectrum of a separably closed field, then any scheme U etale of finite-type over X is uniquely a finite disjoint union U = \ J X ( with X { — X. T he sheaf aP associated with a presheaf P is U^

n

PiXi) = p (x Y>

and 4>(U) is the m ap P(U) -* f ] P (X f) defined by the restriction maps. N ow consider an a rb itrary X. F o r each x e X we choose an x and w rite P f = a(upP) with the above definition of a. Let P* be the sheaf flx e x uxPP h an uxp(u%P) -> uxp(P%), where the first m ap is th at corresponding (by adjointness) to the identity m ap upxP -► upxP and the second is induced by upP -> a(upP) = P f. Let aP be the intersection of all subsheaves of P* containing (P). Let ':P F be an o th er m orphism from P into a sheaf F, and consider the diagram P —

aP c

►P *

where \j/ is induced by the m aps i:Pf -► Fx. The injectivity of F -> F* is a restatem ent o f (2.10). The diagram com m utes because of the functoriality of the m aps P -► uxpP*. As i p ' l (F) is a subsheaf of P* co n ­ taining (P) and hence aP, ij/ induces a m orphism i^0:a P -+ F such th at If ipi'-aP -+ F is a second such m orphism , then ker (\j/0 - t/^) is a subsheaf of P* containing (P) and so m ust contain aP, th at is, i/f0 = This com pletes the first proof. In the case of an a rb itrary site ( C/X) E the theorem may be proved as follows. F o r any U in C / X define P 0{U) to be the set of all s e P ( U ) such th at res^.^ (s) = 0 for all I/,- in som e covering of U. It is easily seen th at P 0 is a su bpresheaf o f P and th at the q u otient presheaf p t = p / p 0 (with P {(U) = P ( U ) / P Q(U)) satisfies (S,). N ow , with the n o tatio n s of (III.2) below, define (aP)(U) = lim /? ° (^ , P x) w here the lim it runs over all coverings °U of [/. Thus, a section s e (aP){U) is represented by a pair o f families ((sf), ((/*)) where (I/, -> U) is a covering of I/, e Pi({7,), and reSU, X U j , U i

( S i)

= reS(A X U j . U j IS 7 )

for all i,7'. Tw o such families ((s^, (C/i)) and ((sj), (C j)) represent the sam e elem ent of (aP)(U) if ((/,) and (Cj) have a com m on refinem ent (Ffc) such th a t the families (fk), tk e P x(Vk) arising by restriction from (st) and (sj) are equal. Define restriction m aps as follows: if V -> U is in C /X and s e (aP)(U) is represented by ((st), ((/,)), then resKtf/ (s) is represented by ((re s^ ^ fo JM F

x v I/.)).

T his clearly m akes aP into a presheaf. Also, any m orphism ':P -> F from P into a sheaf factors uniquely thro u g h the canonical m ap :P -> aP. For, according to (Sj), '(P0) = 0 and so ' factors uniquely th ro u g h P P x. Also according to (S2), any m orphism P { -> F extends to a m orphism aP -+ F (Pf°(^, —) is a functor of presheaves, and H °(;//, F) = F (l/)), and according to (S t ), this exten­ sion is unique. The only real com plication in the p ro o f occurs in show ing th at aP is a sheaf. F o r this we refer the reader to A rtin [1, II.1.4(ii)]. (The condition ( + ) there is our (S ^, and we have denoted P f by aP.) Exercise 2.13. M ake explicit the m ap P(U) -> P*(U), where U is etale over X, th at is defined in the first p ro o f above. Remark 2.14. (a) T he above theorem may be restated as follows: the natural inclusion functor S (X £) P (X £) has a left adjoint a : P ( X E) S (X £).

(b) Let 7i:X'E>-+ X E be a continuous m orphism such th at np takes sheaves to sheaves. Then for any presheaf P on X E and sheaf F on X £>, H om s >(n paP, F) % H om s (aP, npF) % H om P (P, rc^F) « H o n v (7rpP, F) % H om s- (anpP, F), which implies there is a canonical isom orphism 7rpa P % #7rpP. (c) By using the fact th at P e P(xel) is a sheaf if and only if it takes finite sums o f schemes to products of abelian groups, one may verify that up takes sheaves to sheaves. Thus upa « aup, and the canonical m ap P-x (={au>xP)(x)) -> (aP)s ( = (up xaP)(x)) is an isom orphism , th at is, P and aP have the same stalks. T h e o r e m 2.15. (a) The inclusion functor S (X £) P ( X E) is left exact and preserves inverse limits; a : P ( X E) -> S ( X E) is exact and preserves direct limits. (b) The following statements are equivalent: 0 -> F' -+ F F" is exact in S ( X E); 0 -> F' - F - F" is exact in P ( X E); 0 -> F ( U) -> F{U) - F''(U) is exact for all U in C /X . For the etale topology, they are also equivalent to the sequence 0 -+ F ~ -+ F $ Ff

being exact for all geometric points x. (c) (j):F -+ F' is surjective in S(X£;) if and only if for any s g F'(U) there exists a covering ( f /( -> U) o f U and elements s( e F(Ui) such that (friSi) = resUitU (s) for all i. For the etale topology, these statements are also equiva­ lent to Fx —►F'x being surjective for all x. (d) To form arbitrary inverse limits {for example, kernels, products) in S ( X E) form the inverse limit in P (X £), and then the resulting presheaf is a sheaf and is the inverse limit in S (X £). To form arbitrary direct limits {for example, cokernels, sums) in S (X £) form the direct limit in P (X £), and then the associated sheaf is the direct limit in S (X £). (e) The category S (X £) is abelian and satisfies AB5 and AB3* {but not in general AB4*). Proof. N ote that S (X £) is a subcategory of an abelian category and so, in particular, is additive. (a) All statem ents except the left exactness of a are formal consequences of the adjointness properties of the two functors (Appendix A). The pro o f of the left exactness of a will m ake use of the construction of aP. C onsider first the etale site. If P -► P' is injective, then, with the n o ta ­ tion of the p ro o f of (2.11), P* -> P'* is injective (since up is exact) and

aP and aP' are subsheaves of P* and P'*. T hus aP -» aP' is injective in P (X et). Let F be the kernel of aP -> aP' in S (X et). T hen F = 0 because the inclusion functor S (X et) -► P ^ , ) is left exact. N ow consider an arb itrary site. W e assum e th at P -+ P' is injective and show first th at P { -► P \ is injective. Suppose th at s e P(U) m aps to s' e P'0(t/); then there exists a covering ( l / f) of U such th at res^. ^ (s') = 0 for all i. The injectivity of P (l/,) -* P'(Ui) shows th at res^. v (s) = 0 for all i, which implies th at s e P 0(^)T he com m utativity of / ? ° ( « , P i ) c—

►n w * i

H ° W , F ,) c—

►n K M ) i

for a covering % of U shows th at H°(%, P t) -> H°(°1l, P\) is injective. By passing to the limit over all coverings °U of I/, one finds th at aP(U) -+ aP'(U) is injective. T he p ro o f may now be com pleted as before. (b) T h at the first two statem ents are equivalent follows from (a); that the next two are equivalent has already been noted; th at the fourth statem ent follows from the th ird is obvious. To see th at the fourth im ­ plies the third, let s' e F (U) m ap to zero in F(U). F o r any x-point of U, s'z m aps to zero under F* F*, which implies that it is zero as this m ap is injective, which in tu rn implies th at s' = 0 (2.10). N ow let s e F(U) m ap to zero in F'"(U). This im plies th at s* lies in the subgroup F^ of F* for all x-points of U. It follows that, for any a e U, there is an etale m ap Vu -> U whose image contains u and th at is such th at s | Vu lies in the subgroup F'(VU) of F(VU). As F' is a sheaf and (Vu) is a covering of U, this show s th a t s e F'{U). (c) Let F -► F' be a m orphism of sheaves, and let P be its cokernel in P { X E). T he sequence F -* F ' -> aP -> 0 is exact because a is exact. Thus F -» F' is surjective o aP = 0 o P = P 0 (in the n o tatio n of the p ro o f o f (2.11)) the statem ent in (c) holds. O n the etale site, aP = 0 if and only if (aP)t = 0 for all x in X , th at is, if and only if F* -> F^ is surjec­ tive for all x. (d) Both statem ents follow im m ediately from (a). (See A ppendix A4.) (e) T o show th at S ( X F) is abelian, it only rem ains to show that for any m orphism :F-> F' in S, the induced m orphism :coim () -> im (0) is an isom orphism . But P(Ar/:) is an abelian category, a takes isom orphism s to isom orphism s, and 0 :c o im () im ((p) in S is o b ­ tained by applying a to the sam e m ap in P(A'£).

The fact th at S { X E) satisfies AB5 and AB3* follows from (a) and (d). (For AB5 only the left exactness o f direct limits has to be proved.) Exercise 2.16. Let be a family of discrete G-modules where G is a profinite group. Let M * = Y\ in the usual sense, and let M be the subm odule (JM * of M *, where the union runs through all open subgroups H o f G. Show that M is the product of the M, in the category of discrete G-modules. D educe that products are not exact, that is, that AB4* does not hold in the category of discrete G-modules if G is infinite. D educe the same result for with X the spectrum of a nonseparably closed field. Exam ine the situation in S(ArZar), with X any scheme. Remark 2.17. (a) Let (F t) be a pseudofiltered direct system of sheaves on som e site, and let F be the presheaf direct limit of the system. Then F satisfies the sheaf condition (5) for finite coverings (G f U). Indeed, each sequence f \ (U ) -

n FtiUj) =t 11 F W j x Uk) j J.k

is exact. As each product is finite, it may be replaced by a direct sum. As direct limits com m ute with direct sums, on passing to the direct limit, we obtain an exact sequence,

F ( U ) - Y \ F ^ j ) ^ U F (Uj j

j,k

X

U k).

O n som e sites, nam ely, the N oetherian sites, this is enough to show that F is a sheaf (III.3). (b) If X is a Jacobson scheme, then in (2.10), the proof of (2.11), (2.15) and sim ilar situations only geom etric points x lying over closed points of X need be considered. (Recall [E G A . 1.6.4] th at a scheme X is Jacobson if every closed subset of X is the closure of its set of closed points; spec Z and spec of a field are Jacobson, and any scheme th at is locally of finite-type over a Jacobson scheme is Jacobson.) M ore generally, any set S of geom etric points such that { x \x e 5} is very dense in X suffices. (c) Theorem (2.15) implies th at the functors (F i-> Fx): S( Xet) -> Ab, x e X, form a faithful and conservative family: two m orphism s , X

§3. Direct and Inverse Images of Sheaves Suppose th at n: X' -» X defines a m orphism of sites ( C / X ' ) E>-> (C/X)E. The direct image of a sheaf F' on X'E> is defined to be 7i,F' = npF' and the inverse image of a sheaf F on X E is defined to be 7r*F = a(npF). Recall (2.7) th at npF' is a sheaf. T here are canonical isom orphism s H o m 8(*E) (F, 7i,F') *

(irpF, F ) « H om S(, kl) (tt*F, F )

th at show th at 7r, and n* are adjoint functors S(X£«) S (X f:). T hus 71, is left exact and com m utes with inverse limits, and 7r* is right exact and com m utes with direct limits. If np is exact (2.6), then 71* = S(X ^) C* P (X i-) ^

P (X £) A S (X £)

is left exact as well as right exact. Remarks 3.1. (a) If n\ X' X is in C /X , then n * : S ( ( C / X ) E) - + S ( ( C / X ’)E) is simply the restriction functor; we usually write it F h ►F |X '. (We som etim es also use this n o tatio n for 7r*F when n is not of this form.) (b) If E' c E, C / X cz C /X , and n : X -» X is the identity m ap, then it is not necessarily true th at 7c,7r*(F) = F. However, it is true for the identity m ap X fl -> X et (com pare (III.3.lib ) ) . (c) N ote th at 7t* depends on the underlying categories of the sites and not ju st the topologies. F o r exam ple, let 7r:spec k spec k be the m o r­ phism defined by the inclusion of a field k into its algebraic closure k. If we give spec k and spec k their big etale sites, then n* is simply the restriction functor and so com m utes with arbitrary products of sheaves. However, we leave it as an exercise to the reader to show th at if spec k and spec k are given their small etale sites, then 7i* does not com m ute with p roducts (use 2.16). (d) Let 7i:X' -> X be a m orphism ; let G be a group scheme on X, and assum e th at the F-topology is coarser than the canonical topology so th at G defines sheaves Gx and Gx >on X E and X E. The m ap npGx -> Gx>, which sends the elem ent o f T((7', G) represented by (s, g) with g\U' -» U and s e T((7, G) to s g e G xm

= GX.(W) = T { U \ G X.\

factors uniquely thro u g h n*Gx . T hus there is a canonical m ap 0 G:7r*G^ Gx>. This need not be an isom orphism . F o r example, let X = spec k with k a field of characteristic /?, let X = spec A with A a k-algebra with m any nilpotents, let G = oap, and consider n : X fel -► X et. Then (yxp)x = 0,

so n*{xLp)x = 0, but (oolp)x • need not be zero. Thus x 0, and so cj)G is surjective but not injective in this case. There are tw o im p o rtan t situations where cj)G is an isom orphism , namely, where n* is a restriction m ap (that is, X ' -> X is in C/X ) and where G is in C/ X. T o see the second case note that, by definition of 7i*, R G * , F) % r(G * , zr*F) for any sheaf F on X'E. Since, under this iso­ m orphism , hom om orphism s correspond to hom om orphism s, H o m S(X ) (Gx', F) % H o m S(X) (Gx, fl*F). This shows th a t Gx » n*Gx by the uniqueness of adjoints. (e) Let 7r:spec K ' -+ spec K correspond to an inclusion of fields K K'. There is a noncanonical com m utative diagram :

c>

K' ^ X E. Clearly {n'n)^ = 71*71* and 7r*7r'* is adjoint to 7r*7r*, which implies th at 7r*7t'* = (n'n)*. F o r the rem ainder of this section all schemes will have their etale topology unless explicitly stated otherwise. W e first investigate the effect of 7i* and 7t* on stalks. T h e o r e m 3 .2 . Let n : X ' X be a morphism. (a) For any sheaf F on X et and any x' e X', R * F ) .r j ^ F ^ , that is, the stalk o f n*F at x' is isomorphic to the stalk o f F at n(x'). In particular, if u is the canonical morphism 7r:spec (9x ,x X, then

F-x = (n*F)x = H sp ec GXtX, tt*F).

(b) Assume that n is quasi-compact. Let x e X ; let x = spec k(x)sep; let f be the canonical morphism X = spec anc^ Iet X' = X' x x X \ X ' X is an etale neighbor­ h ood o f x —> X, then x -► U factors th ro u g h x -» spec Ox,x (uniquely if U is connected). (See (1.4).) (b) By definition (f'*F)(X') = lim F(U') where the lim it is taken over all com m utative diagram s with U ’ -+ X' etale: U’

X' —

X1

O n the o ther hand, from the definitions of n * and stalks, we see that (tt*F)x = lim F(U ) w here the lim it is taken over all such diagram s th at com e by base extension from a com m utative diagram with U -> X etale: U

T o prove th at these tw o lim its are isom orphic, we m ust show th at the second set of diagram s is cofinal in the first, th at is, th at X ‘ -> V factors thro u g h X ’ -> U{X1 for som e etale U -> X. But X is a limit, X = lim U ,

in which each U is affine and is etale over X. As inverse lim its com m ute with fiber products (in any category) and n is quasi-com pact, X' is a limit, X' = X' x (lim U) = lim (/(Jr), in which each U(X1 is quasi-com pact and the transition m orphism s are all affine. N ow [/' being of finite-type over X' implies th at X' -► U' factors through some U(X1, as follows from the next lemma. Lemma 3.3. Let X be a scheme and let Y = lim Yh where (19 is a filtered inverse system of X-schemes such that the transition morphisms Yi Y{ for some i. More precisely, H o m * (y , Z ) = lim Horn* ( Yh Z). Proof. In the case th a t X = spec A, Yi — spec Bh and Z = spec C all are affine, the lem m a is obvious because it merely asserts th a t if B — lim B{ and C is finitely generated over A, then any ^-h o m o m o rp h ism C -> B factors through som e Bt. The general case may be deduced from this case by the usual patching argum ent [E G A . IV.8.13.1]. Remark 3.4. If F is the sheaf on X'ct defined by a group scheme G th at is locally of finite-type over X ', then ( /'* F )(X ') = G(X'). This follows from (3.3) and the fact th at X' = lim U \ where the limit is taken over all diagram s as at the start o f the p ro o f of (3.2b). C o r o l l a r y 3.5. (a) Let i : Z —►X be a closed immersion, and let F be a sheaf on Z et. Let x e X . Then,

(«*0* = 0

if x f i(Z), and

V*F)x = F*0

\f x = '(*oX *o £ Z.

(b) Let j: U -» X be an open immersion, and let F be a sheaf on Ue{. If e j ( U) , X = j ( x 0), then ( j st,F)x = F Xo. (However, (j^F)x need not be zero ifxtj(U ).) (c) Let n : X ’ -* X be a finite morphism and F a sheaf on X ’ex. For any x e X , ( n j ) x = j ] F f xl where the product is taken over all x' such that n(x') = x, and d(x') is the separable degree of k(x') over k(x). In particular, if n is etale of constant degree d, then (n*F)x = Fx. where x' is any point of X' such that n(x') — x. Proof, (a) W ith the n otation of the theorem Z is em pty if x 4 i(Z) X

and Z = spec ((9x. xl^x. x) = sPec @z,i if '(z) = *• (b) O bvious. (c) As X' is finite over X = spec 0 X X and (9X>X is H enselian, the con­ nected com ponents of X' are in one-to-one correspondence with the

connected com ponents of n l(x) (1.4.2b). Thus X' =

[]

spec (&$%■),

n(x') = x

and so F(X') = f [ F f x'\ C o r o l l a r y 3.6. I f n is a finite morphism, for example, a closed im­ mersion, then 7T* is exact (for the etale topology). Proof. T his follows im m ediately from (3.5) and (2.17c). Exercise 3.7. Let X be an integral schem e with generic point g:rj -> X. Show th at if X is norm al, then g^M^ = M x for any co n stant sheaf M tJ on rj. Show th a t if X is a curve w ith a node i:z X , then there is an exact sequence, 0 M x -+ g * M n -» i *Mz -> 0.

W hat is true in general? (H int: w rite g as the com posite rj X' -> X w here X' is the norm alization of X.) Remark 3.8. F o r a p ro p er m orphism n : X ’ -* X a m uch sharper result th an (3.2b) holds, nam ely, (n%F)x = T(AT, F\X'x) where F is a sheaf on X' and X x c> X' is the geom etric fiber of X ' over x X. Thus the stalks of 7i*F can be com puted on the geom etric fibers of X'/X. After (3.2b), it suffices to prove this with X a strictly local scheme with closed point x ~ x. The stalk of n^F, (n^F)^, is then equal to T ( X, n^F) = T ( X \ F). T hus to prove the above assertion we m ust verify th at if zr: AT' -► X is p roper and X is strictly local, then T ( X \ F) A r(A ^ , F\X'x) for any sheaf F on X' (where X'x is the closed fiber of X'/X). This statem ent is p art o f the pro p er base change theorem and will be considered in (VI.2). W e only m ake som e rem arks here. Firstly, if n:X' X is finite, then it m ay be proved by the sam e argum ent as in (3.5c) above. Secondly, if it is true for tw o m orphism s, then it is true for their com ­ posite. Finally, if F is a con stan t sheaf F — M x , then it follows from the existence of a Stein factorization. Recall [EG A . III.4.3.1] or H arts­ horne [2, I I I .11.5] th at the Stein factorization of a proper m orphism 7i:X' -> X is a sequence X' Y X w ith n2 finite (F = spec n+Ox') and 7ii p ro p er with connected fibers. T hus we need only consider the case th at X is strictly local and th a t X x (and X') are connected. But then r(X 'x, M) = M = r ( X \ M). Example 3.9. Let X be integral and quasi-com pact. Let g :spec K X be the inclusion of the generic point, and let K denote the sheaf defined by G w on spec K. F rom the definition of 0*G W X etale, where R(U) is the ring of rational functions on U.

There is a canonical injection 4>:GmX -* g^Gm>K th at on any U is simply the inclusion T([/, (9 %) c* F (l/)* . T he cokernel of this m ap is defined to be the sheaf of Cartier divisors, Di vx , on X ev In the case that X is regular, C artier divisors may be interpreted as Weil divisors. Let X { be the set of points x on X of codim ension 1, th at is, such that (9X x has dim ension 1 and so is a discrete valuation ring. The sheaf Dx of Weil divisors on X et is defined to be where Z denotes the constant sheaf on x defined by Z and ix \x c* X. I claim that Dx & Div*. F o r any U etale over X,

n

u , d x) =

©

z,

xelii and so we m ay define a m ap il/’- g*GmtK -> Dx by requiring th at / e R(U)* m aps to the family (ordx ( / ) ) JC where ord* is the discrete valuation on R(U) defined by &Ux. T o prove the claim, it suffices to show that

o - Gm.x i g*GnuK ± D x ->0 is exact. But, for any x e X , the sequence of stalks at x is 0 -> A* -> L* -> @ Z -> 0 where A = Cx L is the field of fractions of A, and the sum is over the prim es of height 1. But A is regular (1.4), hence factorial, and so the prim e ideals of height 1 are principal, which shows th at iJ/ is surjec­ tive. The exactness at the other points is trivial. (A lternatively one may simply use the fact th at the sequence is know n to be exact when restricted to l / Zar, for any U etale over X. See M um ford [2, Lecture 9] or H artshorne [ 2, 11.6.1 1 ].) We now consider the situ atio n : X is a schem e; U is an open subschem e of X, and Z is a subschem e of X whose underlying set is the com plem en­ tary closed subset Z = X ~ U. W e denote the inclusion m aps by / and j,

z

x I u.

If F is a sheaf on X et, we get sheaves F x = i*F and F2 = j* F on Z and U. M oreover, since Horn (F9j#j *F) % H orn (j*F,j*F), there is a can o n ­ ical m orphism F -►j* j* F corresponding to the identity m ap of j*F. By applying i* to this, we get a canonical m orphism (j)P: Fl -> i*/*F2. Thus, there is associated with any sheaf F on X et a triple (Fu F 2, (ftp) where F j e S (Z ), F 2 g S (L ), and (j>:Fl -> /*/*F2. W e shall m ake this into an equivalence of categories. Define T(2Q to be the category whose objects are triples ( F 1? F 2, ) with F { e S (Z et), F 2 e S (L/et), and a m orphism F { -+ /*7* F 2. A m o r­ phism (F j, F 2, S(A"et). F o r any F e S ( X ci) the canonical m aps F -> f ^ F and F -* 7*7* F induce a m ap F sf(F). To show th at this is an isom orphism , it suffices to show th at F --------- ►j *j*F

i*i*F -------►i*i*j*j*F is C artesian. Since the functor th at takes a sheaf to its stalk preserves fiber products and the family o f such functors is conservative, we may replace the dia-

§3.

IMA GES O F S H EA VES

gram by its stalk. If x e U, then the diagram becomes F 1 X

►1F-X

0

►0

(see (3.5)), which is clearly C artesian. If x 6 Z, then the diagram becom es F* ----- ►U J * F h

Fx ------ U J * F )x , which is again C artesian. Since s and t induce inverse m aps on m orphism s (this again only has to be checked on the stalks (2.17c)), this proves that the categories are equivalent. If Y is any subschem e of X and F is a sheaf on X et, we say th at F has its support on Y if F* = 0 for any x) where N { e Gk-mod, N 2 6 GK-mod, and -► N !2 is a Gfc-hom om orphism . N ote th at (j) may m ore conveniently be regarded as a hom om orphism N{ N 2 th at is com patible with the actions of Gk and GK. Remark 3.13. A sequence in T (Z )

( F f F'2, ')-+(FU F 2,

(F'i, F ' i . f )

is exact if and only if the sequences F\ - + F X -+ F I

F’i

F2

F2

are exact in S (Z et), S(C/et) respectively. This follows from (3.10), (2.15), and the fact th at s(Fi , F 2, 0 ) , = ( F t )x = (F2)x

xeZ x EU

for any ( F i9 F 2, 4>) in T(X). F o r exam ple, for any sheaf F on X there is an exact sequence in T ( X \ 0 -> (0J * F , 0) -> (i*FJ*F, (f)F) - (i*F, 0, 0) - 0. In the n o tatio n introduced below, this corresponds to the sequence 0 -> j \j *F F in S(X). It is possible to define six functors, i*

• R i +lf (M' ), i > 0, such th at the sequence • ■• is exact;

R ‘f ( M ) -» R ‘f ( M " ) 4 R i+lf ( M ' ) -> R i+lf ( M ) -> • • •

§1.

COHOMOLOGY

83

(d) the association in (c) of the long exact sequence to the short exact sequence is functorial. An object M in A is f-acyclic if R f ( M ) — 0 for all i > 0 . If 0

-

M -> N ° -

N l -> N 2 -> • • •

is a resolution of M by /-acy clic objects N ‘, then the objects R' f( M) are canonically isom orphic to the cohom ology objects of the complex 0

j' N° -► /TV1 -> f N 2 - » • • • ,

In order to apply these results to functors from S ( X E\ we m ust show that it has enough injectives. P r o p o s i t i o n 1 .1 . The category S ( X E) has enough injectives. Proof. We give two proofs, one which w orks for the etale site and one which w orks in general. L e m m a 1 .2 . (a) A product o f injectives is injective. (b) Any functor f : A -> B that has an exact left adjoint g: B -» A pre­ serves injectives; f or example, if n is a continuous morphism o f sites, then 7r* preserves injectives when n* is exact. Proof, (a) Easy. (b) Let / be injective in A. The functor M H om B (M, / / ) is isom or­ phic to the functor M ►H o m A (gM, /), and the latter is exact because it is the com posite of tw o exact functors. N ow assum e th at we are on the etale site, and let ux :x -► X be a geo­ m etric point of X . The category S(3cet) is isom orphic to Ab and so has enough injectives. Let F g S(ATet), and choose for each x e X an em bedding u*F Fx of u*F into an injective sheaf. Define F* = [Xc6* Mx*w*Fand F** = f | xgA t/x* F /T h e n the canonical m aps F -> F* and F* F** are m onom orphism s (II.2), and F** is injective according to (1 .2). N ow consider an arb itrary site. Recall th at a family (Aj)jeJ of objects of a category A is a f amily o f generators for A if, given a m onom orphism B -> A in A that is not an isom orphism , there is a j and a m orphism Aj -► A th at does not factor thro u g h B A. L e m m a 1 .3 . Any abelian category that satisfies A B 5 , A B 3 * and pos­ sesses a family o f generators (Ai) has enough injectives. Proof. Let A = A t. Then M H orn (A , M) is an em bedding of A into the category of left R -m odules, where R is the ring H om A (A, A). The p ro o f proceeds by exam ining this functor (see B ucur-D eleanu [ 1 ,6 .3 2 ] ) .

T o prove (1.1 ), it rem ains to show th at S (X E) has a family of generators. F o r any f : U - * X in C / X define Z v to be the s h e a f /Z , w here Z is the co n stan t sheaf on U E defined by Z (II.3.18). Then H orn* (Zv , F) * H o rn , (Z, F\ U) « F(U), and so a family of generators for S ( X E) can be form ed by taking one sheaf Z v for each o f sufficiently m any isom orphism classes of objects of C/ X. (O f course, it has to be show n th at this may be taken to be a set; com pare with the rem arks following (1 1.2 .2).) Remark 1.4. (a) If we consider the etale site on X = spec K, K a field, then the above con stru ctio n gives (Z [ G / H ] ) h , w here H runs through the open subgroups of G = G al ( K J K ) as a set of generators for Gmod = S (X et). T he earlier con stru ctio n gives an injective em bedding for N g G-m od of the form N -► M G(N') where N -> AT is an injective em ­ bedding of N as an abelian g ro u p and M G(N') = C onts M aps (G, AT) is as in Serre [ 8, p. M 2 ]. (b) If, in (a), G is infinite, then it is not difficult to show th at the only projective object in G-mod = S (X et) is 0. T he categories S (X £) will rarely have enough projectives. W ith this, we are able to define the right derived functors of any left exact functor from S (X £) into an abelian category. D e f i n i t i o n s 1.5. (a) The functor r ( X , —): S( XE) -> A b with F(X, F) F(X), is left exact and its right derived functors are w ritten

=

R T (X , - ) = H \ X , - ) = H \ X e, - ) . The g ro u p H l( X E, F) is called the flh cohomology group of X E with values in F. (b) F o r any U -> X in C / X , the right derived functors of F > F(U): S ( X E) -► Ab are w ritten H l( U, F). They should be distinguished (tem po­ rarily) from the groups H \ U , F | U). (c) The inclusion functor i : S{ XE) -> P { X E) is left exact. Its right derived functors are w ritten F) or simply H'(F). (d) F o r any fixed sheaf F0 on X E, the functor F i-> H o m s (F 0, F) is left exact. Its right derived functors are w ritten R* H o m s (F 0, —) = Ext' (F 0, - ) . (e) If F 0 and F x are sheaves on X E, then Horn (F 0. F x) is the sheaf U h-> H orn (F 0 1(7, F { | U). Fix F 0 and consider the functor F i ►H orn (F 0, F ) :S ( X E)

S { X E).

It is left exact, and its right derived functors are w ritten E xt1 (F 0, F).

(f) F o r any continuous m orphism n\X'E> -> X E, the right derived functors R ln^ of the functor nt :S(X'E>) S ( X E) are defined. The sheaves R'n^F are called the higher direct images of F. The rest of this section will be devoted to the study of these functors. Remark 1.6. (a) By definition of E xt1, a short exact sequence 0 -> F' -» f p" _+ o of sheaves induces a long exact sequence of Exts in the second variable. It also induces such a long exact sequence in the first variable, th at is, for a fixed F0 there is a long exact sequence • • • -> Ext' ( F , F 0) -> E xt/ + 1 (F", F0) -+ Exti + 1 (F, F0) -> • • •. Indeed, if F0 -> F is an injective resolution of F 0, then 0 -> H orn (F"9 F) -

Horn (F, F ) -> Horn (F , F ) -> 0

is an exact sequence of complexes, from which the required long exact sequence can be constructed by the usual process. (b) As in any abelian category, E xt1 (F, F ) may also be interpreted as the group of Y oneda extensions, that is, the group of all exact sequences 0 —►F ' -> F, _!

F 0 -> F —> 0

m odulo a certain equivalence relation; see M itchell [1]. (c) H \ X e, F) is a co n trav arian t functor on X E\ if n * :S (X E) -> S (Y ^) is exact then the m aps H \ X E, F) -> H l(X'E , 7c*F) are induced by the obvious m ap H°(X, F) -+ H ° { X \ n*F) and the universal property of derived functors (B ucur-D eleanu [1, 7.12]). (d) The functors in (1.5) and their right derived functors are, to some extent, dependent only on the category S (Y £) of sheaves. F o r exam ­ ple, if n:X' -> X is a universal hom eom orphism , then S(X'el) « S(ATet) (II.3.17), and if an F on X et corresponds to an F' on X'et, then H l( X et, F) * H \ X ;t, F ) , Exls(xet)

0’ F) % Exts(*it) (F 0, F ),

and n#H i{ X \ F ) - m x , F). (e) The functors in (1.5) are not unrelated. F o r example, there is an iso­ m orphism of functors T(Y, —) « H o m (Z , —) where Z denotes the con­ stant sheaf on X E. T hus H \ X , - ) % Ext' (Z, - ) , th at is, the cohom ology groups are special cases of Ext groups. Also it is easy to show that //'( [ /, F) (take an injective resolution and H '(F ) is the presheaf U com pute the tw o groups f f i(F)((7) and H l( U, F ) ). D eeper relations, in­ volving spectral sequences, will be given later in this section and the next.

Example 1.7. We interpret som e of these functors in the case of the etale site on X = spec K, K a field. Recall S(Aret) % G-mod where G = G al ( K SJ K ) . (a) If F corresponds to the m odule M, then HAT, F) = M° , and so H \ X , F) = H l(G, M) = H \ K , M), where the two right hand groups refer to G alois cohom ology in the sense of Serre [ 8]. (b) If F0 and F correspond to M 0 and M, then Horn (F 0, F) = H o m c (M 0, M) and Ext^ (F 0, F) = E xtG (Af 0, M), where Ext^ refers to Exts in the category G-mod. (c) If F 0 and F x correspond to M 0 and M u then Horn (F 0, F ,) cor­ responds to U H o m Ab (M 0,

= (J H orn,, (M 0) M ,), II

//

w here H runs thro u g h the open (norm al) subgroups of G. (The action of H on H o m Ab (Af0, M j) is M x.) W e denote this gro u p by H o m G (M 0, If M 0 is finitely generated, then H o m G (M 0, = H o m Ab (Af0, M j), but not usually otherwise. The derived functors of M h H o m G (M 0, M) will be w ritten E xt^ (M 0, M). T he next lem m a allows us to com pute right-derived functors using classes other than the class of injective objects. Lemma 1.8. Let f : A -* B be a left exact functor o f abelian categories, and assume that A has enough injectives. Let T be a class of objects such that: (a) every object of A is a subobject of an object of T; (b) // A © A e T, then A e T; (c) 0 A' —►A A" -> 0 is exact and A' and A are in T, then A" e T and 0 -> JA' -> JA -> fA" -> 0 is exact. Then all injectives are in T and all elements of T are f-acyclic. Thus the functors R f can be computed using resolutions by objects in T. Proof. Let / be an injective object of A, and em bed it into an object A of T. Since / is injective, it is a direct sum m and of A and so, according to (b), is in T. Let A e T and form an injective resolution 0 -> A -

7° -► I 1 - > • • • - > T~ 1

Define Z 1 inductively so that the sequences 0 -> A -> 7° -> Z 1 0 -> Z 1' -> V

-> 0,

Z l + 1 -> 0

.

are exact. F rom (c), Z 1 is in T, and by induction it follows that each Z l is in T. Also the sequences 0 - * f Z l - > f f - + f Z i+{ - 0 are all exact, which implies th at R f { A ) = 0, all i > 0. Example 1.9. (a) The class of all injective objects in a category A with enough injectives satisfies the conditions of (1.8) for any / . W e have assum ed (a), and it is easy to see that a direct sum m and of an injective is injective. F o r (c), note th at A being injective implies th at the sequence 0 -» A -> A -> A ' -> 0 splits, hence that A", being a direct sum m and of A, is injective, and finally th at any left exact functor preserves the exact­ ness of the sequence because it preserves direct sums. (b) A sheaf F on a topological space X (in the usual sense) is flabby if the restriction m aps F(X) -> F(U) are all surjective. The class of flabby sheaves on X satisfies the conditions o f (1.8) with / = r ( X , —) etc. (see G odem ent [1,11.3]). (c) A sheaf F on a site ( C/X) E will be said to be flabby if H ({U, F) = 0 for all U in C / X and all i > 0. This agrees with the notion of a flask sheaf in Artin [1, p. 39] (com pare (2.12) below), but disagrees with the notion of flasque sheaf in [SG A . 4, V.4.1]. (The definition in [SGA . 4] is that F is flasque if for any sheaf S of sets on X E, the higher right derived functors of F' i-> H orn (S, F') are zero on F ; thus a sheaf is flabby if it is flasque.) Let T be the class of all flabby sheaves on X E. Clearly T contains all injective sheaves, which implies that it satisfies (a) of (1.8). Also, since cohom ology com m utes with finite direct sum s (a finite direct sum of in­ jective resolutions is again an injective resolution), it satisfies (b). The first p art of (c) is obvious, and the second p art also is obvious for / = r(3f, —) and / = H°{U, —) and hence for It also holds, but less obviously, for / = (1.14). P r o p o s i t i o n 1.10. For any sheaf F on X E and any U -> X in C / X , the groups H l(U, F) and H l( UE, F \ U ) are canonically isomorphic. ( UE means the site ( C/ U)E.) Proof. The functor (F i—►F |C ):S (A r£) -> S ( UE) is exact, and so the proposition follows from the next lemma.

L emma 1.11. For any n : U X in C /X , the functor n* = (F h-> F \ U ) takes injective sheaves to injective sheaves. Proof. The functor tt* has an exact left adjoint nr.S{U) S(X) (II.3.18), and hence preserves injectives ( 1 .2). Remark 1.12. (a) It follows from the proposition th at the functor F h F \ U preserves flabby sheaves. An alternative proof of this fact and hence of the proposition will be given in the next section (2.13).

(b) It follows from the p roposition th at H '(F ) is the presheaf U H \ U E, F \ U ) . W e now consider the problem of com puting the higher direct images of a sheaf and their cohom ology. P r o p o s i t i o n 1.13. Let n:X'E- -* X E be continuous, and let F e S(X£>). Then R'n+F = anp(Hl(F)), that is, R ln^F is the sheaf associated with the presheaf U i-> HfU', F | IF), where we have written U ’ = U x x X'. Proof. By definition, n * = anpi, where i is the inclusion S(Ar') P (X '). Let F -► F be an injective resolution of F in S(Ar/). T hen R'n^F is the ith cohom ology group of the com plex of sheaves anp{iF). But a and 7tp are exact, (II.2.15) and (11.2.6), and so com m ute with the form a­ tion of cohom ology. T hus

R % F = anp( H \ i l ) ) = anp{H\F)). C o ro lla ry 1.14. If F is flabby, then R'n^F = 0 for i > 0. Thus flabby resolutions may be used to compute the functors R ln*. Proof. By definition, HfU', F \ V ) = 0 for any flabby F and any U '. The second statem ent follows from (1.8), using only the fact th a t R l n^F = 0 for F flabby. T h e o r e m 1.15. Let n : Y -> X be a quasi-compact morphism; let F be a sheaf on Yet. Let x be a geometric point of X such that k(x) is the sepa­ rable closure of k(x). Let X = spec (9X let Y = Y x x X, and let F be the inverse image of F on Y:

Y -» (C / X )E, there is a spectral sequence

H P( X E, R ^ F ) => H p +q( X E F), where F is a sheaf on X'E-. (b) For any continuous morphisms X E-X'Esequence (Rpn*)(Rqri*)F => R p+Hnn%F,

X E there is a spectral

where F is a sheaf on X E-. Proof. Since flabby sheaves are acyclic for both V(XE, —) and 7t*, both parts are consequences of (Appendix B l) and the following lemma. 1.19. Let n\X'E>-> X E be continuous. Then 7T* maps flabby sheaves to flabby sheaves. Proof. This is proved in the next section (2.13). Remark 1.20. (a) If in (1.19), n* is exact (this is usually so; see the first p aragraph of (II.3)), then n* has an exact left adjoint and preserves injectives, and it is not necessary to wait until the next section for the p ro o f of (1.18). L em m a

(b) Let G be a profinite group. Recall th at for any abelian group N, M g(N) is the G -m odule { / : G -► N \ f continuous} with G acting by (af)(t) = / { t o ). Such a m odule M G(N) is called an induced G-module. If G = G al (k/k) for som e field k and k = ksep, then the induced G -m odules correspond exactly to the sheaves on X et of the form u* F where u : spec k -> spec k = X. Call such a sheaf u*F induced. Now assum e (1.19). As any sheaf on (spec fc)el is flabby, any induced sheaf will be flabby and hence acyclic for T(X, —), T(U, —), n*, and — Any sheaf on X can be em bedded in an induced sheaf because the canonical m ap P -> u^u*F is injective, and it follows that H \ X , —), H l(U, —), H l( ~ ) and R'n^ m ay be com puted using resolutions by induced sheaves. The dictionary (II. 1.9) allows us to deduce th at sim ilar statem ents holds for induced G-m odules. In p articular, resolutions by induced m odules may be used to com pute H l(G, - ) . In the case that M 0 is p ro ­ jective as an abelian gro u p this is also true for the functors H om G (M 0, - ) and H o m G (M 0, - ) . (Exercise!) (c) The above rem ark suggests the following definition: a sheaf F on X el is induced if it is of the form Y \ x e x llx*(Fx) where Fx is a sheaf on x (we have chosen, for each x e X , a geom etric point ux:x -> X ). W rite X' = ] J . x Gx x (disjoint union of schemes). T o give a sheaf on X'ct is the sam e as to give a family of abelian groups (F x)xeX, where Fx is to be regarded as a constant sheaf on x. If u\X' -► X is the obvious m ap, then which shows th at an induced sheaf is flabby (since every sheaf on X'et is flabby). As F -* u+u*F is injective, resolutions by induced sheaves may be used to com pute cohom ology and higher direct images. An im p o rtan t fact is th at every sheaf F on X et has a canonical resolution by induced sheaves called its Godement resolution, which is defined as follows: (i) C°(F) = u#u*(F); there is a canonical m ap e :F -> C °(F); (ii) C l(F) = C °(coker (e)); there is a canonical m ap d°:C°{F)^> C J( F ) ;

(iii) (Inductively) C*(F) = C °(coker (d 1’" 2)); there is a canonical m ap E x {^{Ve) ( F t | (7, F 2 | U). T o see this, we only have to check the following statem ents. (a) the sheaves agree for p ~ 0; but this is the definition of Horn (Fu F 2); (b) Extp (Fu F 2) = 0 and Extg(l/) (F , 11/, F2 \U) = 0, p > 0, if F 2 is in­ jective; this is true by definition in the first case and follows from (1 .1 1 ) in the second; (c) both functors associate long exact sequences to short exact se­ quences in the second variable. This is p art of the definition of the first family of functors and follows from the fact th at a: P -> S is exact for the second family. As a consequence of this and (1.6a) we find th at associated with any exact sequence 0 -* F' F -> F" 0 of sheaves on X E, there is a long exact sequence of sheaves,

0

Horn ( F #, F 2) -+ • • — E U P (F, F 2) -> E x t' ( F , F2) - + E x t p+ l (F'\ F 2) - > • • • .

We return now to the situation studied in (II.3.10), namely, to z -U X ^ u where i is a closed im m ersion and j is an open im m ersion such th at X is the disjoint union of i(Z) and j(U). F o r any sheaf F on X et, i^iF is the largest subsheaf of F that is zero outside Z. The group r(X , i j F ) = T(Z, vF) = K er (F(X) -

F(U))

is called the g roup of sections of F with support on Z. The functor F f—► T(Z, vF) is left exact, and its right derived functors, HP Z{ X , F), are called the cohomology groups of F with support on Z. The functors H%(X, F) are co n trav arian t in (X , U). (The topologists usually write H P( X , (7, - ) for H pz (X, - ) .)

92

111: P r o p o s i t i o n 1 .2 5 .

0

COHOMOLOGY

For any sheaf F on X el there is a long exact sequence,

(i'F)(Z) -> F(X) -> F(U)

H P(X, F) -+ H P(U, F)

-+ H ^ \X , F ) - F o r any sheaf F on X el, there is an exact sequence

Proof.

0

j \j*F -> F -> i#i*F -> 0

(II.3.13). In particular, if F is the constan t sheaf Z, there is an exact sequence

0

—*

iu

—*

z

—>

z

z

—*

0

where we have w ritten l v and Z z for f f f Z and i*/*Z. C orrespondingly there is a long exact sequence, ••■

E xtp (Z, F) -> E xtp (Zy, F) -> E xtp+1 (Zz , F) -> • • •.

W e have already noted th at E xtp (Z, F) = / / P(X, F). The groups H o m S(X) f l u , F)

and

H o m S(l/) (Z, j*F)

are canonically isom orphic (II.3.14), and so E xtp (ZD, F) is the ilh right derived functor of F

H om S(l/) (Z,j*F) = r ( l / , F | l / ) .

Since j* preserves injectives and is exact, this shows th at Extp (Zv , F) = / / p(t/, F\U). T here are canonical isom orphism s H o m S(J0 (Zz , F) * H o m S(Z) (I, fF) * H°Z(X, F). T hus E xtp (Zz , F) « H P Z(X, F). Remark 1.26. A slight refinem ent of the above argum ent shows that for any triple X =d U =3 F, where U and F are open subschem es of A", and any sheaf F on X et, the following sequence is exact: ■• • - H px . u ( X , F) -

H px „ v (X, F)

//f i_ v(£7# F | 17)

- H JT -K *, F) -> • • • . If F is em pty, this is the sam e as the above sequence. P r o p o s i t i o n 1.27. (Excision). Let Z c= X and Z' c : X' he closed sub­ schemes, and let n \X ' -» X be an etale morphism such that the restriction of n to Z ’ is an isomorphism n |Z ': Z ' A Z and n ( X f — Z ) a X — Z. F/ten Fl^(X, F) -* H?Z{X', n*F) is an isomorphism for all p > 0 and all sheaves F on X er

Proof. Since n* is exact and preserves injectives, it suffices to prove this for p — 0. T here is an exact com m utative diagram : 0 -> H°Z(X, F) -> T(X, F) - H I /, F) | 1 i 0 -> H°Z ( X \ F\X') -+ T(X', F) -> T (f/', F)

{U = X - Z) (Uf = X' - Z \

Let y e HZ{ X , F) m ap to zero in H Z ( X \ F \ X ’). If we regard y as an ele­ m ent of T(X, F), then this m eans th at y \ U = 0 = y | X'. As ((7 X, X ' -* X) is a covering of X , this implies th at y = 0. F o r the surjectivity, let / e H Z ( X \ F |X ') and regard y' as an elem ent of HA"', F). As the sections y’ e T(X', F) and 0 e T (l/, F) agree on X' x x U a U \ they arise from a section y e T(X, F), which m ust in fact be in H Z{ X , F). C o r o l l a r y 1.28. Let z be a closed point o f X. Then H P( X , F ) A H p{spec &h Xt29 F) for any sheaf F on X et. Proof. By the proposition, HP(X, F) = H py(Y, F) for any etale neigh­ borhood (F, y) of z such th at only y m aps to z. A ccording to (1.16),

lim H P(Y, F) = H p(spec (9h x ^ F). Finally in this section we define the cohomology groups with compact support H P( X , F) of an F on a separated variety X. The group of sections of F with com pact su p p o rt is defined to be

rc(X, F)

= (J ker (T(X, F)

T(X - Z, F ))

where the Z run th ro u g h the com plete subvarieties of X. Since r c(X, —) is left exact, it is tem pting to define its right derived functors to be the cohom ology groups of F with com pact support, but these are u n inter­ esting. F o r exam ple, if X is affine, then r c(X, F) = # *(X , F) and so R PTC(F) = ® X6*o H P(X,F). Instead we assum e that X can be em ­ bedded j \ X c* X as an open subvariety of a com plete variety X and define H P( X , F) = H P(X, /,F). (Recall th a t/, is exact but does not preserve injectives.) W e shall see later (VI.3, 11) that if F is torsion, H P(X, F) is independent of the choice of X and satisfies a Poincare duality theorem . 1.29. The assumptions on X, X, and F are as above. (a) H C °(X, F) = r f(X, F). (b) The functors H P(X, - ) form a 3-functor, that is, they functorially associate long exact sequences of abelian groups to short exact sequences of sheaves. (c) For any complete subvariety Z of X, there is a canonical morphism of 3-functors H P Z{X> - ) -> H f(X , - ) . P ro p o s itio n

II I:

94

Proof,

COHOMOLOGY

(a) From the exact sequence of sheaves on X

where i:X — X c+ X, we see that tfc°(X, F) = ker (r(X, F)

T(X - X, i% F )).

But T(X - X, i % F ) = lim F(V x g X, F) where V -* X is etale and contains X — X in its image. Thus Hc(X, F) = U ker (T(X, F) - T(V x , X, F)). Suppose that s e TC(X, F), so that s \ X — Z = 0 for some complete Z c X ; then Z, being complete, is closed in X and s \ V n X = 0 if V = X - Z; thus s e HC °(X, F). Conversely, suppose that s e //°(X , F), so that s \ V x X = 0 for some V; then the image V' of V in X is open and contains X — X; thus Z = X — V is a complete subvariety and X —Z = F n X ; a s K x X F' n X is an etale covering, s \ V x X = 0 implies that s \ X — Z = 0 and s 6 r c(X, F). (b) As y, is exact, an exact sequence 0 F' -> F F" -* 0 on X gives rise to an exact sequence 0 -> f F -+ j,F -+ jiF" -+ 0 on X and hence to an exact cohomology sequence on X. (c) There is a canonical map //°(X , F) -► / / C °(X, F) for any sheaf F, and so (c) follows from the universality of derived functors. Rem ark 1.30. Let Z be a closed subscheme of X. For any sheaf F on X, there is an exact sequence 0

F (F |X - Z ) - > F - > / * ( F | Z ) - + 0

where / and i are the immersions X - Z c ^ X , Z c ^ X (11.3.13). Thus there is a long sequence of cohomology groups, • • • -> //?(X - Z, F) - tf?(X, F) - Hf(Z, F) -> • • •. More generally, if X = X 0 3 X t 3 • • • id X r ^ 0 is a sequence of closed subschemes of X, then there is a spectral sequence, EY =

- X , + 1, F) =*

F).

Exercise 1.31. Write A for a (Noetherian) ring and the sheaf it defines on X et. (a) Show that if F is an injective sheaf of T-modules on X et, then F* is an injective T-module. (Use Godement [ 1 , 1.1.4.1] and (1.11).)

(b) Show th at if F0 is pseudocoherent at x, then m P A (Fo,F)* = K*lP A {Fox, Fx). (Take an injective resolution F / ’ of F and apply (II.3.20b) and (a).) (c) D educe th at Ext^ (F, A) = 0, p > 0, if F is a locally free sheaf of /4-modules of finite rank (that is, F\ U{ % An for each C/f in som e covering (Ui) of X) o r if F is pseudocoherent at all geom etric points x of X and A is an injective /1-module, for exam ple, A = Z/(/i). §2. Cech Cohomology It is possible to define a Cech cohom ology theory that isclosely an alo ­ gous to the usual theory over a topological space. Let Ql = (Ui X) ieI be a covering of X in the £-topology on X. F o r any (p + l)-tuple (i0, . . . , ip) with the ij in / we write Uio * x ' '' x x Uip = Uio... /p. Let P be a presheaf on X F. The canonical projection ^ i lno

• • • lip

L/;*0

... 1Ji .... l Pi = U:* 0

X

’’•

X

(/•l j

- I

X

( 7:l j

+ 1

X

*’ '

X

[/;l p

induces a restriction m orphism P ( U lo. . . h . . . lp) ^ P ( U lo. . . tp) which we write am biguously as res,. Define a complex C \ % P) = (C'( H P( Y , P).

Lemma 2.1. T/te map p ( Y , H, t) does not depend on t or the r\j. Proof. Suppose th at x\ with family (>;'•), is an o th er such m ap J -* /. F o r 5 e Cp(%, P) define (^ )io

-

jp -\

= X ( ~ ^ reSfJj0 x ' ' ' x tov»rtr>x ‘ ■' x fl j p - i^Oo • • • r j r x ' j r • • • x ' j p - j )•

T hen kp is a hom om orphism Cp(il, P) -> Cp_1( ^ \ P) with the property that d p' l kp + kp+ l dp = t ' p - r p O n passing to the cohom ology, we find this equation becomes p ( Y , i f t') -

p (tT ,

t) = 0.

Hence, if Y is a refinem ent of i f we get a hom om orphism p ( Y , ®): H P(H, P) -> H P( Y , P) depending only on Y and i l. It follows that if i l, Y , i f are three coverings of X such th at i f is a refinem ent of Y and Y is a refinem ent of i f then p ( i f \ il) = p ( i f , Y ) p ( Y , il). T hus we may define the Cech cohomology groups of P over X to be H P( X E, P ) = lim R P(H9 P), where the limit is taken over all coverings i l of X. Remark 2.2. (a) T he category over which the above limit is taken is not cofiltered. However, let J x be the set of coverings of X m odulo the equivalence relation: H = Y if each is a refinem ent of the other. The notio n of refinem ent induces a partial ordering on J x , which is even filtered because any two coverings i l = (U/) and Y = (Vf have a com ­ m on refinem ent ( l / f x J/). A ccording to (2.1), the functor H h* H p( ^ , P) factors through J* , and limit may be taken over J x . (b) If U -* X is in C / X and P is a presheaf on (C/X)E, then, as above, we may define cohom ology groups H p( i l / U , P) and H P{ U, P) = lim / / p(# /(7 , P) where °U now denotes a covering of U. N ote th at H P(U, P) is defined intrinsically in term s of P. Thus, for exam ple, H P( U , P) is obviously the sam e as H P(U, P\U). The m apping U H P(U, P) extends to give a functor C / X -> Ab, th at is, a presheaf on (C/X)E, which we denote by H p( X e, P) or simply H P{P).

(c) There is a canonical m ap P -* H°{P) whose kernel is the presheaf P 0 defined in the p ro o f of (II.2.11), th at is, P 0( ^ ) is the group of s e P(U) that restrict to zero on som e covering of U. Thus P -> £f°(P) is injective if and only if P is a separated presheaf. H °(P) is separated, and H°(P) aP for any separated P. In general H°{H°(P)) = aP (see the abovem entioned proof). (d) It is not true that Cech cohom ology can be com puted using alter­ nating cochains unless the m aps l / f X are universal m onom orphism s, for exam ple, open im m ersions. In particular, the cohom ology groups arising from a covering (V -» U) consisting of a single m orphism are not necessarily zero for p > 0 (see (e) below, or (2.6)). The proof in Serre [1] fails in our situation because the restriction m aps pss , s' a subsim plex of s, are not well-defined. (e) In the case that X - spec A is affine,