Grothendieck Duality and Base Change

302 Pages·2000·1.496 MB·English

by  Brian Conrad

Grothendieck Duality and Base Change
Brian Conrad

iii
Preface
Grothendieck duality theory on noetherian schemes, particularly the no-
tion of a dualizing sheaf, plays a fundamental role in contexts as diverse as the
arithmetictheoryofmodularforms[DR],[M]andthestudyofmodulispacesof
curves[DM]. Thegoalofthetheoryistoproduceatracemapintermsofwhich
onecanformulatedualityresultsforthecohomologyofcoherentsheaves. Inthe
‘classical’ case of Serre duality for a proper, smooth, geometrically connected,
n-dimensional scheme X over a (cid:12)eld k, the trace map amounts to a canonical
k-linear map t : Hn(X;Ωn ) ! k such that (among other things) for any
X X=k
locallyfreecoherentsheafF onX withdualsheafF_ =HomOX(F;OX),the
cup product yields a pairing of (cid:12)nite-dimensional k-vector spaces
Hi(X;F)(cid:10)Hn(cid:0)i(X;F_(cid:10)Ωn )!Hn(X;Ωn )(cid:0)t!X k
X=k X=k
which is a perfect pairing for all i. In particular, using F = O and i = 0,
X
we see that dim Hn(X;Ωn ) = 1 and t is non-zero, so t must be an iso-
k X=k X X
morphism. Grothendieck duality extends this to a relative situation, but even
the relative case where the base is a discrete valuation ring is highly non-trivial.
The foundations of Grothendieck duality theory, based on residual complexes,
are worked out in Hartshorne’s Residues and Duality (hereafter denoted [RD]).
These foundations make the duality theory quite computable in terms of differ-
ential forms and residues, and such computability can be very useful (e.g., see
Berthelot’s thesis [Be, VII, x1.2] or Mazur’s pioneering work on the Eisenstein
ideal [M, II, p.121]).
In the construction of this theory in [RD] there are some essential compat-
ibilities and explications of abstract results which are not proven and are quite
difficult to verify. The hardest compatibility in the theory, and also one of the
mostimportant, isthebasechangecompatibilityofthetracemapinthecaseof
properCohen-Macaulaymorphismswithpurerelativedimension(e.g., (cid:13)atfam-
ilies of semistable curves). Ignoring the base change question, there are simpler
methods for obtaining duality theorems in the projective CM case (see [AK1],
[K],whichalsohaveresultsintheprojectivenon-CMcase). However,theredoes
not seem to be a published proof of the duality theorem in the general proper
CM case over a locally noetherian base, let alone an analysis of its behavior
with respect to base change. For example, the rather important special case of
compatibility of the trace map with respect to base change to a geometric (cid:12)ber
isnotatallobvious,evenifwerestrictattentiontodualityforprojectivesmooth
maps. This was our original source of motivation in this topic and (amazingly)
even this special case does not seem to be available in the published literature.
The aim of this book is to prove the hard unproven compatibilities in the
foundations given in [RD], particularly base change compatibility of the trace
map,andtoexplicatesomeimportantconsequencesandexamplesoftheabstract
theory. This book should be therefore be viewed as a companion to [RD], and
is by no means a logically independent treatment of the theory from the very
beginning. Indeed, we often appeal to results proven in [RD] along the way,
iv
rather than reprove everything from scratch (and we are careful to avoid any
circular reasoning). More precisely, we will give the de(cid:12)nitions of most of the
basic constructions we need from [RD] (aside from a few cases in which the
de(cid:12)nitions are very elaborate, in which case we refer to speci(cid:12)c places in [RD]
for the relevant de(cid:12)nitions), and we will sometimes refer to [RD] for proofs of
various properties of these basic constructions. It is our hope that by providing
a detailed explanation of some of the more difficult aspects of the foundations,
Grothendieck’s work on duality for coherent sheaves will be better understood
by a wider audience.
There is a different approach to duality, and particularly the base change
problem for the trace map, which should be mentioned. In [LLT], Lipman
worksoutavastgeneralizationofGrothendieck’stheory,usingDeligne’sabstract
constructionofatracemap[RD,Appendix]inplaceofGrothendieck’s‘concrete’
approachviaresidualcomplexes(asinthemaintextof[RD]). Lipman’stheory
requires a lot more preliminary work with derived categories than is needed
in [RD], but it yields a more general theory without noetherian conditions or
boundednesshypothesesonderivedcategories(thoughthe‘old’theoryin[RD]is
adequatefornearlyallpracticalpurposes). Intheseterms, Lipmansaysthathe
candeducethebasechangecompatibilityofthetracemapintheproperCohen-
Macaulay case. However, it seems unwise to ignore the foundations based on
residual complexes, because of their usefulness in calculations.
In any case, it is unlikely that Lipman’s powerful abstract methods lead to
a much shorter proof that the trace map is compatible with base change. The
reason is that ultimately one wants to have statements in terms of sheaves of
differentials or (at least in the projective case) their Ext’s, with concrete base
change maps. Translating Deligne’s more abstract approach into these terms is
anon-trivialmatterwhichcannotbeignored,andthisappearstocanceloutany
appearance of brevity in the proofs. Either one builds the concreteness directly
into the foundations (as in [RD] and this book) and then one needs to check a
lot of commutative diagrams, or else one uses abstract foundations and has to
do a lot of hard work to make the results concrete. To quote Lipman on the
issue of the choice of foundations,
\::: TheabstractapproachofDeligneandVerdier,andthemore
recent one of Neeman, seem on the surface to avoid many of the
grubby details; but when you go beneath the surface to work out
the concrete interpretations of the abstractly de(cid:12)ned dualizing
functors, it turns out to be not much shorter. I don’t know of
any royal road ::: "
Contents
Chapter 1. Introduction 1
1.1. Overview and Motivation 1
1.2. Notation and Terminology 5
1.3. Sign Conventions 7
Chapter 2. Basic Compatibilities 21
2.1. General Nonsense 21
2.2. Smooth and Finite Maps 28
2.3. Projective Space and the Trace Map 32
2.4. Proofs of Properties of the Projective Trace 39
2.5. The Fundamental Local Isomorphism 52
2.6. Proofs of Properties of the Fundamental Local Isomorphism 56
2.7. Compatibilities between ((cid:1))♯ and ((cid:1))♭ 76
2.8. Gluing ((cid:1))♯ and ((cid:1))♭ 99
Chapter 3. Duality Foundations 105
3.1. Dualizing Complexes 105
3.2. Residual Complexes 125
3.3. The Functor ((cid:1))! and Residual Complexes 133
3.4. The Trace Map Tr and Grothendieck-Serre Duality 146
f
3.5. Dualizing Sheaves and CM maps 153
3.6. Base Change for Dualizing Sheaves 164
Chapter 4. Proof of Main Theorem 175
4.1. Case of an Artinian Quotient 175
4.2. Case of Artin Local Base Schemes 182
4.3. Duality for Proper CM Maps in the Locally Noetherian Case 189
4.4. Conclusion of Proof of Duality Theorem 201
Chapter 5. Examples 217
5.1. Higher Direct Images 217
5.2. Curves 225
Appendix A. Residues and Cohomology with Supports 237
A.1. Statement of Results 237
A.2. Proofs 241
Appendix B. Trace Map on Smooth Curves 271
v
vi CONTENTS
B.1. Motivation 271
B.2. Preparations 273
B.3. The Proof 277
B.4. Duality on Jacobians 283
Bibliography 291
Index 293
CHAPTER 1
Introduction
1.1. Overview and Motivation
Let f : X ! Y be a proper, surjective, smooth map of schemes, with all
(cid:12)bersequidimensionalwithdimensionn,andlet! =Ωn . Grothendieck’s
X=Y X=Y
duality theory [RD, VII, 4.1] produces a trace map
(1.1.1) (cid:13)f :Rnf(cid:3)(!X=Y)!OY
which is an isomorphism when f has geometrically connected (cid:12)bers. When
n=0, this is just the usual trace map f(cid:3)(OX)!OY.
Oneaspectofthe‘duality’isthatifF isanylocallyfreesheafof(cid:12)niterank
on X with dual F_ =HomOX(F;OX), the cup product pairing
Rif(cid:3)(F)(cid:10)Rn(cid:0)if(cid:3)(F_(cid:10)!X=Y)(cid:0)[!Rnf(cid:3)(!X=Y)(cid:0)(cid:13)!f OY
(where [ denotes cup product) induces a map
(1.1.2) Rn(cid:0)if(cid:3)(F_(cid:10)!X=Y)!HomOY(Rif(cid:3)(F);OY)
which is an isomorphism if Rjf(cid:3)(F) is locally free (necessarily of (cid:12)nite rank)
for all j. To make the proofs work, one actually establishes a more general
isomorphism on the level of derived categories in the locally noetherian case.
Animportantpropertyofthetracemapisthatitisofformationcompatible
with arbitrary (e.g., non-(cid:13)at) base change. More precisely, recall that if
X′ u′ //X
f′ f
(cid:15)(cid:15) (cid:15)(cid:15)
//
Y′ Y
u
isacartesiandiagramofschemes,thenthereisanaturalbasechangemorphism
u(cid:3)Rnf(cid:3)(!X=Y)!Rnf(cid:3)′(u′(cid:3)!X=Y)≃Rnf(cid:3)′(!X′=Y′):
Since f is proper and (cid:12)nitely presented with n-dimensional (cid:12)bers, it follows
from direct limit arguments and Grothendieck’s theorem on formal functions
thatRnf(cid:3) isarightexactfunctoronquasi-coherentsheaves,sothisbasechange
1
2 1. INTRODUCTION
morphism is an isomorphism. The compatibility of the trace map with base
change means that the diagram
(1.1.3) u(cid:3)Rnf(cid:3)(!X=Y) ≃ //Rnf(cid:3)′(!X′=Y′)
u(cid:3)((cid:13)f) (cid:13)f′
(cid:15)(cid:15) (cid:15)(cid:15)
u(cid:3)(OY) OY′
commutes. One of the main goals of this book is to prove this commutativity.
Thisisneededovercertainbasesin[RD]inordertode(cid:12)nethetracemap(cid:13) over
f
an arbitrary base. More importantly, the commutativity of (1.1.3) is crucial in
the proof that (cid:13) is an isomorphism when f has geometrically connected (cid:12)bers.
f
The standard references [RD], [Verd] ignore the veri(cid:12)cation that (1.1.3)
commutes. In [RD] this is left to the reader and [Verd] only checks the case
of (cid:13)at base change. From the point of view of either of these references, the
analysis of (1.1.3) is a non-trivial matter. The very de(cid:12)nition of the trace map
(cid:13) in [RD] involves a series of intermediate steps for which general base change
f
makes no sense; general base change maps are only meaningful for the ‘outer
pieces’ Rnf(cid:3)(!X=Y) and OY left at the end of the construction. This makes the
commutativity of (1.1.3) seem like a miracle. The methods in [Verd] take place
in derived categories with \bounded below" conditions. This leads to technical
problems for a base change such as p : Spec(A=m) ,! Spec(A) with (A;m) a
non-regularlocalnoetherianring,inwhichcasetherightexactp(cid:3) doesnothave
(cid:12)nite homological dimension (so Lp(cid:3) does not make sense as a functor between
\bounded below" derived categories). Moreover, Deligne’s construction of the
trace map in [RD, Appendix], upon which [Verd] is based, is so abstract that
itisanon-trivialtasktorelateDeligne’sconstructiontothesheafRnf(cid:3)(Ωn ).
X=Y
However, a direct relation between the duality theorem and differential forms is
essential for many important calculations (e.g., [M, x6, x14(p.121)]).
Iinitiallytriedtoverifythecommutativityof(1.1.3)byadirectcalculation
withExt sheavesinthesmoothprojectivecase. Thisapproachquicklygetsstuck
onthefactthatbasechangemapsforExtO ’s[AK2, x1]areonlyde(cid:12)nedwhen
X
thesheavesinvolvedsatisfycertain(cid:13)atnessandquasi-coherenceconditions,and,
more importantly, this de(cid:12)nition is local on X and not local over Y (unless X
is (cid:12)nite over Y). When I asked Deligne about this difficulty, he agreed that
theprojectivesmoothcaseseemedpuzzlingifonetriedtoanalyzeitdirectlyvia
Ext-sheaves. Asubsequentdiscussionaboutthegeneralcaseof(1.1.3)withsome
otherexpertswasalsoinconclusive. Despitethefactthateveryonebelievesthat
(1.1.3) commutes, no published proof seems to exist. Nevertheless, it is widely
used. Aproofofthecommutativityof(1.1.3)willbegiveninthisbook;itmakes
essential use of a generalization to the proper Cohen-Macaulay case, using the
foundations of duality theory in [RD].
Many of the unveri(cid:12)ed compatibilities in [RD] are not hard to check (and
compatibilities with respect to translations, (cid:13)at base change, and composites of
schememorphismsareoftentrivialtoverify),butsomeunveri(cid:12)edcompatibilities
in [RD] are genuinely difficult to prove and their truth depends in an essential
1.1. OVERVIEW AND MOTIVATION 3
way on a correct choice of sign conventions. Thus, in order to construct a
global theory and to make explicit calculations, we must (cid:12)x once and for all a
correct and consistent choice of sign conventions in the main constructions of
Grothendieckdualitytheory(e.g.,forKoszulcomplexes,residues,etc.). Anyone
who has ever used an argument of the form \x = (cid:0)x, so therefore x = 0" (in
a Z[1=2]-module) can appreciate the importance of eliminating sign ambiguities
in the foundations, even if one admits that a global theory ought to exist.
Priortothestatementandproofofthemaindualitytheorem[RD,VII,3.4],
nearlyallofthedifficultcompatibilityproblemsin[RD]areinthefoundational
chapter [RD, III]. Most of the remaining omitted proofs and omitted compat-
ibilities are quite straightfoward to (cid:12)ll in. Thus, we devote approximately the
(cid:12)rst half of the book (Chapters 1 and 2) to justifying certain difficult compat-
ibilities which arise without proof in [RD, III]. Chapters 3 and 4 are devoted
to developing the theory of the dualizing sheaf for Cohen-Macaulay morphisms
and proving that the base change diagram (1.1.3), as well as a generalization to
theproperCohen-Macaulaycase, commutes. WeconcludeinChapter5andthe
appendices by giving some important consequences and examples of the general
theory.
There are two observations that enable us to successfully analyze the base
change question for the trace map. First of all, for a technical reason to be
explained shortly, we relax the smoothness condition to the condition that the
propermapf bea(proper)CMmap|thatis,locally(cid:12)nitelypresentedand(cid:13)at
withCohen-Macaulay(cid:12)bers(soiff is(cid:12)niteand(cid:12)nitelypresented,thenf isCM
ifandonlyiff is(cid:13)at)|andthen! =Ωn hastobereplacedbya‘relative
X=Y X=Y
dualizingsheaf’! . Secondandmoreimportantly, weignoreadirecttreatment
f
of the projective case. Instead, we study the de(cid:12)nition in [RD, VI, VII] of !
f
and the trace map (cid:13) for proper morphisms f : X ! Y to noetherian schemes
f
Y whichadmitadualizingcomplex(thisincludesY of(cid:12)nitetypeoverZorover
a (cid:12)eld, or more generally over a complete local noetherian ring). This de(cid:12)nition
of (! ;(cid:13) ) uses the theory of residual complexes, to be discussed in Chapter 3,
f f
and the de(cid:12)nition of (cid:13) is built up from derived category trace maps associated
f
to certain (cid:12)nite morphisms which are ‘supported’ at closed points in the (cid:12)bers
of f. The auxiliary (cid:12)nite maps arising in this ‘residual complex’ de(cid:12)nition of (cid:13)
f
cannotgenerallybechosentobesmooth(i.e.,(cid:19)etale)iff issmooth,buttheycan
be chosen to be CM (i.e., (cid:13)at) if f is CM. Thus, our goal is to use the ‘residual
complex’de(cid:12)nitionofthetracemapandthebasechangetheoryofExt’s[AK2,
x1]inordertoformulateabasechangetheoryfor(! ;(cid:13) )intheCMcaseandto
f f
reducethebasechangeproblemforproperCMmapstothespecialcaseof(cid:12)nite
(cid:13)at maps, in which case a direct calculation is possible. We emphasize that this
method forces us to go outside of the category of smooth maps, but one wants
a duality theorem for possibly non-smooth Cohen-Macaulay maps anyway (e.g.,
for the study of (cid:13)at families of semistable curves).
We now describe the basic idea that makes reduction to the (cid:12)nite case
plausible. Recall the following fact [EGA, IV , 19.2.9]: if f : X ! Y is a CM
4
mapofschemesandx2X d=eff(cid:0)1(y)isaclosedpointinthe(cid:12)berovery, there
y
4 1. INTRODUCTION
exists a commutative diagram of locally (cid:12)nitely presented maps
i //
Z @ X
@
@
@
@
g @@ (cid:15)(cid:15)f
Y
in which g is quasi-(cid:12)nite, separated, and (cid:13)at (hence CM) and i is an immersion
which passes through x. Moreover, i can be chosen to contain any desired
in(cid:12)nitesimalthickeningSpec(O =mn)offxgalongX . Tode(cid:12)nei,wesimply
Xy;x x y
choose a system of parameters in the local Cohen-Macaulay ring O which
Xy;x
lie in mn and we lift these to a small neighborhood of x in X; these lifted
x
functions then de(cid:12)ne the subscheme Z. Note that if we restricted ourselves to
the case of smooth Y-schemes, such a quasi-(cid:12)nite smooth g exists if and only
if the (cid:12)nite extension k(x)=k(y) is separable, and such a Z never contains the
higher in(cid:12)nitesimal neighborhoods of x in f(cid:0)1(y).
The theory of residual complexes (used in the de(cid:12)nitions of ! , (cid:13) ) is a
f f
priori compatible with base change to a henselization of the local ring at any
point in Y, so it is easy to reduce the general base change question to the case
of a local henselian Y. When Y is local henselian with closed point y, it follows
from Zariski’s Main Theorem [EGA, IV , 18.12.13] that Z as above breaks up
4
asadisjointunionofapartZ whichis(cid:12)nite((cid:13)at)overY andapartZ′ which
(cid:12)n
has empty closed (cid:12)ber (so Z′ does not contain x and therefore can be ignored).
By letting x run through the closed points in the (cid:12)bers of f and choosing such
Z’s as above containing larger and larger in(cid:12)nitesimal neighborhoods of x, we
are almost able to reduce to the study of the Z ’s in place of X, which would
(cid:12)n
reduceustothe(cid:12)nite(cid:13)atcase. Tobemoreprecise,wewillusesuchi’sasabove
to reduce the general base change question (1.1.3) to the following two cases:
(cid:15) Y, Y′ are local artin schemes and f is (cid:12)nite (cid:13)at,
(cid:15) Y =Spec(A) with A a local noetherian ring admitting a dualizing com-
plex (e.g., a complete local noetherian ring or a local ring of a (cid:12)nite type
Z-algebra) and the base change is to an artinian quotient of A.
WhenthedetailsarecarriedoutinChapter4,wewillusetheKrullIntersection
Theorem to bypass the need to use henselizations or Zariski’s Main Theorem,
but the above idea was our source of motivation.
Inx3.1{x3.4,wereviewtheconstructionofdualitytheoryintermsofresidual
complexes and in x3.5{x3.6 we use this to formulate the base change question in
terms of residual complexes and Ext-sheaves rather than in terms of sheaves of
relative differentials. The advantage of this reformulation is that it makes sense
for proper CM maps, not just proper smooth maps. In particular, we can work
with(cid:12)nite(cid:13)atmaps, whichare\usually"notsmooth. Itsufficestoconsiderthe
twospecialcasesabove,andthesearetreatedbydirectcalculationswithresidual
complexesinx4.1{x4.2. ThisgivesadualitytheoremforproperCohen-Macaulay
mapsf :X !Y withpurerelativedimensionoverabaseY whichisnoetherian
andadmitsadualizingcomplex,andweprovethatthecorrespondingtracemap
is compatible with any base change Y′ ! Y where Y′ is also noetherian and