Class field theory for arithmetic schemes [Elektronische Ressource] / vorgelegt von Walter Hofmann

Class Field Theory for Arithmetic SchemesDen Naturwissenschaftlichen Fakult atender Friedrich-Alexander-Universit at Erlangen-Nurn bergzurErlangung des Doktorgradesvorgelegt vonWalter Hofmannaus Nurn bergAls Dissertation genehmigtvon den Naturwissenschaftlichen Fakult atender Universit at Erlangen-Nurn bergTag der mundlic hen Prufung: 12. Juni 2007Vorsitzender derder Promotionskommission: Prof. Dr. E. B anschErstberichterstatter: Prof. Dr. W.-D. GeyerZweitberichterstatter: Prof. Dr. H. LangeDrittberichterstatter: Prof. Dr. U. JannsenZusammenfassung der ErgebnisseDie vorliegende Arbeit besch aftigt sich mit der Klassenk orpertheorie arithmetischer Schemata, d.h. mitseparierten, reduzierten und zusammenh angenden Schemata X , die von endlichem Typ ub er SpecZoder allgemeiner ub er einem o enen Teil S des Spektrums eines Ganzheitsrings eines algebraischenZahlk orpers liegen.In [6,24] wurde zun achst fur Fl achen und dann fur Schemata beliebiger Dimension eine Klassengruppede niert:0 1M M M M @ AC := coker (C) ! Z (C) : (1)X vC X x2X C X v2C1hierbei durchlauftC alle irreduziblen Kurven aufX , undxauftl ub er alle abgeschlossenen Punkte von X .Die StellenmengeC enth alt gerade diejenigen Stellen von (C), die nicht Punkten der Normalisierung1von C entsprechen. Die Abbildung wird in De nition 16 erl autert.
Publié le : lundi 1 janvier 2007
Lecture(s) : 15
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Source : WWW.OPUS.UB.UNI-ERLANGEN.DE/OPUS/VOLLTEXTE/2007/667/PDF/DISSERTATION.PDF
Nombre de pages : 25
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Class Field Theory for Arithmetic Schemes
DenNaturwissenschaftlichenFakult¨aten derFriedrich-Alexander-Universit¨atErlangen-N¨urnberg zur Erlangung des Doktorgrades
vorgelegt von Walter Hofmann ausNu¨rnberg
Als Dissertation genehmigt vondenNaturwissenschaftlichenFakulta¨ten derUniversita¨tErlangen-Nu¨rnberg
Tagdermu¨ndlichenP¨ufung: r Vorsitzender der der Promotionskommission: Erstberichterstatter: Zweitberichterstatter: Drittberichterstatter:
12. Juni 2007 Prof.Dr.E.B¨ansch Prof. Dr. W.-D. Geyer Prof. Dr. H. Lange Prof. Dr. U. Jannsen
Zusammenfassung der Ergebnisse DievorliegendeArbeitbescha¨ftigtsichmitderKlassenko¨rpertheoriearithmetischerSchemata,d.h.mit separierten,reduziertenundzusammenha¨ngendenSchemataXpecTmehcildSrebu¨py,enonevdiZ oderallgemeineru¨bereinemoenenTeilSdes Spektrums eines Ganzheitsrings eines algebraischen Zahlko¨rpersliegen. In[6,24]wurdezun¨achstfu¨rFla¨chenunddannf¨urSchematabeliebigerDimensioneineKlassengruppe definiert: CX:= cokerCM⊆Xκ(C)×xMXZCM⊆XvMCκ(C)v×.(1) hierbeidurchl¨auftCalle irreduziblen Kurven aufX, undxllaeebartfu¨¨luaPuenensslochesbgnovetknX. Die StellenmengeCnehta¨ltgeradediejenigtSneellenovnκ(C), dienichtPunkten der Normalisierung vonCentchenspreibbAeiD.riwgnudlnienDdier16onti.aul¨rtte ¨ DieseKlassengruppebeschreibtdieendlichen´etalenUberlagerungenvonX: IstXregetni,ra¨luger undachu¨berS, so gibt es einen Isomorphismus CX/CX◦=πab1(X)(2) wobeiCXdie Zusammenhangskomponente der Null vonCXundπ1abdie abelsch gemachte Fundamental-gruppeist.DiesesErgebniswurdezuna¨chstfu¨rdimX= 2 und modulon]g[6ineta¨psdnu,tgiezerwie hier verwendet in [24]. ZieldieserArbeitistesnun,dieReziprozita¨tsabbildung(2)imFalleeinesnichtregula¨renSchemasX zuuntersuchen.Dazuwerdenzuna¨chstgewisseKohomologiegruppenHKq(XF)egeinuhf¨,irtntealrhVehr unterprojektivenLimitenuntersuchtundeineVerbindungzue´talenKohomologiegruppenHte´(XF) hergestellt. Mit Hilfe dieser wird dann in Theorem 38 eine exakte Sequenz aufgestellt, die die Abweichung derAbbildung(2)voneinemIsomorphismusbeschreibt.W¨ahrendzun¨achstTheorem38modulon formuliert ist, wird in Theorem 46 die Aussage dann allgemein gezeigt. 1
Abstract
Thispresentworkdescribesthe´etalefundamentalgroupofpossiblysingulararithmeticschemes.Let S ⊆SpecObe an open subscheme of the spectrum of the ring of integersOof an algebraic number field, and letXscheme which is flat, proper and of finite type overbe a separated, reduced and connected S. Denote byπ1ab(Xd´etnizeundaaleflargemtndneluoaptt)ailebaehCXbheet`eidcllegssapuorfoXas defined in [24].CXis connected component of zero ofCX describe the kernel and cokernel of the. We mapCX/CXπ1ab(X) by means of embedding it in an exact sequence: ˆ He´t2(XQ/Z)H2K(XˆZ)→ CX/CXπ1ab(X)H1K(XZ)0
HqKare certain cohomology groups defined in this work.
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Contents 1 Introduction 4 2 Preliminaries 6 2.1 Notes on Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 ´ 2.3 Etale Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 ´ 2.4 Finiteness of Etale Cohomology Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 A Leray Spectral Sequence for Cohomology with Compact Support . . . . . . . . . . . . . 7 2.6 De Jong’s Theory of Alterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Class Field Theory for Regular Arithmetic Schemes 9 4 The Kato Complex and its Cohomology 11 4.1 A Projective System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 The Kato Complex and Kato Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.3 Limits of Cohomology Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 ´ 4.4 Kato Cohomology and Etale Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5 Singular Class Field Theory 17 5.1 Class Field Theory for Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.2 The Theory modulon 18. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6 Acknowledgements 21 7 References 21 3
1 Introduction In 1783 Leonard Euler stated, and in 1796 Carl Friedrich Gauß was the first to prove what became known as Gauß’ quadratic reciprocity law: qp qp= (1)p21q2 1.(3) Together with two supplementary theorems it provides an easy way to compute the Legendre symbol (pq reciprocity laws were studied by Gauß, Jacobi, Eisenstein and Kummer in the 19th century.). Higher About seventy years later, while examining the work of Niels Henrik Abel, Leopold Kronecker ob-served that one can obtain abelian extensions of imaginary quadratic number fields by adjoining special values of certain automorphic functions related to elliptic functions. Kronecker asked whether all abelian extensions of an algebraic number fieldK idea became known ascould be obtained in this manner. This Kroneckers Jugendtraum. For the rational numbersQ, he had conjectured earlier the following theorem. Theorem 1 (Kronecker-Hilbert)Every abelian extension ofQis contained in a cyclotomic extension ofQ. Kronecker contributed ideas to the proof, and Heinrich Weber proposed a “proof” which turned out to be wrong much later. It was David Hilbert who gave the first complete proof of this theorem. While the work of Euler, Gauß, Jacobi, Eisenstein, Kummer, Kronecker and Weber provided the first elements of class field theory, it was Hilbert who saw the complete picture: the theory of abelian extensions. At the International Congress of Mathematicians in Paris in 1900, Hilbert posed in his famous speech a number of problems, two of which focus on class field theory: Hilbert’s 9th Problem:To develop the most general reciprocity law in an arbitrary number field, generalizing Gauß’ law of quadratic reciprocity. Hilbert’s 12th Problem:Extend Kronecker’s theorem on the generation of abelian extensions of the rational numbers to any base number field. While Hilbert’s 12th problem remains open, for abelian extensions the 9th problem has found its solution in Emil Artin’s reciprocity law. LetKbe a number field andSa finite set of places ofK, such thatScontains all infinite places ofK by. DenoteKvthe completion ofKat a placev. For every discrete valuationv6∈S, we consider the mapv:K×Z, while for valuationsvSwe use the embeddingK×,Kv×into the completion to define theS-class groupofK: CS:= cokerv6∈S vSv(4) K×MZMK×It is an abelian topological group. For a groupG, defineGabto be the maximal abelian factor group of G, i.e.Gab=G/G0whereG0is the commutator group ofG. Theorem 2LetKSbe the maximal algebraic extension ofK, unramified outsideS. Then there is a reciprocity map ρ:CSGal(KS|K)ab(5) which is surjective and its kernel is the connected component of 0. The generalization of Theorem 2 to the non-abelian case remains open and is subject ofLanglands Program, a set of far-reaching conjectures set forth by Robert Langlands in 1967. There is, however, another direction in which Theorem 2 can be generalized, namely to higher di-mensions.Tostatetheresults,weneedtointroducethee´talefundamentalgroup.LetXbe a connected scheme andx¯Xa geometric point onXupronemaglatelatdnuf.The´eπ1et´(X¯x), introduced by AlexanderGrothendieck,classiesthenitee´talecoveringsofX. IfX= SpecK, then choosing a geo-metric point ¯xamounts to choosing a separably closed field Ω containingK, andπ1´et(X) = Gal(Ksep|K), whereKsepis the separable closure ofK a description of Hencein Ω.π´et1(X¯x) orπ1ab(X) generalizes Theorem 2. (Note that a change of the base point ¯xchangesπ1´et(X¯x) by an isomorphism, which is 4
canonical up to an inner automorphism. Hence there is no need to specify a base point when we discuss π1ab(X).) One interesting class of schemes studied arearithmetic schemes. LetXbe a regular, integral and separated scheme flat and of finite type over SpecZ. Thenπ1ab(X) can be described by higher dimensional Milnor K-Theory, as was done by Spencer Bloch, Kazuya Kato and Shuji Saito in [1, 9–12]. There is a fairlycomplicateddescriptionoftheid`eleclassgroupbythetheoryofParshinchains.However,there isalsoasecondapproachbyG¨otzWiesendandtheauthorwhichonlyusesK0andK1groups, i.e.Z and multiplicative groups. In [6], we give a theory in case of dimX= 2, describing both the abelian and non-abelian case. In the abelian case, the description ofπba1(X) in this paper is only modulon. For the non-abelian case, the higher dimensional theory is reduced to the (currently unknown) one dimensional theory. The results of [6] are greatly extended in later papers of Wiesend. In [23], a non-abelian theory for arithmetic schemes of general dimension is given. In [24], the corresponding abelian class field theory is given for arithmetic schemes of general dimension. While all the approaches mentioned in the preceding paragraph are for regularX, there are also results for more generalXPeter Stevenhagen generalized one-dimensional class field theory to [22], . In orders in number fields, while a one-dimensional local theory was given four years earlier by Saito in [19]. In [14], Kazuya Matsumi, Kanetomo Sato and Masanori Asukura explore class field theory for (pos-sibly singular) normal, proper and geometrically integral surfacesXdefined over finite fields. They describe when the reciprocity map CH0(X)πba1(X) is injective using resolution of singularities and a cohomological Hasse principle for smooth proper surfaces. One year later, Alexander Schmidt and Michael Spieß published [20] a theory of tamely ramified covers of varieties over finite fields. Another approach is given by Uwe Jannsen and Shuji Saito in [7]. They use an approach first suggested by Grothendieck and then studied by Bloch and Arthus Ogus [2]. LetKbe a non-archimedean local field andVa proper variety over Spec(K by the works of Bloch, Kato and Saito, there is). Then, a reciprocity map ρV:SK1(V)π1ab(V)(6) where SK1(V) := cokerxMV1K2(x)xMV0K1(x)!(7) Vi:={xV|dim{x}=i}(8) k(x) := residue field ofxand (9) Kq(x) :=q-th algebraicK-group ofk(x), (10) andis the boundary map fromK-theory. For a positive integern, prime to char(K) let ρV,n:SK1(V)/nπ1ab(V)/n(11) be the induced map. Then Jannsen and Saito prove the existence of an exact sequence H2K(VZ/n)SK1(V)/nρV,nπ1ab(V)/nH1K(VZ/n)0(12) whereHqKis theq-th Kato homology group ofV, defined as the homology of a complexCr,s(VZ/n) of Bloch-Ogus type (see [8,§1] for details). Thispresentworksetsouttodescribethee´talefundamentalgroupofpossiblysingulararithmetic schemes over SpecZ Letusing the class field theory of Wiesend.S ⊆SpecObe an open subscheme of the spectrum of the ring of integersOof an algebraic number field, and letXbe a separated, reduced and connected scheme of finite type over SpecZ regular. ForXflat overS, the Main Theorem of [24] asserts CX/CX0=πab(X) (13) 1 whereCXuorgssalcele`dieisthfpoXand in Definition 16 below andas defined in [24] CXis the connected component of zero. For singularX However, we can describe, such an isomorphism cannot be expected. kernel and cokernel of the mapCX/CX0πab1(Xmeans of embedding it in an exact sequence similar) by to (12). Among the groups occuring in this sequence are certain cohomology groups. While these are 5
not the same groups originally defined by Kato in [8], we retain the name Kato cohomology groups for them, as they are constructed similarly. Theorem 46 in this work gives the sequence describingπab1(X): H2´et(XQ/Z)H2K(XZˆ)→ CX/CXπ1ab(X)H1K(XZ)ˆ0 (14) Note that while similar to (12), this sequence has some significant improvements. While Jannsen and Saito prove their sequence only for arithmetic surfaces, the sequence given here is also valid in higher dimensions. Also, we describe the full groupπ1ab(X) and not just the quotientsπab1(V)/n. Finally, there is one more term in the sequence. We do however have to retain the assumption thatXis proper over Sto get this result. This work is arranged as follows: We start by introducing our notation and reviewing basic facts about´etalecohomology.Thensometoolsusedlaterintheproofsareintroduced:ALerayspectral sequence and de Jong’s theory of alterations. Wiesend’s class field theory for regular arithmetic schemes is reviewed. This includes the definition of the class groupCX whole next chapter is used to. The define the necessary prerequisites for proving sequence (14) modulon complex is defined and its: A homology taken, which we will call Kato cohomology. The behaviour of Kato cohomology when taking projective limits over certain subschemes is studied in detail. There is a natural transformation from Kato cohomology to etale cohomology. An analysis of the connections between Kato cohomology and ´ ´etalecohomologythenyieldssequence(14)modulonwe use alterations to get the general . Finally, result (without modulon).
2 Preliminaries 2.1 Notes on Notation We use roman letters (X,Y . . ), . general schemes, and script letters ( forX,Y, . . . ) for arithmetic schemes. Ifi:X → Yarithmetic schemes (usually just an embedding) andis a morphism of Fis a sheaf onYthen to ease notation, we are supressingi For example, wein many places throughout this work. writeiFinstead ofiiF. 2.2 Schemes LetO Throughoutbe the ring of integers in a number field. this work, we will always denote by S ⊆SpecOan open subscheme of SpecO. Definition 3Anarithmetic schemeXis a separated, reduced and connected scheme of finite type over SpecO that, in [24], also flatness over Spec. (NoteOis required.) Definition 4AcurveConXis a closed and reduced subscheme ofXof dimension 1. ´ 2.3 Etale Sheaves Themainreferencefor´etalecohomologyusedinthisworkisthebookofMilne[15].Amorphismof schemes (or rings) ise´telatepotelaylogoimaW.deyasetahtovacinerorgfe´thifitisatandunr isasurjectivefamilyofe´talemorphismsUiX. This defines theetopolog´etaly(in the sense of a Grothendieck topology) on a schemeX. There is a theory of presheaves and sheaves for this topology, callede´talepresheavesande´talesheaves,cf.[15,II.3].Inthispaperwewillmostofthetimeuselocally constant or constructible sheaves. Moreover, the values of a sheaf are at least abelian groups. Definition 5A sheafFis calledconstantif it is a sheaf associated to a constant presheaf. A sheafF is calledlocally constantif there is a coveringUiXsuch thatF |Uiis constant for alli all. IfF |Ui are finite, we say thatFisfinite locally constant. Definition 6A sheafFon a schemeXis calledconstructible, ifXcan we written as a union of finitely many locally closed subschemesYXfor whichF |Y (is finite locally constant.Yis calledlocally closedif it is the intersection of an open and a closed subscheme.)
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