Lectures on Etale Cohomology

202 Pages·1998·1.444 MB·English

by  Milne J.S.

Lectures on
´
Etale Cohomology
J.S. Milne
Version2.21
March22,2013
These are the notes for a course taught at the University of Michigan in 1989 and 1998.
In comparison with my book, the emphasis is on heuristic arguments rather than formal
proofs and on varieties rather than schemes. The notes also discuss the proof of the Weil
conjectures(GrothendieckandDeligne).
BibTeX information
@misc{milneLEC,
author={Milne, James S.},
title={Lectures on Etale Cohomology (v2.21)},
year={2013},
note={Available at www.jmilne.org/math/},
pages={202}
}
v2.01 (August9,1998). Firstversionontheweb;197pages.
v2.10 (May20,2008). Fixedmanyminorerrors;changedTeXstyle;196pages
v2.20 (May3,2012). Corrected;minorimprovements;202pages
v2.21 (March22,2013). Minorcorrections. 202pages
Please send comments and corrections to me at the address on my website www.jmilne.
org/math/
ThephotographisofFallscreek,Fiordland,NewZealand.
Copyright(cid:13)c 1998,2008,2012,2013J.S.Milne.
Singlepapercopiesfornoncommercialpersonalusemaybemadewithoutexplicitpermis-
sionfromthecopyrightholder.
Table of Contents
TableofContents 3
I BasicTheory 7
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 E´taleMorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 TheE´taleFundamentalGroup . . . . . . . . . . . . . . . . . . . . . . . . 25
4 TheLocalRingfortheE´taleTopology . . . . . . . . . . . . . . . . . . . . 31
5 Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6 SheavesfortheE´taleTopology . . . . . . . . . . . . . . . . . . . . . . . . 42
7 TheCategoryofSheavesonX . . . . . . . . . . . . . . . . . . . . . . . . 50
et
8 DirectandInverseImagesofSheaves. . . . . . . . . . . . . . . . . . . . . 58
9 Cohomology: DefinitionandtheBasicProperties. . . . . . . . . . . . . . . 64
10 CˇechCohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
11 PrincipalHomogeneousSpacesandH1. . . . . . . . . . . . . . . . . . . . 76
12 HigherDirectImages;theLeraySpectralSequence. . . . . . . . . . . . . . 81
13 TheWeil-DivisorExactSequenceandtheCohomologyofG . . . . . . . . 84
m
14 TheCohomologyofCurves. . . . . . . . . . . . . . . . . . . . . . . . . . 89
15 CohomologicalDimension. . . . . . . . . . . . . . . . . . . . . . . . . . . 105
16 Purity;theGysinSequence. . . . . . . . . . . . . . . . . . . . . . . . . . . 108
17 TheProperBaseChangeTheorem. . . . . . . . . . . . . . . . . . . . . . . 114
18 CohomologyGroupswithCompactSupport. . . . . . . . . . . . . . . . . 118
19 FinitenessTheorems;SheavesofZ -modules. . . . . . . . . . . . . . . . . 123
`
20 TheSmoothBaseChangeTheorem. . . . . . . . . . . . . . . . . . . . . . 127
21 TheComparisonTheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . 130
22 TheKu¨nnethFormula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
23 TheCycleMap;ChernClasses . . . . . . . . . . . . . . . . . . . . . . . . 138
24 Poincare´ Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
25 LefschetzFixed-PointFormula. . . . . . . . . . . . . . . . . . . . . . . . 147
II ProofoftheWeilConjectures. 151
26 TheWeilConjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
27 ProofoftheWeilConjectures,exceptfortheRiemannHypothesis . . . . . 154
28 PreliminaryReductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
29 TheLefschetzFixedPointFormulaforNonconstantSheaves. . . . . . . . . 163
30 TheMAINLemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
31 TheGeometryofLefschetzPencils . . . . . . . . . . . . . . . . . . . . . . 184
32 TheCohomologyofLefschetzPencils . . . . . . . . . . . . . . . . . . . . 186
33 CompletionoftheProofoftheWeilConjectures. . . . . . . . . . . . . . . 193
34 TheGeometryofEstimates . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Index 201
3
Notationsandconventions
The conventions concerning varieties are the same as those in my notes on Algebraic Ge-
ometry. For example, an affine algebra over a field k is a finitely generated k-algebra A
suchthatA˝ kal ishasnononzeronilpotentsforone(henceevery)algebraicclosurekal
k
of k — this implies that A itself has no nilpotents. With such a k-algebra, we associate
a ringed space Specm.A/ (topological space endowed with a sheaf of k-algebras), and an
affine variety over k is a ringed space isomorphic to one of this form. A variety over k is
aringedspace.X;O /admittingafiniteopencoveringX D SU suchthat.U ;O jU /
X i i X i
is an affine variety for each i and which satisfies the separation axiom. We often use X to
denote.X;O /aswellastheunderlyingtopologicalspace. Aregularmapofvarietieswill
X
sometimesbecalledamorphismofvarieties.
For those who prefer schemes, a variety is a separated geometrically reduced scheme
X offinitetypeoverafieldk withthenonclosedpointsomitted.
IfX isavarietyoverkandK (cid:27) k,thenX.K/isthesetofpointsofX withcoordinates
in K and X or X is the variety1 over K obtained from X. For example, if X D
K =K
Specm.A/,thenX.K/ D Hom .A;K/andX D Specm.A˝ K/. Ingeneral,X.K/
k-alg K k
is just a set, but I usually endow X.C/ with its natural complex topology. A separable
closure ksep of a field k is a field algebraic over k such that every separable polynomial
withcoefficientsink hasarootinksep.
Ourterminologyconcerningschemesisstandard,exceptthatIshallalwaysassumethat
ourringsareNoetherianandthatourschemesarelocallyNoetherian.
Ourterminologyconcerningringsisstandard. Inparticular,ahomomorphismA ! B
of rings maps 1 to 1. A homomorphism A ! B of rings is finite, and B is a finite A-
algebra,ifB isfinitelygeneratedasanA-module. WhenAisalocalring,Ioftendenoteits
(unique)maximalidealbym . Alocalhomomorphismoflocalringsisahomomorphism
A
fWA ! B suchthatf(cid:0)1.m / D m (equivalently,f.m / (cid:26) m ).
B A A B
Generally, when I am drawing motivation from the theory of sheaves on a topological
space,IassumethatthespacesareHausdorff,i.e.,notsomeweirdspaceswithpointswhose
closureisthewholespace.
X (cid:26) Y X isasubsetofY (notnecessarilyproper);
X Ddef Y X isdefinedtobeY,orequalsY bydefinition;
X (cid:25) Y X isisomorphictoY;
X ' Y X andY arecanonicallyisomorphic(orthereisagivenoruniqueisomorphism)
PrerequisitesandReferences
Homological algebra I shall assume some familiarity with the language of abelian cate-
gories and derived functors. There is a summary of these topics in my Class Field Theory
notespp69–76,andcompletepresentationsinseveralbooks,forexample,in
Weibel,C.A.,AnIntroductiontoHomologicalAlgebra2,CambridgeU.P.,1994.
We shall not be able to avoid using spectral sequences — see pp 307–309 of my book
onEtaleCohomologyforabriefsummaryofspectralsequencesandChapter5ofWeibel’s
bookforacompletetreatment.
1OurterminologyfollowsthatofGrothendieck.Thereisaconflictingterminology,basedonWeil’sFoun-
dationsandfrequentlyusedbyworkersinthetheoryofalgebraicgroups,thatwritesthesetheotherwayround.
2Onp9,WeibeldefinesCŒp(cid:141)n D Cn(cid:0)p. Thecorrectoriginaldefinition, universallyusedbyalgebraic
andarithmeticgeometers,isthatCŒp(cid:141)nDCnCp(seeHartshorne,R.,ResiduesandDuality,1966,p26).Also,
inWeibel,a“functorcategory”neednotbeacategory.
4
Sheaf theory Etale cohomology is modelled on the cohomology theory of sheaves in the
usualtopologicalsense. Muchofthematerialinthesenotesparallelsthatin,forexample,
Iversen,B.,CohomologyofSheaves,Springer,1986.
Algebraic geometry I shall assume familiarity with the theory of algebraic varieties, for
example,asinmynotesonAlgebraicGeometry(Math. 631). Also,sometimesIwillmen-
tion schemes, and so the reader should be familiar with the basic language of schemes as,
forexample,thefirst3sectionsofChapterIIofHartshorne,AlgebraicGeometry,Springer
1977,thefirstchapterofEisenbudandHarris,Schemes,Wadsworth,1992,orChapterVof
Shafarevich,BasicAlgebraicGeometry,2ndEdition,Springer,1994.
Forcommutativealgebra,Iusuallyreferto
Atiyah,M.,andMacDonald,I.,IntroductiontoCommutativeAlgebra,Addison-Wesley,
1969.
Etalecohomology Therearethefollowingbooks:
Freitag,E.,andKiehl,R.,EtaleCohomologyandtheWeilConjecture,Springer,1988.
Milne,J.,EtaleCohomology,PrincetonU.P.1980(citedasEC).
Tamme,IntroductiontoEtaleCohomology,Springer.
Theoriginalsourcesare:
Artin,M.,GrothendieckTopologies,LectureNotes,HarvardUniversityMath. Dept. 1962.
Artin,M.,The´ore`mesdeRepre´sentabilite´pourlesEspaceAlge´briques,Pressesdel’Universite´
deMontre´al,Montre´al,1973
Grothendieck,A.,etal.,Se´minairedeGe´ome´trieAlge´brique.
SGA4 (with Artin, M., and Verdier, J.-L.). The´orie des topos et cohomologie e´tale des
sche´mas(1963–64). SpringerLectureNotes1972–73.
SGA41/2 (by Deligne, P., with Boutot, J.-F., Illusie, L., and Verdier, J.-L.) Cohomologie
e´tale. SpringerLectureNotes1977.
SGA5 Cohomologiel-adiqueetfonctionsL(1965–66). SpringerLectureNotes1977.
SGA7 (withDeligne,P.,andKatz,N.)Groupesdemonodromieenge´ome´triealge´briques
(1967–68). SpringerLectureNotes1972–73.
ExceptforSGA41,thesearethefamousseminarsledbyGrothendieckatI.H.E.S..
2
Irefertothefollowingofmynotes.
FT FieldandGaloisTheory(v4.402013).
AG AlgebraicGeometry(v5.222012).
AV AbelianVarieties(v2.002008).
CFT ClassFieldTheory(v4.012011).
Acknowledgements
I thank the following for providing corrections and comments for earlier versions of these
notes: Martin Bright; Rex Cheung; Sungmun Cho; Ming-chang Kang; Michiel Kosters;
AkhilMatthew;CarlMautnerandothersatUTAustin;BehroozMirzaii;EricMoorhouse;
SeanRostami;StevenSpallone;ImmanuelStampfli;SunShenghao;ZachTeitler;Bhupen-
draNathTiwar;RaviVakil.
5
Comment.
Themajortheoremsine´talecohomologyareprovedinSGA4andSGA5,butoftenunder
unnecessarilyrestrictivehypotheses. Someofthesehypotheseswereremovedlater,butby
proofsthatusedmuchofwhatisinthoseseminars. Thusthestructureofthesubjectneeds
tobere-thought. Also,algebraicspacesshouldbemorefullyincorporatedintothesubject
(seeArtin1973). ItislikelythatdeJong’sresolutiontheorem(Smoothness,semi-stability
and alterations. Inst. Hautes E´tudes Sci. Publ. Math. No. 83 (1996), 51–93) will allow
many improvements. Finally, such topics as intersection cohomology and Borel-Moore
homologyneedtobeaddedtotheexposition. Noneofthiswillbeattemptedinthesenotes.
IthinkonereasonwhyGrothendieck,afterSerre’stalkattheChevalleysemi-
nar in 1958, was confident that e´tale localization would give the correct Hi’s
is that once you had the correct cohomology of curves, then by fibrations in
curves and de´vissage you should also reach the higher Hi’s.” (Illusie NAMS
2010.)
Grothendieck came to Harvard for the first time in 1958. Artin: Yes. At that
time, I still didn’t have any idea what a scheme was. But the second time,
in ’61, I had heard that he had this idea for e´tale cohomology, and when he
arrived I asked him if it was all right if I thought about it, and so that was the
beginning. Hewasn’tworkingonitthen—hehadtheideabuthadputitaside.
Hedidn’tworkonituntilIprovedthefirsttheorem. Hewasextremelyactive,
but this may have been the only thing in those years that he really didn’t do
right away, and it’s not clear why. I thought about it that fall, and we argued
aboutwhatthedefinitionshouldbe[laughs]. AndthenIgaveaseminar,fallof
’62. (Recountings: conversationswithMITmathematicians,p.358.)
6
Chapter I
Basic Theory
1 INTRODUCTION
ForavarietyX overthecomplexnumbers,X.C/acquiresatopologyfromthatonC,andso
onecanapplythemachineryofalgebraictopologytoitsstudy. Forexample,onecandefine
theBettinumbersˇr.X/ofX tobethedimensionsofthevectorspacesHr.X.C/;Q/,and
suchtheoremsastheLefschetzfixedpointformulaareavailable.
For a variety X over an arbitrary algebraically closed field k, there is only the Zariski
topology, which is too coarse (i.e., has too few open subsets) for the methods of algebraic
topology to be useful. For example, if X is irreducible, then the groups Hr.X;Z/, com-
putedusingtheZariskitopology,arezeroforallr > 0.
Inthe1940s,Weilobservedthatsomeofhisresultsonthenumbersofpointsoncertain
varieties (curves, abelian varieties, diagonal hypersurfaces..) over finite fields would be
explained by the existence of a cohomology theory giving vector spaces over a field of
characteristiczeroforwhichaLefschetzfixedpointformulaholds. Hisresultspredicteda
formula for the Betti numbers of a diagonal hypersurface in PdC1 over C which was later
verifiedbyDolbeault.
About1958,Grothendieckdefinedthee´tale“topology”ofascheme,andthetheoryof
e´talecohomologywasworkedoutbyhimwiththeassistanceofM.ArtinandJ.-L.Verdier.
The whole theory is closely modelled on the usual theory of sheaves and their derived
functorcohomologyonatopologicalspace. ForavarietyX overC,thee´talecohomology
groups Hr.X ;(cid:3)/ coincide with the complex groups Hr.X.C/;(cid:3)/ when (cid:3) is finite, the
et
ringof`-adicintegersZ ,orthefieldQ of`-adicnumbers(butnotfor(cid:3) D Z). WhenX
` `
is the spectrum of a field K, the e´tale cohomology theory for X coincides with the Galois
cohomology theory of K. Thus e´tale cohomology bridges the gap between the first case,
whichispurelygeometric,andthesecondcase,whichispurelyarithmetic.
Asweshallseeinthecourse,e´talecohomologydoesgivetheexpectedBettinumbers.
Moreover,itsatisfiesanaloguesoftheEilenberg-Steenrodaxioms,thePoincare´ dualitythe-
orem,theLefschetzfixedpointformula,theLerayspectralsequence,etc.. Theintersection
cohomologyofGoreskyandMacPhersonhasane´taleanalogue,whichprovidesaPoincare´
dualitytheoremforsingular varieties. Etalecohomologyhasbeenbrilliantlysuccessfulin
explainingWeil’sobservation.
AlgebraicTopology
Webrieflyreviewtheoriginsofthetheoryonwhiche´talecohomologyismodelled.
7
8 CHAPTERI. BASICTHEORY
Algebraic topology had its origins in the late 19th century with the work of Riemann,
Betti, and Poincare´ on “homology numbers”. After an observation of Emmy Noether, the
focusshiftedto“homologygroups”. Bythe1950stherewereseveraldifferentmethodsof
attaching(co)homologygroupstoatopologicalspace,forexample,therewerethesingular
homology groups of Veblen, Alexander, and Lefschetz, the relative homology groups of
Lefschetz, the Vietoris homology groups, the C˘ech homology groups, and the Alexander
cohomologygroups.
ThesituationwasgreatlyclarifiedbyEilenbergandSteenrod19531,whichshowedthat
forany“admissible”categoryofpairsoftopologicalspaces,thereisexactlyonecohomol-
ogy theory satisfying a certain short list of axioms. Consider, for example, the category
whose objects are the pairs .X;Z/ with X a locally compact topological space and Z a
closed subset of X, and whose morphisms are the continuous maps of pairs. A cohomol-
ogy theory on this category is a contravariant functor attaching to each pair a sequence of
abeliangroupsandmaps
(cid:1)(cid:1)(cid:1) ! Hr(cid:0)1.U/ ! Hr.X/ ! Hr.X/ ! Hr.U/ ! (cid:1)(cid:1)(cid:1) ; U D X (cid:114)Z;
Z
satisfyingthefollowingaxioms:
(a) (exactnessaxiom)theabovesequenceisexact;
(cid:3)
(b) (homotopyaxiom)themapf dependsonlyonthehomotopyclassoff;
(c) (excision)ifV isopeninX anditsclosureisdisjointfromZ,thentheinclusionmap
.X (cid:114)V;Z/ ! .X;Z/inducesanisomorphismsHr.X/ ! Hr.X (cid:114)V/;
Z Z
(d) (dimensionaxiom)ifX consistsofasinglepoint,thenHr.P/ D 0forr ¤ 0.
The topologists usually write Hr.X;U/ for the group Hr.X/. The axioms for a ho-
Z
mology theory are similar to the above except that the directions of all the arrows are re-
versed. If .X;Z/ 7! Hr.X/ is a cohomology theory such that the Hr.X/ are locally
Z Z
compact abelian groups (e.g., discrete or compact), then .X;Z/ 7! Hr.X/_ (Pontryagin
Z
dual)isahomologytheory. Inthisapproachthereisimplicitlyasinglecoefficientgroup.
In the 1940s, Leray attempted to understand the relation between the cohomology
groups of two spaces X and Y for which a continuous map Y ! X is given. This led
himtotheintroductionofsheaves(localsystemsofcoefficientgroups),sheafcohomology,
and spectral sequences (about the same time as Roger Lyndon, who was trying to under-
standtherelationsbetweenthecohomologiesofagroupG,anormalsubgroupN,andthe
quotientgroupG=N).
Derived functors were used systematically in Cartan and Eilenberg 19562, and in his
1955 thesis a student of Eilenberg, Buchsbaum, defined the notion of an abelian category
andextendedtheCartan-Eilenbergtheoryofderivedfunctorstosuchcategories. (Thename
“abeliancategory”isduetoGrothendieck).
Finally Grothendieck, in his famous 1957 Tohoˆku paper3, showed that the category
of sheaves of abelian groups on a topological space is an abelian category with enough
injectives,andsoonecandefinethecohomologygroupsofthesheavesonaspaceX asthe
right derived functors of the functor taking a sheaf to its abelian group of global sections.
One recovers the cohomology of a fixed coefficient group (cid:3) as the cohomology of the
1FoundationsofAlgebraicTopology,Princeton.
2HomologicalAlgebra,Princeton.
3Surquelquespointsd’alge`brehomologique,ToˆhokuMath.J.9,119–221.
1. INTRODUCTION 9
constant sheaf it defines. This is now the accepted definition of the cohomology groups,
and it is the approach we follow to define the e´tale cohomology groups. Instead of fixing
the coefficient group and having to consider all (admissible) pairs of topological spaces in
order tocharacterize the cohomologygroups, we fix thetopological space butconsider all
sheavesonthespace.
Briefreviewofsheafcohomology
Let X be a topological space. We make the open subsets of X into a category with the
inclusionsastheonlymorphisms,anddefineapresheaftobeacontravariantfunctorfrom
thiscategorytothecategoryAbofabeliangroups. Thus,suchapresheafF attachestoevery
opensubsetU ofX anabeliangroupF.U/andtoeveryinclusionV (cid:26) U arestrictionmap
(cid:26)UWF.U/ ! F.V/insuchwaythat(cid:26)U D id and,wheneverW (cid:26) V (cid:26) U,
V U F.U/
(cid:26)U D (cid:26)V ı(cid:26)U:
W W V
For historical reasons, the elements of F.U/ are called the sections of F over U, and the
elementsofF.X/theglobalsectionsofF. Also,onesometimeswrites(cid:0).U;F/forF.U/
andsjV for(cid:26)U.s/.
V
ApresheafF issaidtobeasheafif
(a) a section f 2 F.U/ is determined by its restrictions (cid:26)Ui.f/ to the sets of an open
U
covering.Ui/i2I ofU;
(b) afamilyofsectionsfi 2 F.Ui/for.Ui/i2I anopencoveringofU arisesbyrestric-
tionfromasectionf 2 F.U/iff jU \U D f jU \U foralli andj.
i i j j i j
Inotherwords,F isasheafif,foreveryopencovering.Ui/i2I ofanopensubsetU ofX,
thesequence
Y Y
F.U/ ! F.U / ⇒ F.U \U /;
i i j
i2I .i;j/2I(cid:2)I
is exact — by definition, this means that the first arrow maps F.U/ injectively onto the
subset of QF.U / on which the next two arrows agree. The first arrow sends f 2
i
F.U/ to the family .fjUi/i2I, and the next two arrows send .fi/i2I to the families
.fijUi \Uj/.i;j/2I(cid:2)I and .fjjUi \Uj/.i;j/2I(cid:2)I respectively. Since we are considering
only(pre)sheavesofabeliangroups,wecanrestatetheconditionas: thesequence
0 ! F.U/ ! Q F.U / ! Q F.U \U /
i2I i .i;j/2I(cid:2)I i j
f 7! .fjU /;
i
.f / 7! .f jU \U (cid:0)f jU \U /
i j i j i i j
is exact. When applied to the empty covering of the empty set, the condition implies that
F.;/ D 0.4
For example, if (cid:3) is a topological abelian group (e.g., R or C), then we can define
a sheaf on any topological space X by setting F.U/ equal to the set of continuous maps
U ! (cid:3)andtakingtherestrictionmapstobetheusualrestrictionoffunctions.
When (cid:3) has the discrete topology, every continuous map fWU ! (cid:3) is constant on
eachconnectedcomponentofU,andhencefactorsthrough(cid:25) .U/,thespaceofconnected
0
4Theemptyset;iscoveredbyafamilyofmapsindexedbytheemptyset,andsothesheafconditionsays
thatF.;/isequaltoaproductofabeliangroupsindexedby;.Butineverycategory,aproductoveranempty
indexingsetisthefinalobject(whenitexists),which,inthecaseofabeliangroups,iszero.
10 CHAPTERI. BASICTHEORY
componentsofU. Whenthislastspaceisdiscrete,F.U/isthesetofallmaps(cid:25) .U/ ! (cid:3),
0
i.e.,F.U/ D (cid:3)(cid:25)0.U/. Inthiscase,wecallF theconstantsheafdefinedbytheabeliangroup
(cid:3).
Grothendieckshowedthat,withthenaturalstructures,thesheavesonX formanabelian
category. Thus, we have the notion of an injective sheaf: it is a sheaf I such that for any
subsheaf F0 of a sheaf F, every homomorphism F0 ! I extends to a homomorphism
F ! I. Grothendieck showed that every sheaf can be embedded into an injective sheaf.
The functor F 7! F.X/ from the category of sheaves on X to the category of abelian
groups is left exact but not (in general) right exact. We define Hr.X;(cid:1)/ to be its rth right
derivedfunctor. Thus,givenasheafF,wechooseanexactsequence
0 ! F ! I0 ! I1 ! I2 ! (cid:1)(cid:1)(cid:1)
with each Ir injective, and we set Hr.X;F/ equal to the rth cohomology group of the
complexofabeliangroups
I0.X/ ! I1.X/ ! I2.X/ ! (cid:1)(cid:1)(cid:1)
Whileinjectiveresolutionsareusefulfordefiningthecohomologygroups,theyarenot
convenient for computing it. Instead, one defines a sheaf F to be flabby if the restriction
mapsF.U/ ! F.V/aresurjectiveforallopenU (cid:27) V,andshowsthatHr.X;F/ D 0if
F isflabby. Thus,resolutionsbyflabbysheavescanbeusedtocomputecohomology.
TheinadequacyoftheZariskitopology
Aswenotedabove,formanypurposes,theZariskitopologyhastoofewopensubsets. We
listsomesituationswherethisisevident.
Thecohomologygroupsarezero RecallthatatopologicalspaceX issaidtobeirreducible
if any two nonempty open subsets of X have nonempty intersection, and that a variety (or
scheme)issaidtobeirreducibleifitisirreducibleasa(Zariski)topologicalspace.
THEOREM 1.1 (GROTHENDIECK’S THEOREM) If X is an irreducible topological space,
thenHr.X;F/ D 0forallconstantsheavesandallr > 0.
PROOF. Clearly, every open subset U of an irreducible topological space is connected.
Hence, if F is the constant sheaf defined by the group (cid:3), then F.U/ D (cid:3) for every
nonemptyopenU. Inparticular,F isflabby,andsoHr.X;F/ D 0forr > 0. (cid:50)
REMARK 1.2 The C˘ech cohomology groups are also zero. Let U D .Ui/i2I be an open
covering of X. Then the C˘ech cohomology groups of the covering U are the cohomology
groupsofacomplexwhoserthgroupis
Y
(cid:3)
.i0;:::;ir/2IrC1; Ui0\:::\Uir¤;
withtheobviousmaps. ForanirreduciblespaceX,thiscomplexisindependentofthespace
X; in fact, it depends only on the cardinality of I (assuming the U are nonempty). It is
i
easytoconstructacontractinghomotopyforthecomplex,andsodeducethatthecomplex
isexact.