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Higher Dimensional Class Field Theory: the Variety Case Higher Dimensional Class Field Theory: The variety case Linda M. Gruendken A Dissertation in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2011 Florian Pop Supervisor of Dissertation Jonathan Block Graduate Group Chairperson Dissertation Committee Florian Pop, Samuel D. Schack Professor of Algebra, Mathematics David Harbater, Christopher H. Browne Distinguished Professor in the School of Arts and Sciences, Mathematics Robin Pemantle, Merriam Term Professor of Mathematics HIGHER DIMENSIONAL CLASS FIELD THEORY: THE VARIETY CASE COPYRIGHT 2011 LINDA MEIKE GRUENDKEN Acknowledgments I would like to thank my advisor, Florian Pop, for suggesting this topic, and for his continued support throughout both the research and writing phases of this thesis. I am also very grateful to David Harbater, Rachel Pries, Scott Corry, Andrew Obus, Adam Topaz for their helpful discussions in various phases of completion of this thesis. I would also like to thank Ching-Li Chai and Ted Chinburg for many interesting discussions in the early stages of the UPenn Ph.D. program. Finally, I would like to thank my mother and my father for their love and support. iii ABSTRACT Higher Dimensional Class Field Theory: The variety case Linda M. Gruendken Prof. Dr. Florian Pop, Advisor n Let k be a finite field, and suppose that the arithmetical variety X ⊂ Pk is an open subset in projective space. Suppose that CX is the Wiesend id`eleclass ab ab group of X, π1 (X) the abelianised fundamental group, and ρX : CX −! π1 (X) the Wiesend reciprocity map. We use the Artin-Schreier-Witt and Kummer Theory of affine k-algebras to prove a full reciprocity law for X. We find necessary and sufficent conditions for a subgroup H < CX to be a norm subgroup: H is a norm subgroup if and only if it is open and its induced covering datum is geometrically bounded. We show that ρX is injective and has dense image. We obtain a one- to-one correspondence of open geometrically bounded subgroups of CX with open ab 00 subgroups of π1 (X). Furthermore, we show that for an ´etalecover X −! X with maximal abelian subcover X0 −! X, the reciprocity morphism induces an 0 isomorphism CX =NCX00 ' Gal(X =X). iv Contents 1 Introduction 1 2 Preliminaries 5 2.1 Basic Facts and Generalities . 5 2.2 Fundamental groups . 10 2.3 Covering Data . 17 3 The Wiesend Id`eleClass Group 34 3.1 Definitions and some Functiorial Properties . 34 3.2 The Reciprocity Homomorphism . 44 3.3 The Base Cases of the Induction Argument: Reciprocity for Regular Schemes of Dimension One and Zero . 49 4 From the Main Theorem to the Key Lemma 55 4.1 Introducing induced covering data . 56 4.2 Geometrically bounded covering data . 60 v 4.3 Realisable subgroups of the class groups . 68 4.4 The Key Lemma . 73 5 Wildly ramified Covering Data 78 5.1 A Review of Artin-Schreier-Witt Theory . 80 5.1.1 Artin-Schreier Theory . 80 5.1.2 Artin-Schreier-Witt Theory . 85 5.2 Covering data of Prime Index . 94 n 5.2.1 Affine n-space Ak ........................ 94 n 5.2.2 The Case of Open Subsets of Pk . 102 5.3 Covering data of Prime Power Index . 113 6 Tamely ramified covering data revisited 122 6.1 A review of Kummer Theory . 123 6.2 Tamely ramified covering data of cyclic factor group . 129 vi Chapter 1 Introduction Class Field Theory is one of the major achievements in the number theory of the first half of the 20h century. Among other things, Artin reciprocity showed that the unramified extensions of a global field can be described by an abelian object only depending on intrinsic data of the field: the Class Group. In the language of Grothendieck's algebraic geometry, the theorems of classical global class field theory [12, Ch. VI] can be reformulated as theorems about one- dimensional arithmetical schemes, whose function fields are precisely the global fields. (A summary of results in convenient notation is presented in Section 3.3). The class field theory for such schemes X then turns into the question of describing the unramified abelian covers of the schemes, i.e. describing the fundamental group ab π1 (X). It is therefore natural to ask for a generalisation of class field theory to arithmetical schemes of higher dimensions. 1 Several attempts at a Higher Class Field Theory have already been made, with different generalisations of the class group to higher dimensional schemes: Katz- Lang [4] described the maximal abelian cover of a projective regular arithmetic scheme and Serre [15] gave a description of the abelian covers of schemes over Fp in terms of generalised Jacobians. Finally, Parshin and Kato, followed by several others, proposed getting a higher dimensional Artin reciprocity map using alge- braic K-Theory and cohomology theories. Although promising, these approaches become quite technical, and the heavy machinery involved makes the results very complicated and difficult to apply in concrete situations. It was G. Wiesend [21] who had the idea to reduce the higher dimensional class field theory to the well developed and known class field theory for arithmetical curves. He defines an "id`eleclass group" CX in terms of the arithmetical curves and closed points contained in X, and gets a canonical homomorphism ab ρX : CX −! π1 (X) in the hope of establishing properties similar to that of the Artin reciprocity map. Wiesend's work was supplemented by work by M. Kerz and A. Schmidt, where more details of Wiesend's approach were given [5]. Most notably, in the so-called flat case of an arithmetical scheme X over Spec Z (c.f. Definition 2.1.2), they were able to prove surjectivity of the canonical homomorphism, and they provided a concrete description of the norm subgroups in CX . The focus of this thesis is on the still-open regular variety case, where X is a 2 regular arithmetical variety over some finite field k of characteristic p. In this case, Kerz and Schmidt have already proved a higher dimensional reci- ab;t procity law for the abelianised tame fundamental group π1 (X), which classifies all the tame ´etaleabelian covers of X. A generalised Artin map can be defined between t ab;t ρX;t : CX −! π1 (X) ; t where CX denotes the tame class group, as defined by Wiesend [21]. As in the flat case, the proof crucially relies on finiteness theorems for the geometric part of the tame fundamental group [4]. These finiteness results are known to be false for the full fundamental group due to the presence of wild ramification: For any affine variety of dimension ≥ 1 over a finite field, the p-part of the fundamental group is infinitely generated. n Let k be a finite field, and X ⊂ Pk an open subvariety, then we prove: n Theorem 1.0.1. Let X 2 Pk be an open subvariety, and let ab ρX : CX −! π1 (X) be the Wiesend recprocity morphism. Then the following hold: 1) There exists a one-to-one correspondence between open and geometrically bounded ab −1 subgroups of CX and open subgroups N of π1 (X); it is given by ρX (N) 7! N. The ab reciprocity morphism ρX is a continuous injection with dense image in π1 (X). 2) A subgroup of CX is a norm subgroup iff it is open, of finite index and geometri- cally bounded. 3 3) If X00 −! X is an ´etaleconnected cover, X0 −! X the maximal abelian subcover, then NCX00 = NCX0 and the reciprocity map gives rise to an isomorphism ' 0 CX =NCX00 −! Gal(X =X): This thesis is organised as follows: We begin by introducing notation and re- viewing basic facts from the theories of arithmetical schemes, fundamental groups and covering data. We then review Wiesend's definition of the id`eleclass group and the reciprocity homomorphism, giving the right definitions as to guarantee commu- tativity of all relevant diagrams. (Unfortunately, in all the published works so far, the given morphisms do not make all the diagrams commute as stated.) We also review the results of classical class field theory as the "base" case of our induction argument in the proof of the Main Theorem 4.3.4. All of Chapter 4 is devoted to assembling and refining the necessary tools for the proof of the Main Theorem 4.3.4, and in the process, the proof of the Main Theorem 4.3.4 is reduced to the Key Lemma 4.4.4. Theorem 1.0.1 is shown as a corollary to the Main Theorem 4.3.4. n The Key Lemma is shown for open subvarieties X ⊂ Pk in two steps. In Chapter 5, we analyse the behaviour of index-pm wildly ramified covering data on X to show Part 1). The second part of the Key Lemma, which was already known from the results of Wiesend, Kerz and Schmidt ([21], [5]), is reproven in Chapter 6 without making explicit use of geometric finiteness results. 4 Chapter 2 Preliminaries 2.1 Basic Facts and Generalities In this thesis, we shall be concerned with arithmetical schemes X over Spec Z: Definition 2.1.1. X is said to be an arithmetical scheme over Z if it is integral, separated and of finite type over Spec Z. We assume that all schemes are arithmetical, unless otherwise stated, and dis- tinguish two cases: Definition 2.1.2. If the structural morphism X −! Spec Z has open image, X will be called flat. If this is not the case, then the image of X −! Spec Z is a closed point p 2 Spec Z, and X is a variety over the residue field of the point k = k(p) = Fp.
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