Almost Ring Theory

313 Pages·2003·2.21 MB·English

by  Ofer Gabber

1800
Lecture Notes in Mathematics
Editors:
J.--M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
3
Berlin
Heidelberg
NewYork
HongKong
London
Milan
Paris
Tokyo
Ofer Gabber Lorenzo Ramero
Almost Ring Theory
1 3
Authors
OferGabber LorenzoRamero
IHES InstitutdeMathe´matiques
LeBois-Marie Universite´deBordeauxI
35,routedeChartres 351,coursdelaLibe´ration
91440Bures-sur-Yvette,France 33405Talence,France
e-mail:[email protected] e-mail:[email protected]
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ISSN0075-8434
ISBN3-540-40594-1Springer-VerlagBerlinHeidelbergNewYork
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CONTENTS
1. Introduction 1
1.1. Motivationsandalittlehistory 1
1.2. Themethodofalmoste´taleextensions 3
1.3. Contentsofthisbook 7
1.4. Theviewfromabove 9
1.5. Acknowledgements 10
2. Homologicaltheory 11
2.1. Somering-theoreticpreliminaries 11
2.2. Categoriesofalmostmodulesandalgebras 16
2.3. Uniformspacesofalmostmodules 23
2.4. Almosthomologicalalgebra 31
2.5. Almosthomotopicalalgebra 40
3. Almostringtheory 50
3.1. Flat,unramifiedande´talemorphisms 51
3.2. Nilpotentdeformationsofalmostalgebrasandmodules 53
3.3. Nilpotentdeformationsoftorsors 61
3.4. Descent 71
3.5. Behaviourofe´talemorphismsunderFrobenius 82
4. Finestudyofalmostprojectivemodules 92
4.1. Almosttraces 92
4.2. EndomorphismsofG(cid:1) . 100
m
4.3. Modulesofalmostfiniterank 107
4.4. Localizationintheflatsite 113
4.5. Constructionofquotientsbyflatequivalencerelations 123
5. Henselizationandcompletionofalmostalgebras 130
5.1. Henselianpairs 131
5.2. Criteriaforunramifiedmorphisms 135
5.3. Topologicalalgebrasandmodules 141
5.4. Henselianapproximationofstructuresoveradicallycompleterings 148
5.5. Liftingtheoremsforhenselianpairs 162
5.6. Smoothlocusofanaffinealmostscheme 171
5.7. Quasi-projectivealmostschemes 180
5.8. Liftinganddescentoftorsors 189
6. Valuationtheory 195
6.1. Orderedgroupsandvaluations 196
6.2. Basicramificationtheory 206
6.3. Algebraicextensions 210
6.4. Logarithmicdifferentials 220
6.5. Transcendentalextensions 226
6.6. Deeplyramifiedextensions 234
7. Analyticgeometry 242
7.1. Derivedcompletionfunctor 243
7.2. Cotangentcomplexforformalschemesandadicspaces 251
vi Contents
7.3. Deformations of formal schemes and adic spaces 262
7.4. Analytic geometry over a deeply ramified base 272
7.5. Semicontinuity of the discriminant 278
8. Appendix 287
8.1. Simplicial almost algebras 287
8.2. Fundamental group of an almost algebra 292
References 301
Index 304
Lebruitdesvaguese´taitencoreplusparesseux,plus
e´talequ’amidi.C’e´taitlemeˆmesoleil,lameˆme
lumie`resurlemeˆmesablequiseprolongeaitici.
A.Camus–L’e´tranger
1. INTRODUCTION
1.1. Motivationsandalittlehistory. Almostmathematicsmadeitsofficialdebut
in Faltings’ fundamental article [33], the first of a series of works on the subject
ofp-adicHodgetheory,culminatingwith[34].Althoughalmostringtheoryisde-
veloped here as an independent branch of mathematics, stretching somewhere in
between commutative algebra and category theory, the original applications to p-
adicHodgetheorystillprovidethemainmotivationandlargelydrivetheevolution
ofthesubject.
Indeed,oneof thechiefaimsof ourmonographisto supplyadequatefounda-
tionsfor[34],andtopavethewaytofurtherextensionsofFaltings’methods(espe-
cially,ofhisdeep“almostpurity”theorem),thatweplantopresentinafuturework.
Forthesereasons,itisfittingtobeginthisintroductionwithsomebackground,lead-
inguptoareviewoftheresultsof[33].(Besides,wesuspectthatallbutthemost
dedicatedexpertof p-adic Hodgetheorywill requiresome inducementbeforede-
cidingtoplungeintocloseto300pagesoffoundationalarcana.)
The starting point of p-adic Hodge theory can be located in Tate’s paper [74]
onp-divisiblegroups.Animportantexampleofp-divisiblegroupisthe p-primary
torsionsubgroupAp∞ ofanabelianschemeAdefinedoverthevaluationringK+
of a completediscretely valuedfield K of characteristiczero.We assume thatthe
residuefieldκofK isperfectofcharacteristicp > 0;also,letπ beauniformizer
ofK+,Ka thealgebraicclosureofK anddenotebyC thecompletionofKa;the
Galois groupG := Gal(Ka/K) acts linearly on the e´tale cohomologyof A, and
actually Ap∞ and the Galois moduleH´e1t(AKa,Zp) determine each other. G also
actssemilinearlyonC,whenceanaturalcontinuoussemilinearactionofGonthe
tensorproductofGaloismodules
H´e•t(AKa,C):=H´e•t(AKa,Zp)⊗Zp C.
At first sight, it would seem that, in replacing a linear Galois representation by a
semilinearone,wearetradingasimplerobjectbyamorecomplicatedone.Infact,
theoppositeholds:asaconsequenceofhisgeneralstudyofp-divisiblegroups,Tate
showedthatforeveryi≤2dim(A)thereexistsanaturalequivariantisomorphism
(cid:2)
(1.1.1) Hi (A ,C)(cid:4) Hj(A,Ωk)⊗ C(−k)
´et Ka A K
j+k=i
where, for every integer j ∈ Z, we define C(j) := Qp(j)⊗Qp C, and Qp(j) is
thej-thtensorpoweroftheone-dimensionalp-adicrepresentationQ (1)onwhich
p
G actsasthe p-primarycyclotomiccharacter.Tate conjecturedthatanequivariant
decompositionsuchas(1.1.1)shouldexistforanysmoothprojectivevarietydefined
O.GabberandL.Ramero:LNM1800,pp.1–10,2003.
(cid:1)c Springer-VerlagBerlinHeidelberg2003
2 Chapter1:Introduction
overK.Toputthingsinperspective,letusturntoconsiderthearchimedeancounter-
partof(1.1.1):ifX isasmooth,propercomplexalgebraicvariety,onecancombine
deRham’stheoremwithGrothendieck’stheoremonalgebraicdeRhamcohomology
[42],todeduceanaturalisomorphism
(1.1.2) H•(Xan,Z)⊗ZC(cid:4)Hd•R(X)
betweenthesingularanddeRhamcohomologies.Thetwosidesof(1.1.2)contribute
complementary information on X; namely, singular cohomology supplies an in-
tegral structure for H•(Xan,R) (the lattice of periods) and deRham cohomology
givestheHodgefiltration:neitherofthesetwostructuresisreducibletotheother.
The above conjecture of Tate is rather startling because it implies that in the
non-archimedeancase, e´tale cohomologyand deRham cohomologyshouldnotbe
complementary: rather, e´tale cohomology, viewed as a Galois module, would al-
readydetect,ifnotquitetheHodgefiltration,atleastitsassociatedgradedsubquo-
tients,eachofthemclearlyrecognizablebythedifferentweight(orTatetwist)with
whichitappearsinH•(A ,C)(nowthisgradedGaloismoduleisknownasthe
´et Ka
Hodge-TatecohomologyandoftendenotedH (−)).
HT
On the other hand, working aroundthe same time as Tate, Grothendieckreal-
izedthatthedeRhamcohomologyofanabelianschemecarriesmorestructurethan
itwouldappearatfirstsight:usinghiscrystallineDieudonne´theoryheshowedthat
H1 (A) comes with a canonical K -structure (where K is the field of fractions
dR 0 0
oftheringW(κ)ofWittvectorsofκ),namelytheK -vectorspaceM ⊗ K
0 W(κ) 0
whereM istheDieudonne´moduleofthespecialfibreofAp∞ (see[43]).Further-
more,thisK -vectorspaceisendowedwithanautomorphismφwhichissemilinear,
0
i.e.compatiblewiththeFrobeniusautomorphismofK .Grothendieckevenproved
0
that Ap∞ is determined up to isogeny by Hd1R(A) together with its Hodge filtra-
tion,K -structureandautomorphismφ. Takingintoaccounttheabovetheoremof
0
Tate,hewasthenledtoaskthequestionofdescribinganalgebraicprocedurethat
wouldallowto passdirectlyfromH1 (A) to H1(A ,Q )withoutthe interme-
dR ´et Ka p
diaryofthep-divisiblegroupAp∞;healso expectedthatsuchaprocedureshould
existforthecohomologyinarbitrarydegree(hebaptized1thisastheproblemofthe
“mysteriousfunctor”).
The question in degree one was finally solved by Fontaine, severalyears later
([37]);heactuallyconstructedafunctorintheoppositedirection,i.e.fromthecat-
egoryof p-adic Galois representationsto the categoryof filtered K-moduleswith
additionalstructureasabove.TheconstructionofFontainehingesonaremarkable
ring(actuallyawholehierarchyofincreasinglycomplexrings),endowedwithboth
a Galois action and a filtration (and eventually,additional subtler structures). The
simplestofsuchringsisthegradedringBHT := ⊕i∈ZC(i),withitsobviousmul-
tiplication;withitshelp,Tate’sdecompositioncanberewrittenasanisomorphism
ofgradedK-vectorspaces:
1Thenameenteredthefolklore,eventhough Grothendieckapparentlyonlyeverusedit
orally,andwecouldfindnotraceofitinhiswritings
§1.2:Themethodofalmoste´taleextensions 3
(cid:2)
(H´eit(AKa,Qp)⊗Qp BHT)G (cid:4) Hj(A,ΩkA).
j+k=i
Fontaine proved that his functor solves Grothendieck’s problem for the H1, and
proposedapreciseconjecture(theC conjecture)inarbitrarydegree,forschemes
cris
X thatareproperandsmoothoverK andhavegoodreductionoverK+.
1.2. The method of almost e´tale extensions. The C conjecture is now com-
cris
pletelyproved,aswellassomelaterextensions(e.g.toschemeswithnotnecessarily
goodreduction,ornotnecessarilyproper).Thereareactuallyatleasttwoverydif-
ferentmethodsthatbothhaveledtoaproof:one–duetoFontaine,Kato,Messing
andcrownedbyTsuji’swork[75]–reliesonso-calledsyntomiccohomologyanda
delicatestudyofvanishingcycles;theother,duetoFaltings,isbasedonhistheory
of almost e´tale extensions (for the case of varieties of good reduction, Niziol has
foundyetanothermethod,thatusesacomparisontheoremfrome´talecohomology
toK-theoryasago-between:see[63]).
Wewon’tsayanythingaboutthefirstandthethirdapproaches,butwewishto
givearoughoverviewofthemethodofalmoste´taleextensions,whichwasfirstpre-
sentedin[33],whereFaltingsusedittoprovethesoughtcomparisonwithHodge-
Tatecohomology;insubsequentpapersthemethodhasbeenrefinedandamplified,
and its latest incarnationis contained in [34]. However,many importantideas are
alreadyfoundin[33],soitisonthelatterthatwewillfocusinthisintroduction.
ForsimplicitywewillassumethatourvarietieshavegoodreductionoverK+,
hence we let X be a smooth, connected and projective K+-scheme. The idea is
to constructan intermediatecohomologyH(X), with valuesin C-vectorspaces,
receiving maps from both e´tale and Hodge-Tate cohomology, and prove that the
resultingnaturaltransformations
H´et(XKa,Zp)⊗Zp C →H(X) and HHT(XK)→H(X)
are isomorphisms of functors. In order to motivate the definition of H(X), it is
instructivetoconsiderfirstthecaseofapoint,i.e.X =SpecK+.Inthiscasee´tale
cohomologyreducestoGaloiscohomology,andthecalculationofthelatterwasthe
maintechnicalresultin[74].
Tate’s calculation can be explained as follows. The valuation v of K extends
uniquelyto any algebraicextension,and we wantto normalizethe value groupin
such a way that v(p) = 1 in every such extension. Let E be a finite Galois ex-
tension of K, with Galois group G . Typically, one is given a discrete E+[G ]-
E E
moduleM (suchthattheΓ-actiononM issemilinear,thatis,compatiblewiththe
G -action on E+), and is interested in studying the (modified) Tate cohomology
E
H(cid:1)i := H(cid:1)i(G ,M) (for i ∈ Z). (Recall that H(cid:1)i agrees with Galois cohomology
E
RiΓGEM for i > 0, with Galois homology for i < −1, and for i = 0 it equals
MGE/Tr (M),theG -invariantsdividedbytheimageofthetracemap).
E/K E
In such a situation, the scalar multiplication map E+ ⊗Z M → M induces
naturalcupproductpairings
H(cid:1)i(GE,E+)⊗ZH(cid:1)j →H(cid:1)i+j.
4 Chapter1:Introduction
(cid:1)
Especially,theactionof(E+)GE = K+ onHi factorsthroughK+/Tr (E+);
E/K
inotherwords,theimageofE+ underthetracemapannihilatesthemodifiedTate
cohomology.
IfnowtheextensionE istamelyramifiedoverK,thenTr (E+)=K+,so
E/K
theH(cid:1)ivanishforalli∈Z.Evensharperresultscanbeachievedwhentheextension
isunramified.Indeed,insuchcaseE+isaG -torsorforthee´taletopologyofK+,
E
hence,somebasicdescenttheorytellsusthatthenaturalmap
E+⊗ RΓGEM →M[0]
K+
is an isomorphism in the derived category of the category of E+[G ]-modules
E
(where we have denoted by M[0] the complex consisting of M placed in degree
zero).
InTate’spaper[74]thereoccursavariantoftheabovesituation:insteadofthe
finiteextensionE oneconsidersthealgebraicclosureKa ofK,sothatG = G
Ka
is the absolute Galois group of K, and the discrete G-module M is replaced by
the topological module C(χ), obtained by “twisting” the natural G-action on C
via a continuouscharacterχ : G → K×. Thenthe relevantH• is the continuous
GaloiscohomologyH• (G,C(χ)), whichisdefinedingeneralasthehomology
cont
of a complex of continuous cochains. Under the present assumptions, Hi can be
computedbytheformula:
Hciont(G,C(χ)):=(l←im− Hi(G,Ka+(χ)⊗ZZ/pnZ))⊗ZQ.
n
LetnowK∞beatotallyramifiedGaloisextensionwithGaloisgroupHisomorphic
toZp.Taterealizedthat,forcohomologicalpurposes,theextensionK∞ playsthe
roleofamaximaltotallyramifiedGaloisextensionofK.Moreprecisely,letLbe
anyfiniteextensionofK,andsetLn := L·Kn,whereKn isthesubfieldofK∞
fixedbyHpn (cid:4) pn·Z . TheextensionK ⊂ L isunramifiedifandonlyifthe
p n n
differentidealD :=D equalsL+.Incasethisfails,thevaluationv(δ )of
n L+n/Kn+ n n
ageneratorδ ofD willbeastrictlypositiverationalnumber,givingaquantitative
n n
measurefortheramification.Withthisnotation,[74,§3.2,Prop.9]reads
(1.2.1) lim v(δ )=0
n
n→∞
(indeed, v(δ ) approaches zero about as fast as p−n). In this sense, one can say
n
that the extension K∞ ⊂ L∞ := L·K∞ is almost unramified. One immediate
consequenceisthatthemaximalidealmofK+ iscontainedinTr (L+).If,
∞ L∞/K∞ ∞
additionally,LisaGaloisextensionofK,wecanconsiderthesubgroup
G∞ :=Gal(L∞/K∞)⊂Gal(L/K)
and the foregoing implies that m annihilates Hi(G∞,M), for every i > 0, and
everyL+∞[G∞]-moduleM.Moreprecisely,thehomologyoftheconeofthenatural
morphism
(1.2.2) L+ ⊗ RΓG∞M →M[0]
∞ K∞+
isannihilatedbyminalldegrees,i.e.itisalmostzero. Equivalently,onesaysthat
themapsonhomologyinducedby(1.2.2)arealmostisomorphismsinalldegrees.