Class Field Theory for Arithmetic Schemes Den Naturwissenschaftlichen Fakult¨aten der Friedrich-Alexander-Universit¨atErlangen-N¨urnberg zur Erlangung des Doktorgrades vorgelegt von Walter Hofmann aus N¨urnberg Als Dissertation genehmigt von den Naturwissenschaftlichen Fakult¨aten der Universit¨atErlangen-N¨urnberg Tag der m¨undlichen Pr¨ufung: 12. Juni 2007 Vorsitzender der der Promotionskommission: Prof. Dr. E. B¨ansch Erstberichterstatter: Prof. Dr. W.-D. Geyer Zweitberichterstatter: Prof. Dr. H. Lange Drittberichterstatter: Prof. Dr. U. Jannsen Zusammenfassung der Ergebnisse Die vorliegende Arbeit besch¨aftigtsich mit der Klassenk¨orpertheorie arithmetischer Schemata, d.h. mit separierten, reduzierten und zusammenh¨angendenSchemata X , die von endlichem Typ ¨uber Spec Z oder allgemeiner ¨uber einem offenen Teil S des Spektrums eines Ganzheitsrings eines algebraischen Zahlk¨orpers liegen. In [6,24] wurde zun¨achst f¨ur Fl¨achen und dann f¨urSchemata beliebiger Dimension eine Klassengruppe definiert: 0 1 M × M M M × CX := coker @ κ(C) ! Z ⊕ κ(C)v A : (1) C⊆X x2X C⊆X v2C1 hierbei durchl¨auft C alle irreduziblen Kurven auf X , und x l¨auft¨uber alle abgeschlossenen Punkte von X . Die Stellenmenge C1 enth¨altgerade diejenigen Stellen von κ(C), die nicht Punkten der Normalisierung von C entsprechen. Die Abbildung wird in Definition 16 erl¨autert. Diese Klassengruppe beschreibt die endlichen ´etalen Uberlagerungen¨ von X : Ist X regul¨ar,integer und flach ¨uber S, so gibt es einen Isomorphismus ◦ ∼= ab CX =CX −! π1 (X ); (2) ◦ ab wobei CX die Zusammenhangskomponente der Null von CX und π1 die abelsch gemachte Fundamental- gruppe ist. Dieses Ergebnis wurde zun¨achst f¨urdim X = 2 und modulo n in [6] gezeigt, und sp¨aterwie hier verwendet in [24]. Ziel dieser Arbeit ist es nun, die Reziprozit¨atsabbildung(2) im Falle eines nicht regul¨arenSchemas X q zu untersuchen. Dazu werden zun¨achst gewisse Kohomologiegruppen HK(X ; F) eingef¨uhrt,ihr Verhalten unter projektiven Limiten untersucht und eine Verbindung zu ´etalenKohomologiegruppen H´et(X ; F) hergestellt. Mit Hilfe dieser wird dann in Theorem 38 eine exakte Sequenz aufgestellt, die die Abweichung der Abbildung (2) von einem Isomorphismus beschreibt. W¨ahrendzun¨achst Theorem 38 modulo n formuliert ist, wird in Theorem 46 die Aussage dann allgemein gezeigt. 1 Abstract This present work describes the ´etalefundamental group of possibly singular arithmetic schemes. Let S ⊆ Spec O be an open subscheme of the spectrum of the ring of integers O of an algebraic number field, and let X be a separated, reduced and connected scheme which is flat, proper and of finite type over S. ab Denote by π1 (X ) the abelianized ´etalefundamental group and let CX be the id`eleclass group of X as ◦ defined in [24]. CX is connected component of zero of CX . We describe the kernel and cokernel of the ◦ ab map CX =CX ! π1 (X ) by means of embedding it in an exact sequence: 2 ∗ K ^ ◦ ab K ^ H´et(X ; Q=Z) ! H2 (X ; Z) !CX =CX ! π1 (X ) ! H1 (X ; Z) ! 0 K Hq are certain cohomology groups defined in this work. 2 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 modulo n ...................................... 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: p q p−1 q−1 = (−1) 2 2 : (3) q p Together with two supplementary theorems it provides an easy way to compute the Legendre symbol p ( q ). Higher reciprocity laws were studied by Gauß, Jacobi, Eisenstein and Kummer in the 19th century. 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 field K could be obtained in this manner. This idea became known as Kroneckers Jugendtraum. For the rational numbers Q, he had conjectured earlier the following theorem. Theorem 1 (Kronecker-Hilbert) Every abelian extension of Q is contained in a cyclotomic extension of Q. 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. Let K be a number field and S a finite set of places of K, such that S contains all infinite places of K. Denote by Kv the completion of K at a place v. For every discrete valuation v 62 S, we consider the map v : K× ! Z, while for valuations v 2 S we use the × × embedding K ,! Kv into the completion to define the S-class group of K: 0 1 × M M × CS := coker @K ! Z ⊕ Kv A ; (4) v62S v2S It is an abelian topological group. For a group G, define Gab to be the maximal abelian factor group of G, i.e. Gab = G=G0 where G0 is the commutator group of G. Theorem 2 Let KS be the maximal algebraic extension of K, unramified outside S. Then there is a reciprocity map ab ρ: CS ! Gal(KSjK) ; (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 of Langlands 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. To state the results, we need to introduce the ´etalefundamental group. Let X be a connected ´et scheme andx ¯ ! X a geometric point on X. The ´etalefundamental group π1 (X; x¯), introduced by Alexander Grothendieck, classifies the finite ´etalecoverings of X. If X = Spec K, then choosing a geo- ´et sep metric pointx ¯ amounts to choosing a separably closed field Ω containing K, and π1 (X) = Gal(K jK), sep ´et ab where K is the separable closure of K in Ω. Hence a description of π1 (X; x¯) or π1 (X) generalizes ´et Theorem 2. (Note that a change of the base pointx ¯ changes π1 (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 ab π1 (X).) One interesting class of schemes studied are arithmetic schemes. Let X be a regular, integral and ab separated scheme flat and of finite type over Spec Z. Then π1 (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 fairly complicated description of the id`eleclass group by the theory of Parshin chains. However, there is also a second approach by G¨otzWiesend and the author which only uses K0 and K1 groups, i.e. Z and multiplicative groups. In [6], we give a theory in case of dim X = 2, describing both the abelian and ab non-abelian case. In the abelian case, the description of π1 (X) in this paper is only modulo n. 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 regular X , there are also results for more general X . In [22], Peter Stevenhagen generalized one-dimensional class field theory to 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 surfaces X defined over finite fields. They ab describe when the reciprocity map CH0(X ) ! π1 (X ) is injective using resolution of singularities and a cohomological Hasse principle for smooth proper surfaces.