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Almost Ring Theory PDF

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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] Cataloging-in-PublicationDataappliedfor BibliographicinformationpublishedbyDieDeutscheBibliothek DieDeutscheBibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataisavailableintheInternetathttp://dnb.ddb.de Thecoverfigureentitled"Prescano"isreproducedbykindpermissionof MichelMende`sFrance MathematicsSubjectClassification(2000): 13D10,13B40,12J20,14G22,18D10,13D03 ISSN0075-8434 ISBN3-540-40594-1Springer-VerlagBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965, initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer-Verlag.Violationsare liableforprosecutionundertheGermanCopyrightLaw. Springer-VerlagBerlinHeidelbergNewYorkamemberofBertelsmannSpringer Science+BusinessMediaGmbH http://www.springer.de (cid:1)c Springer-VerlagBerlinHeidelberg2003 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:Camera-readyTEXoutputbytheauthor SPIN:10896419 41/3142/du-543210-Printedonacid-freepaper 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.

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This book develops thorough and complete foundations for the method of almost etale extensions, which is at the basis of Faltings' approach to p-adic Hodge theory. The central notion is that of an "almost ring". Almost rings are the commutative unitary monoids in a tensor category obtained as a quot
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