Class Field Theory for Arithmetic Schemes

Den Naturwissenschaftlichen Fakultatender Friedrich-Alexander-Universitat Erlangen-Nurnberg

zurErlangung des Doktorgrades

vorgelegt vonWalter Hofmann

aus Nurnberg

Als Dissertation genehmigtvon den Naturwissenschaftlichen Fakultaten

der Universitat Erlangen-Nurnberg

Tag der mundlichen Prufung: 12. Juni 2007Vorsitzender derder Promotionskommission: Prof. Dr. E. BanschErstberichterstatter: Prof. Dr. W.-D. GeyerZweitberichterstatter: Prof. Dr. H. LangeDrittberichterstatter: Prof. Dr. U. Jannsen

Zusammenfassung der Ergebnisse

Die vorliegende Arbeit beschaftigt sich mit der Klassenkorpertheorie arithmetischer Schemata, d.h. mitseparierten, reduzierten und zusammenhangenden Schemata X , die von endlichem Typ uber Spec Zoder allgemeiner uber einem offenen Teil S des Spektrums eines Ganzheitsrings eines algebraischenZahlkorpers liegen.

In [6,24] wurde zunachst fur Flachen und dann fur Schemata beliebiger Dimension eine Klassengruppedefiniert:

CX := coker

⊕C⊆X

κ(C)× →⊕x∈X

Z⊕⊕C⊆X

⊕v∈C∞

κ(C)×v

. (1)

hierbei durchlauft C alle irreduziblen Kurven auf X , und x lauft uber alle abgeschlossenen Punkte von X .Die Stellenmenge C∞ enthalt gerade diejenigen Stellen von κ(C), die nicht Punkten der Normalisierungvon C entsprechen. Die Abbildung wird in Definition 16 erlautert.

Diese Klassengruppe beschreibt die endlichen etalen Uberlagerungen von X : Ist X regular, integerund flach uber S, so gibt es einen Isomorphismus

CX /CX∼=−→ πab

1 (X ), (2)

wobei CX die Zusammenhangskomponente der Null von CX und πab1 die abelsch gemachte Fundamental-

gruppe ist. Dieses Ergebnis wurde zunachst fur dimX = 2 und modulo n in [6] gezeigt, und spater wiehier verwendet in [24].

Ziel dieser Arbeit ist es nun, die Reziprozitatsabbildung (2) im Falle eines nicht regularen Schemas Xzu untersuchen. Dazu werden zunachst gewisse Kohomologiegruppen Hq

K(X ,F) eingefuhrt, ihr Verhaltenunter projektiven Limiten untersucht und eine Verbindung zu etalen Kohomologiegruppen Het(X ,F)hergestellt. Mit Hilfe dieser wird dann in Theorem 38 eine exakte Sequenz aufgestellt, die die Abweichungder Abbildung (2) von einem Isomorphismus beschreibt. Wahrend zunachst Theorem 38 modulo nformuliert ist, wird in Theorem 46 die Aussage dann allgemein gezeigt.

1

Abstract

This present work describes the etale fundamental group of possibly singular arithmetic schemes. LetS ⊆ SpecO 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.Denote by πab

1 (X ) the abelianized etale fundamental group and let CX be the idele class group of X asdefined in [24]. CX is connected component of zero of CX . We describe the kernel and cokernel of themap CX /CX → πab

1 (X ) by means of embedding it in an exact sequence:

H2et(X ,Q/Z)∗ → HK

2 (X , Z)→ CX /CX → πab1 (X )→ HK

1 (X , Z)→ 0

HKq are certain cohomology groups defined in this work.

2

Contents

1 Introduction 4

2 Preliminaries 62.1 Notes on Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Etale Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Finiteness of Etale Cohomology Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 A Leray Spectral Sequence for Cohomology with Compact Support . . . . . . . . . . . . . 72.6 De Jong’s Theory of Alterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Class Field Theory for Regular Arithmetic Schemes 9

4 The Kato Complex and its Cohomology 114.1 A Projective System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 The Kato Complex and Kato Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 Limits of Cohomology Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.4 Kato Cohomology and Etale Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Singular Class Field Theory 175.1 Class Field Theory for Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2 The Theory modulo n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3 The General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Acknowledgements 21

7 References 21

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1 Introduction

In 1783 Leonard Euler stated, and in 1796 Carl Friedrich Gauß was the first to prove what becameknown as Gauß’ quadratic reciprocity law:(

p

q

)(q

p

)= (−1)

p−12

q−12 . (3)

Together with two supplementary theorems it provides an easy way to compute the Legendre symbol(pq ). 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 specialvalues of certain automorphic functions related to elliptic functions. Kronecker asked whether all abelianextensions of an algebraic number field K could be obtained in this manner. This idea became known asKroneckers 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 extensionof Q.

Kronecker contributed ideas to the proof, and Heinrich Weber proposed a “proof” which turned out tobe 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 thefirst elements of class field theory, it was Hilbert who saw the complete picture: the theory of abelianextensions. At the International Congress of Mathematicians in Paris in 1900, Hilbert posed in hisfamous 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 therational numbers to any base number field.

While Hilbert’s 12th problem remains open, for abelian extensions the 9th problem has found itssolution 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. Forevery discrete valuation v 6∈ S, we consider the map v : K× → Z, while for valuations v ∈ S we use theembedding K× → K×v into the completion to define the S-class group of K:

CS := coker

K× →⊕v 6∈S

Z⊕⊕v∈S

K×v

, (4)

It is an abelian topological group. For a group G, define Gab to be the maximal abelian factor group ofG, i.e. Gab = G/G′ where G′ is the commutator group of G.

Theorem 2 Let KS be the maximal algebraic extension of K, unramified outside S. Then there is areciprocity map

ρ : CS → Gal(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 of LanglandsProgram, 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 etale fundamental group. Let X be a connectedscheme and x → X a geometric point on X. The etale fundamental group πet

1 (X, x), introduced byAlexander Grothendieck, classifies the finite etale coverings of X. If X = SpecK, then choosing a geo-metric point x amounts to choosing a separably closed field Ω containing K, and πet

1 (X) = Gal(Ksep|K),where Ksep is the separable closure of K in Ω. Hence a description of πet

1 (X, x) or πab1 (X) generalizes

Theorem 2. (Note that a change of the base point x changes πet1 (X, x) by an isomorphism, which is

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canonical up to an inner automorphism. Hence there is no need to specify a base point when we discussπab

1 (X).)One interesting class of schemes studied are arithmetic schemes. Let X be a regular, integral and

separated scheme flat and of finite type over Spec Z. Then πab1 (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 afairly complicated description of the idele class group by the theory of Parshin chains. However, thereis also a second approach by Gotz Wiesend and the author which only uses K0 and K1 groups, i.e. Zand multiplicative groups. In [6], we give a theory in case of dimX = 2, describing both the abelian andnon-abelian case. In the abelian case, the description of πab

1 (X) in this paper is only modulo n. For thenon-abelian case, the higher dimensional theory is reduced to the (currently unknown) one dimensionaltheory. The results of [6] are greatly extended in later papers of Wiesend. In [23], a non-abelian theoryfor arithmetic schemes of general dimension is given. In [24], the corresponding abelian class field theoryis given for arithmetic schemes of general dimension.

While all the approaches mentioned in the preceding paragraph are for regular X , there are alsoresults for more general X . In [22], Peter Stevenhagen generalized one-dimensional class field theory toorders 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. Theydescribe when the reciprocity map CH0(X ) → πab

1 (X ) is injective using resolution of singularities anda cohomological Hasse principle for smooth proper surfaces. One year later, Alexander Schmidt andMichael 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 firstsuggested by Grothendieck and then studied by Bloch and Arthus Ogus [2]. Let K be a non-archimedeanlocal field and V a proper variety over Spec(K). Then, by the works of Bloch, Kato and Saito, there isa reciprocity map

ρV : SK1(V )→ πab1 (V ), (6)

where

SK1(V ) := coker

(⊕x∈V1

K2(x) ∂→⊕x∈V0

K1(x)

), (7)

Vi := x ∈ V | dim x = i, (8)k(x) := residue field of x and (9)

Kq(x) := q-th algebraic K-group of k(x), (10)

and ∂ is the boundary map from K-theory. For a positive integer n, prime to char(K) let

ρV,n : SK1(V )/n→ πab1 (V )/n (11)

be the induced map. Then Jannsen and Saito prove the existence of an exact sequence

HK2 (V,Z/n)→ SK1(V )/n

ρV,n−→ πab1 (V )/n→ HK

1 (V,Z/n)→ 0, (12)

where HKq is the q-th Kato homology group of V , defined as the homology of a complex Cr,s(V,Z/n) of

Bloch-Ogus type (see [8, §1] for details).This present work sets out to describe the etale fundamental group of possibly singular arithmetic

schemes over Spec Z using the class field theory of Wiesend. Let S ⊆ SpecO be an open subscheme ofthe spectrum of the ring of integers O of an algebraic number field, and let X be a separated, reducedand connected scheme of finite type over Spec Z. For regular X flat over S, the Main Theorem of [24]asserts

CX /C0X∼= πab

1 (X ) (13)

where CX is the idele class group of X as defined in [24] and in Definition 16 below and CX is the connectedcomponent of zero. For singular X , such an isomorphism cannot be expected. However, we can describekernel and cokernel of the map CX /C0

X → πab1 (X ) by means of embedding it in an exact sequence similar

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 forthem, as they are constructed similarly. Theorem 46 in this work gives the sequence describing πab

1 (X ):

H2et(X ,Q/Z)∗ → HK

2 (X , Z)→ CX /CX → πab1 (X )→ HK

1 (X , Z)→ 0 (14)

Note that while similar to (12), this sequence has some significant improvements. While Jannsen andSaito prove their sequence only for arithmetic surfaces, the sequence given here is also valid in higherdimensions. Also, we describe the full group πab

1 (X ) and not just the quotients πab1 (V )/n. Finally, there

is one more term in the sequence. We do however have to retain the assumption that X is proper overS to get this result.

This work is arranged as follows: We start by introducing our notation and reviewing basic factsabout etale cohomology. Then some tools used later in the proofs are introduced: A Leray spectralsequence and de Jong’s theory of alterations. Wiesend’s class field theory for regular arithmetic schemesis reviewed. This includes the definition of the class group CX . The whole next chapter is used todefine the necessary prerequisites for proving sequence (14) modulo n: A complex is defined and itshomology taken, which we will call Kato cohomology. The behaviour of Kato cohomology when takingprojective limits over certain subschemes is studied in detail. There is a natural transformation fromKato cohomology to etale cohomology. An analysis of the connections between Kato cohomology andetale cohomology then yields sequence (14) modulo n. Finally, we use alterations to get the generalresult (without modulo n).

2 Preliminaries

2.1 Notes on Notation

We use roman letters (X, Y , . . . ) for general schemes, and script letters (X , Y, . . . ) for arithmeticschemes. If i : X → Y is a morphism of arithmetic schemes (usually just an embedding) and F is a sheafon Y then to ease notation, we are supressing i∗ in many places throughout this work. For example, wewrite i∗F instead of i∗i∗F .

2.2 Schemes

Let O be the ring of integers in a number field. Throughout this work, we will always denote byS ⊆ SpecO an open subscheme of SpecO.

Definition 3 An arithmetic scheme X is a separated, reduced and connected scheme of finite type overSpecO. (Note that, in [24], also flatness over SpecO is required.)

Definition 4 A curve C on X is a closed and reduced subscheme of X of dimension 1.

2.3 Etale Sheaves

The main reference for etale cohomology used in this work is the book of Milne [15]. A morphism ofschemes (or rings) is etale if it is flat and unramified. We say that a covering for the etale topologyis a surjective family of etale morphisms Ui → X. This defines the etale topology (in the sense of aGrothendieck topology) on a scheme X. There is a theory of presheaves and sheaves for this topology,called etale presheaves and etale sheaves, cf. [15, II.3]. In this paper we will most of the time use locallyconstant or constructible sheaves. Moreover, the values of a sheaf are at least abelian groups.

Definition 5 A sheaf F is called constant if it is a sheaf associated to a constant presheaf. A sheaf Fis called locally constant if there is a covering Ui → X such that F|Ui is constant for all i. If all F|Uiare finite, we say that F is finite locally constant.

Definition 6 A sheaf F on a scheme X is called constructible, if X can we written as a union of finitelymany locally closed subschemes Y ⊆ X for which F|Y is finite locally constant. (Y is called locallyclosed if it is the intersection of an open and a closed subscheme.)

6

2.4 Finiteness of Etale Cohomology Groups

The functor Γ(X, · ) of global sections on X from the category of etale sheaves on X to the categoryof abelian groups is left exact. As the category of etale sheaves has enough injectives, we can define,for each left exact functor f , an essentially unique sequence of of functors Rif , called the right derivedfunctors of f . The Rif are characterized by the following properties:

(i) R0f = f ,

(ii) Rif(I) = 0 if i > 0 and I injective,

(iii) for any exact sequence 0→ F ′ → F → F ′′ → 0 of etale sheaves on X, we can assign, in a functorialway, morphisms ∂i : Rif(F ′′)→ Ri+1f(F ′) for i ≥ 0, such that the sequence

· · · → Rif(F)→ Rif(F ′′) ∂i→ Ri+1f(F ′)→ Ri+1f(F)→ · · · (15)

is exact.

Definition 7 The right derived functors of Γ(X, · ) are written

RiΓ(X, · ) = Hiet(X, · ). (16)

The group Hiet(X,F) is called the i-th etale cohomology group of X with values in F .

The following proposition will allow us to take projective limits of exact sequences of etale cohomologygroups.

Proposition 8 Let X be an arithmetic scheme, proper over S. For a constructible etale sheaf F on X ,the group Hq

et(X ,F) is finite.

Proof. Let π : X → S be proper. By [15, Theorem III.1.18a] there is a Leray spectral sequence

Ep,q2 : Hpet(S, R

qπ∗F)⇒ Hp+qet (X ,F). (17)

As π is proper, [15, Theorem VI.2.1] shows that Rqπ∗F is a constructible sheaf on S. Thus [16, TheoremII.3.1] and the remark that follows show that Hp

et(S, Rqπ∗F) is finite. As the only terms of the spectralsequence are in the quadrant p, q ≥ 0, this shows the claim of the proposition.

2.5 A Leray Spectral Sequence for Cohomology with Compact Support

The main tool used to define cohomology with compact support is to embed a given scheme X in acomplete scheme and then extend the given etale sheaf by zero outside X. For the latter, consider anetale morphism f : X → Y . We construct a functor f! from the category of sheaves on X to the categoryof sheaves on Y .

Definition 9 Let F be a sheaf of abelian groups on X. We denote by f!(F) the sheaf on Y associatedto the presheaf

(V → Y ) 7→⊕

φ∈HomY (V,X)

F(φ) (18)

with natural restriction maps. The direct sum is taken over all factorizations

V //

φ

Y

X.

>>

(19)

As said above, most of the time f will be the embedding of X in a complete scheme. If j is an openembedding, then the restriction of j!(F) to X is isomorphic to F , and the restriction of j!(F) to thecomplement of X in Y vanishes. These two properties characterize j!. In this situation we call j!(F) theextension by zero of F outside X.

We define cohomology groups with compact support and higher direct images with compact supportusing j!:

7

Definition 10 Let π : U → S be a compactifiable morphism, i.e. a morphism that can be embedded ina commutative diagram

U j //

π @@@@@@@ X

π~~~~~~~

S,

(20)

with j an open immersion, π proper, and F a torsion sheaf on U . Then, generalizing Milne [15, III.1.29],we define

Hqc (U,F) = Hq

et(X, j!F) (21)

and call this group the q-th cohomology group with compact support of X with values in F . Similarly,the sheaf

Rqcπ∗F = Rqπ(j!F) (22)

is called the q-th higher direct image with compact support.

Remark 11 By the proper base change theorem [15, Corollary VI.2.3], Hqc (U,F) and Rqcπ∗F are inde-

pendent of the choice of X.

Remark 12 The notation used here is standard, but misleading. One often writes Rqπ! for Rqcπ∗, butRqπ! is not the q-th derived functor of R0π!. Similarly, Hq

c (X,F) is not the i-th derived functor ofΓc(X,F) := H0

c (X,F).

We can now prove a Leray spectral sequence for cohomology with compact support.

Lemma 13 Let U ′, U be Noetherian schemes over S, α : U ′ → U a separated morphism over S of finitetype, and F a torsion sheaf on U ′. Then there is a spectral sequence

Ep,q2 : Hpc (U,Rqcα∗F)⇒ Hp+q

c (U ′,F). (23)

Proof. By [18, Theorem 2], we can compactify the maps to get to the situation of the following diagram.

U ′ j′ //

α

X ′ j′′ //

π′

proper

~~|||||||||||||X ′′

π′′

proper

U

j// X

proper

S

(24)

We start by noting that, by the definition of Rqc , we have

R0cj′′∗ = j′′! , (25)

Rpcj′′∗ = 0 for all p > 0, (26)

R0c j∗ = j!, (27)

Rpc j∗ = 0 for all p > 0, (28)Rqcπ

′′∗ = Rqπ′′∗ for all q and (29)

Rqcπ′∗ = Rqπ′∗ for all q. (30)

The spectral sequence of [15, Theorem VI.3.2c] now gives

(Rqcπ′′∗ )(Rpcj

′′∗ )⇒ Rq+pc (π′′ j′′)∗ and (31)

(Rpc j∗)(Rqcπ′∗)⇒ Rp+qc (j π′)∗ . (32)

8

It follows from (26) and (28) that their terms are actually equal to the limit term. Taking p = 0 andusing (25–30) as well as the commutativity of (24) gives

(Rqπ′′∗ )j′′! = Rqc(π′′ j′′)∗ = Rqc(j π′)∗ = j!Rqπ′∗ . (33)

To get the spectral sequence of the lemma, we start by using a Leray spectral sequence for etale coho-mology [15, Theorem III.1.18a]:

Hpet(X,R

qπ′′∗ (j′′ j′)!F)⇒ Hp+qet (X ′′, (j′′ j′)!F) (34)

Using (33) and the definitions of Rc and Hc finishes the proof.

Hpet(X, j!R

qπ′∗j′!F)⇒ Hp+q

et (X ′′, (j′′ j′)!F) (35)

Hpc (U,Rqcα∗F)⇒ Hp+q

c (U ′,F) (36)

2.6 De Jong’s Theory of Alterations

This section reviews a theorem of de Jong [3].

Definition 14 An alteration of an integral scheme X is an integral scheme X ′ together with a morphismϕ : X ′ → X which is surjective, proper and such that, for a suitable open dense set U ⊆ X, the inducedmorphism ϕU : ϕ−1(U)→ U is finite.

Let K be a global field, R ⊂ K a Dedekind domain with fraction field K. Let S = SpecR. For afinite extension K ⊂ K ′ of fields, let R′ be the integral closure of R in K ′ and set S′ = SpecR′. We calla morphism S′ → S obtained in this way a finite extension of Dedekind schemes.

An S-variety is an integral scheme X over S with X → S separated of finite type and flat. Then deJong proves the following theorem [3, Theorem 8.2]:

Theorem 15 Let X be an S-variety and Z ⊂ X a proper closed subset. There exists a diagram

X1

X1j1oo

ϕ1 // X

S1

idS1

ψ1 // S

(37)

where:

1. ψ1 : S1 → S is a finite extension of Dedekind schemes,

2. ϕ1 is an alteration, j1 is an open immersion and X1 is projective over S,

3. the scheme X1 is regular and the closed subset j1(ϕ−11 (Z)) ∪ (X1 \ j1(X1)) is a strict normal

crossings divisor in X1.

3 Class Field Theory for Regular Arithmetic Schemes

The main reference for this chapter is the paper of Wiesend [24].We want to define the idele class group for a reduced separated scheme X of finite type over Spec Z.

For an irreducible curve C on X , the function field κ(C) is a global field. Those places of κ(C) that donot correspond to points on the normalization of C will be denoted by C∞. Note that C∞ can containfinite places, and always contains all infinite places.

Definition 16 The idele group of X is

IX :=⊕x∈X

Z⊕⊕C⊆X

⊕v∈C∞

κ(C)×v . (38)

9

Here C runs over all irreducible curves on X , and x runs through all closed points of X . IX is atopological group: A subset is open if the intersection with each finite partial sum is open. The classgroup of X is

CX := coker

⊕C⊆X

κ(C)× → IX

. (39)

The map from a term κ(C) corresponding to a curve C into the term Z corresponding to a smoothclosed point x ∈ C is just the valuation at x. If x is a singular point of C, we consider all branchesb of C through x. Let vb be the valuation in x along the branch b, κ(xb) the residue field of κ(b) andfb = [κ(xb) : κ(x)]. Then the map κ(C)× → Z is given by

∑b fbvb. The map κ(C)× → κ(C)×v is just

the embedding of the field in its completion. The remaining maps are zero. CX carries the quotienttopology: a subgroup of CX is open if its preimage in IX is open.

Remark 17 C is a covariant functor from the category of reduced, separated schemes of finite type overSpec Z to the category of abelian topological groups [24, Lemma 2].

If f : Y → X is a morphism of such schemes, the map CY → CX is defined term by term. A closedpoint y ∈ Y is mapped to a closed point x = f(y). The map between the terms Z belonging to thesepoints is multiplication by [κ(y) : κ(x)]. Let C ⊆ Y be an irreducible curve on Y and v ∈ C∞ a valuationassociated to C.

If C is mapped to a closed point x ∈ X , then we can define a map from κ(C)×v to the term Z ofX associated to x by taking the normalized valuation of κ(C)v and multiplying it by the degree of theresidue field extension [κ(v) : κ(x)], where κ(v) is the residue field of the local field κ(C)v.

If the image of C in X is not a closed point, then it is an open subscheme of an irreducible curveD ⊆ X . Let f(v) be the image of the valuation v. We have a finite extension κ(C)v/κ(D)f(v) of localfields and an associated norm map κ(C)×v → κ(D)×f(v). There are two possibilities for the valuation f(v):If f(v) ∈ D∞, then we will use the norm map between the terms associated to C and D. If f(v) 6∈ D∞,then f(v) belongs to a point x ∈ D of the normalization D of D. The point x lies over a point x ∈ X .

We use the map κ(C)×v −→ κ(D)×f(v)

f(v)−→ Z ·[κ(x):κ(x)]−→ Z to get to the term Z of CX associated to x ∈ X .The functoriality gives canonical isomorphisms

IX = lim−→YIY and CX = lim−→

YCY (40)

as topological groups, where Y runs over all curves on X .

The theory for singular arithmetic schemes in this paper banks on the Main Theorem of [24]:

Theorem 18 Let X be a regular and integral arithmetic scheme, flat over S, with etale fundamentalgroup πab

1 (X ). Then there is a surjective continuous map

ρX : CX → πab1 (X ), (41)

which has kernel CX , the connected component of 0. This reciprocity map induces a bijection between theopen subgroups of πab

1 (X ) and the open subgroups of CX that lie above CX .

Proof. See §3 of [24].

Remark 19 Let X be a variety over a finite field Fq. We have defined a class group CX for X inDefinition 16. By local class field theory, elements in the image of CX → πab

1 (X )→ GFq are multiples ofthe Frobenius isomorphism of Fq. (See [24, Proof of Lemma 2] for the definition of ρX . The factors Z mapvia Z → Z ∼= πab

1 (x) → πab1 (X ). The factors κ(C)×v map via κ(C)×v → πab

1 (Spec(κ(C)v)) → πab1 (X ).)

Hence for a variety X over a finite field, the corresponding reciprocity map cannot be surjective. Adescription of the image can be found in [24, Theorem 1(c)].

We will make use of the following corollary [24, Corollary 1]:

Corollary 20 Let X be a regular integral arithmetic scheme and let ϕ : Y → X be finite, etale andGalois. Let Cϕ : CY → CX be the map of class groups associated to ϕ. Then

CX /CϕCY ∼= Gal(Y|X )ab. (42)

10

4 The Kato Complex and its Cohomology

Let X be an arithmetic scheme of dimension d, proper over S. Denote by F a constructible etale sheafon X . We want to define a complex in the spirit of Kato’s definition (see e.g. [7, Definition 1.1] or [8, §1]).It is conjectured that the definition given here matches Kato’s if the arithmetic scheme X is proper overSpec Z.

4.1 A Projective System

Let 0 ≤ k ≤ d and consider two closed reduced subschemes Dk−1 ⊂ Dk of X of dimension k − 1 and k,respectively. For the case k = 0, we will always set D−1 = ∅. Denote by jDk\Dk−1 : Dk \Dk−1 → Dk theopen embedding. We write jDk\Dk−1!F for the extension of F by zero outside the open set Dk \Dk−1.(Note that we write F as an abbreviation for the restriction of F to Dk \Dk−1. This will not be madeexplicit in the notation.)

Consider the etale cohomology groupHqet(Dk, jDk\Dk−1!F). We want to construct a projective system.

If Ek−1 ⊂ Ek are two closed reduced subschemes of X of dimension k − 1 and k, respectively, such thatEk−1 ⊇ Dk−1 and Ek ⊇ Dk, then we have the following maps:

Hqet(Ek, jEk\Ek−1!F)→ Hq

et(Dk, jEk\Ek−1!F|Dk) = Hqet(Dk, jDk\Ek−1!F)→ Hq

et(Dk, jDk\Dk−1!F). (43)

The first map is the contravariance of the cohomology functor in the first argument. The second map isthe covariance of the cohomology functor in the second argument applied to the embedding jDk\Ek−1!F →jDk\Dk−1!F of etale sheaves. The groupsHq

et(Dk, jDk\Dk−1!F) and the maps (43) form a projective systemof finite groups

(Hqet(Dk, jDk\Dk−1!F))Dk,Dk−1 . (44)

We will consider its projective limit

lim←−Dk,Dk−1

Hqet(Dk, jDk\Dk−1!F). (45)

Let D = (D0, . . . , Dk−1, Dk, . . . , Dd) denote a complete flag of X , i.e. Di−1 ⊂ Di and dimDi = i fori = 1, . . . , d. Then we can also take the projective limit over all flags D of X :

lim←−Dk,Dk−1

Hqet(Dk, jDk\Dk−1!F) = lim←−

DHq

et(Dk, jDk\Dk−1!F). (46)

We use this equivalent notation when we apply the projective limit to a sequence where terms of differentdimension occur.

4.2 The Kato Complex and Kato Cohomology

Proposition 21 For 0 ≤ k ≤ d− 1, there is a canonical map

lim←−DHk

et(Dk, jDk\Dk−1!F)→ lim←−DHk+1

et (Dk+1, jDk+1\Dk!F). (47)

Taken together, these maps form a complex of profinite groups

lim←−DH0

et(D0,F)→ lim←−DH1

et(D1, jD1\D0!F)→ lim←−DH2

et(D2, jD2\D1!F)→ · · ·

· · · → lim←−DHd

et(Dd, jDd\Dd−1!F)→ 0.(48)

Proof. To give a map between the projective limits it suffices to give, for all Dk−1 ⊂ Dk ⊂ Dk+1, maps

Hket(Dk, jDk\Dk−1!F)→ Hk+1

et (Dk+1, jDk+1\Dk!F). (49)

which commute with the maps of the projective system (43). To construct such maps, consider theinclusion map ik−1 : Dk−1 → Dk. The sequence

0→ jDk\Dk−1!j∗Dk\Dk−1

F → i∗kF → ik−1∗i∗k−1F → 0 (50)

11

is exact. Taking long exact sequences for k and k + 1 gives maps

Hket(Dk, jDk\Dk−1!j

∗Dk\Dk−1

F)→ Hket(Dk, i

∗kF) (51)

and the boundary map

Hket(Dk+1, ik∗i

∗kF)→ Hk+1

et (Dk+1, jDk+1\Dk!j∗Dk+1\DkF). (52)

We have Hket(Dk, i

∗kF) = Hk

et(Dk+1, ik∗i∗kF). Combining this with (51) and (52) gives the desired map.

Functoriality of the construction guarantees that these maps commute with the maps (43).The resulting sequence (48) is a complex, as the concatenation of two maps in this sequence involves

two successive maps in the long exact sequences associated to (50).

Definition 22 We call sequence (48) the Kato complex. Let F be a constructible sheaf on X . We definethe Kato cohomology groups

HqK(X ,F) (53)

of X with values in F as the cohomology of complex (48).

Remark 23 The construction of the Kato cohomology is functorial.

4.3 Limits of Cohomology Groups

Proposition 24 Let F be a constructible sheaf on X . For q ≤ d− 2, we have

HqK(X ,F) ∼= lim←−

Y⊂XHq

K(Y,F), (54)

where Y ⊂ X runs over all reduced connected subschemes of dimension d− 1.

Proof. We take the defining complex of HqK(X ,F) and compare it with lim←−

Y⊂Xof the complex for

HqK(Y,F), where Y ⊂ X runs over all reduced connected subschemes of dimension d− 1. We use D ⊆ X

and D ⊆ Y to denote complete flags of X and Y, respectively.

0

0

· · · 0

0

0

0 //

0 //

· · · // 0 //

lim←−D⊆X

Hdet(Dd, jDd\Dd−1!F) // 0

lim←−D⊆X

H0et(D0,F) // lim←−

D⊆XH1

et(D1, jD1\D0!F) // · · · // lim←−D⊆X

Hd−1et (Dd−1, jDd−1\Dd−2!F) // lim←−

D⊆XHd

et(Dd, jDd\Dd−1!F) //

0

lim←−D⊆YY⊂X

H0et(D0,F) //

lim←−D⊆YY⊂X

H1et(D1, jD1\D0!F) //

· · · // lim←−D⊆YY⊂X

Hd−1et (Dd−1, jDd−1\Dd−2!F) //

0 //

0

0 0 · · · 0 0 0

(55)

By [13, Satz 1.9], this is a short exact sequence of complexes which gives rise to a long exact sequence ofcohomology groups. For q ≤ d − 2, every third term of the resulting sequence is zero. Thus we get thedesired equations.

Remark 25 The terms for degree d− 1 and d of (55) yield the following sequence.

0→ Hd−1K (X ,F)→ lim←−

Y⊂XdimY=d−1

Hd−1K (Y,F)→ lim←−

Y⊂XdimY=d−1

Hdet(X , jX\Y!F)→ Hd

K(X ,F)→ 0 (56)

12

Lemma 26 Let X be a scheme, F be a locally constant etale sheaf on X, j : U → X an open immersionand i : Z = X \ U → X the complement. Assume that for all connected components Xi of X, we haveU ∩Xi 6= Xi. Then Γ(X, j!F) is zero.1

Proof. By considering each connected component of X separately, we can assume without loss ofgenerality that X is connected. Then the assumption on U reads U 6= X. From the exact sequence

0→ j!F → F → i∗F → 0 (57)

we get

0→ Γ(X, j!F)→ Γ(X,F)→ Γ(X, i∗F). (58)

As F is locally constant, we can consider it to be an πet1 (X)-module. Denote a connected component of

Z by Z ′. Via the map πet1 (Z ′) → πet

1 (X), the sheaf F can be considered an πet1 (Z ′)-module. Then the

map Γ(X,F)→ Γ(Z ′, i∗F) is just the inclusion Fπet1 (X) → (i∗F)π

et1 (Z′) of fixed modules. As Γ(Z ′, i∗F)

is a direct summand in Γ(Z, i∗F), the last map of (58) is injective. Hence Γ(X, j!F) = 0.

Proposition 27 Let X be a separated and reduced scheme of dimension d, of finite type and proper overS. For q ≤ d− 1 and any constructible sheaf F on X , we have

lim←−Y⊂X

dimY=d−1

Hqet(X , jX\Y!F) = 0, (59)

where Y runs through the closed reduced subschemes of dimension d− 1.

Proof. We proceed by induction over d = dimX . If d = 1 the claim follows from Lemma 26. Assumenow that this claim has already been proven for all W with dimW ≤ d − 1. Let Xi be one irreduciblecomponent of X , and denote its function field by Ki.

If Xi is a variety over a finite field k, then Ki has a transcendence basis consisting of d elements.Choosing d − 1 elements gives a rational map ϕi : Xi → Pd−1

k =: Wi. If Xi has mixed characteristic,Ki has a transcendence basis consisting of d − 1 elements. Choose d − 2 of these elements. Such achoice induces a rational map ϕi : Xi → Pd−2

Z =: Wi. In both cases ϕi induces a morphism on an opennonempty subscheme of Xi. Let Γi ⊂ Xi × Wi be the closure of the graph of this morphism. Theprojection πi : Γi →Wi is a proper surjective morphism. Let X be the disjoint union of all Γi and let Wbe the disjoint union of all Wi. As Γi is birational to Xi, and since cohomology with compact supportdoes not depend on the compactification, we can replace X by X without affecting the claim, and writeagain X for this modified scheme.

The claim is equivalent to

lim←−Y0⊆Y⊂X

dimY=d−1

Hqet(X , jX\Y!F) = 0 (60)

for an arbitrary closed reduced subscheme Y0 of dimension d − 1. Choose such an Y0 which π mapssurjectively onto W.

Define α0 by the following diagram

X \ Y0 //

α0

""FFFFFFFFFF X

π

W

(61)

and consider the etale sheaf Rqcα0∗F onW. It is constructible [15, Theorem VI.3.2d]. By [15, PropositionV.1.8b], there is a nonempty open subset U ⊆ W such that Rqcα0!F|U is locally constant. Define closedreduced subschemes Y1 ⊂ X and Y ′1 ⊂ W of dimension d− 1 and d− 2, respectively, by

Y ′1 := (W \ U) ∪ π(vertical components of Y0), (62)

Y1 := Y0 ∪ π−1(Y ′1). (63)1In [15] it is claimed on p.78 that for a (not necessarily locally constant) etale sheaf F on U and an open immersion

j : U → X we have (j!F)(V ) = F(V ) if V → X factors through U , and zero otherwise. This is wrong: Consider globalsections of a skyscraper sheaf F = (Z/n)x for a point x ∈ U . Then the second term in sequence (58) is non-zero, the thirdis zero, hence the first term must be nonzero.

13

Here, horizontal divisors of X are those that map surjectively on W, and vertical divisors are all others.The resulting new set of maps is given by diagram (64):

X \ Y1 //

α1

""FFFFFFFFFFF

X

π

W \ Y ′1

// W

(64)

Let 0 ≤ p ≤ d − 2 be a given integer. The sheaf jW\Y′1!Rqcα1∗F is locally constant on W \ Y ′1 ⊆ U

because on this subscheme it is equal to Rqcα0∗F . Hence by the induction hypothesis, applied to W, wehave

lim←−Y′1⊆Y

′⊂WdimY′=d−2

Hpc (W \ Y ′, Rqcα1∗F) = 0. (65)

We claim jW\Y′1!(R0cα1∗)F = 0 for q = 0. This can be checked on the stalks. Let w ∈ W be a

geometric point. If w ∈ Y ′1, then (jW\Y′1!(R0cα1∗)F)w = 0 by definition of j!. If w ∈ W \ Y ′1, then

(jW\Y′1!(R0cα1∗)F)w =(jW\Y′1!(R0π∗)jX\Y1!F)w = ((R0π∗)jX\Y1!F)w (66)

=H0(Xw, jX\Y1!F) = Γ(Xw, jX\Y1!F) = 0 (67)

by [15, Corollary VI.2.5] (proper base change) and by Lemma 26, as every fiber Xw contains a point ofY1.

Together with (65), we have now shown that

lim←−Y′1⊆Y

′⊂WdimY′=d−2

Hpc (W \ Y ′, (Rqcα1∗)F) = 0 (68)

for p+ q ≤ d− 1. By Lemma 13 there is a spectral sequence

Ep,q2 : lim←−Y′1⊆Y

′⊂WdimY′=d−2

Hpc (W \ Y ′, (Rqcα1∗)F)⇒ lim←−

Y1⊆Y⊂XdimY=d−1

Hp+qc (X \ Y,F). (69)

(The spectral sequence commutes with the projective limit as the projective limit is exact in our situa-tion.) Thus

lim←−Y1⊆Y⊂X

dimY=d−1

Hqc (X \ Y,F) = lim←−

Y1⊆Y⊂XdimY=d−1

Hqet(X , jX\Y!F) = 0 for q ≤ d− 1, (70)

which completes the induction.

Corollary 28 Let F be a constructible sheaf on X , and X proper over S. There are isomorphisms fornonnegative integers q < r < d

HqK(X ,F) ∼= lim←−

Y⊂XdimY=r

HqK(Y,F) (71)

and

Hqet(X ,F) ∼= lim←−

Y⊂XdimY=r

Hqet(Y,F). (72)

Proof. For r = d− 1, the isomorphisms (71) are the same as in Proposition 24. Let i : Y → X be theinclusion. By the preceding proposition, every third term (up to degree d− 1) of lim←−

Y⊂XdimY=d−1

of the long exactsequence of

0→ jX\Y!F → F → i∗F → 0, (73)

is zero. The remaining terms and the maps between adjacent pairs of them are exactly the isomor-phisms (72) for r = d− 1.

14

We are left with proving the claim for 0 < r < d− 1. As the proof is the same for both cases we onlyexplain the first isomorphism (71). We proceed by decreasing induction in r. Let r > q + 1 and assumethat the claim for r was already proved. Let X ′ be an r-dimensional closed reduced subscheme of X .By what we proved at the beginning,

HqK(X ′,F) ∼= lim←−

Y⊂X′dimY=r−1

HqK(Y,F). (74)

Applying the projective limit over all X ′ ⊂ X of dimension r gives

HqK(X ,F) ∼= lim←−

X′⊂XdimX′=r

HqK(X ′,F) ∼= lim←−

Y⊂XdimY=r−1

HqK(Y,F). (75)

The first isomorphism is the induction hypothesis, both give the induction step.

4.4 Kato Cohomology and Etale Cohomology

Proposition 29 Let X be an arithmetic scheme, proper over S, and let F be a constructible sheaf onX . There is a natural transformation of functors on the category of constructible etale sheaves on X

HqK(X ,F)→ Hq

et(X ,F) (q ≥ 0). (76)

In particular,

H0K(X ,F) ∼= H0

et(X ,F). (77)

Proof. Let d = dimX . We have the following diagram:

0 // Hd−1K (X ,F) //

lim←−Y⊂X

dimY=d−1

Hd−1K (Y,F) //

lim←−Y⊂X

dimY=d−1

Hdet(X , jX\Y!F) // Hd

K(X ,F) //

0

0 // Hd−1et (X ,F) // lim←−

Y⊂XdimY=d−1

Hd−1et (Y,F) // lim←−

Y⊂XdimY=d−1

Hdet(X , jX\Y!F) // Hd

et(X ,F) // lim←−Y⊂X

dimY=d−1

Hdet(Y,F)

(78)

The first line of this diagram was taken from Remark 25. The second line is lim←− of the long exactsequence of (73). The zero in the first column of the second line is due to Proposition 27.

For d = 1, the drawn through vertical arrow is the identity map, as the two groups coincide. Hencethere is an isomorphism H0

K(X ,F) ∼= H0et(X ,F). For d ≥ 2, we have

H0K(X ,F) ∼= lim←−

Y⊂XdimY=d−1

H0K(Y,F) ∼= lim←−

Y⊂XdimY=d−1

H0et(Y,F) ∼= H0

et(X ,F) (79)

by Corollary 28 and induction.Map (76) will be constructed by induction in d. If d = 1, only the case q = 1 is left. In this case

the map is the right dashed map in diagram (78), which is given by a diagram chase. Constructing thismap by a diagram chase will automatically ensure that the new diagram with this map is commutative.Assume now that the natural transformation has already be constructed for all Y ⊂ X with dimY = d−1.We use diagram (78) again. The drawn through vertical map is given by the induction hypothesis. Thena diagram chase gives the two dashed maps, completing the commutative diagram. This concludes theinduction for q = d− 1 and q = d. For q ≤ d− 2, we have

HqK(X ,F) ∼= lim←−

Y⊂XdimY=d−1

HqK(Y,F)→ lim←−

Y⊂XdimY=d−1

Hqet(Y,F) ∼= Hq

et(X ,F) (80)

by Corollary 28 and the induction hypothesis.

Corollary 30 Under the assumptions of the proposition, there is an exact sequence

0→ Hd−1K (X ,F)→ Hd−1

et (X ,F)⊕ lim←−Y⊂X

dimY=d−1

Hd−1K (Y,F)

+/−−→ lim←−Y⊂X

dimY=d−1

Hd−1et (Y,F)→

→ HdK(X ,F)→ Hd

et(X ,F)→ lim←−Y⊂X

dimY=d−1

Hdet(Y,F).

(81)

(+/− indicates that we negate the map on the second summand.)

15

Proof. This exact sequence follows by a diagram chase in diagram (78).

Corollary 31 Under the assumptions of the proposition, there are exact sequences

0→ HqK(X ,F)→ Hq

et(X ,F)⊕ lim←−Y⊂X

dimY=q

HqK(Y,F)→ lim←−

Y⊂XdimY=q

Hqet(Y,F)→

→ Hq+1K (X ,F)→ Hq+1

et (X ,F)→ lim←−Y⊂X

dimY=q

Hq+1et (Y,F)

(82)

for all 0 ≤ q ≤ d− 1.

Proof. We proceed by induction in d = dimX . For d = 1, everything is in Corollary 30. Assume nowthat the claim has already been established for all X ′ with dimX ′ = d− 1. If q = d− 1 then the claimis again in Corollary 30. Assume now q ≤ d− 2. Let X ′ ⊂ X be a (d− 1)-dimensional subscheme of X .By the induction hypothesis,

0→ HqK(X ′,F)→ Hq

et(X′,F)⊕ lim←−

Y⊂X′dimY=q

HqK(Y,F)→ lim←−

Y⊂X′dimY=q

Hqet(Y,F)→

→ Hq+1K (X ′,F)→ Hq+1

et (X ′,F)→ lim←−Y⊂X′

dimY=q

Hq+1et (Y,F).

(83)

is exact. Applying lim←−X′⊂X

dimX′=d−1

to this sequence and using Corollary 28 gives the claim if q < d− 2. If q = d− 2,

we cannot apply Corollary 28 to the terms in the second line of (83). For this case, consider diagram (84).

lim←−Y⊂X

dimY=d−2

Hd−2et (Y,F)

0 // Hd−1

K (X ,F) //

lim←−X′⊂X

dimX′=d−1

Hd−1K (X ′,F) //

lim←−X′⊂X

dimX′=d−1

Hdet(X , jX\X ′!F)

0 // Hd−1et (X ,F) // lim←−

X′⊂XdimX′=d−1

Hd−1et (X ′,F) //

lim←−X′⊂X

dimX′=d−1

Hdet(X , jX\X ′!F)

lim←−Y⊂X

dimY=d−2

Hd−1et (Y,F)

(84)

The horizontal lines are the same as in (78). Hence this diagram is commutative. The vertical columnis part of (83) after lim←−

X′⊂XdimX′=d−1

was applied. A diagram chase gives the exact sequence

lim←−Y⊂X

dimY=d−2

Hd−2et (Y,F)→ Hd−1

K (X ,F)→ Hd−1et (X ,F)→ lim←−

Y⊂XdimY=d−2

Hd−1et (Y,F). (85)

This is the remaining part of (83) that we still had to prove. The exactness at the joint is easy to verify.

Remark 32 For an arithmetic scheme X , proper over S, and q = 0 we can use Proposition 29 to seethat the first three terms of the six term sequence of the corollary collapse. We will use the constantsheaf F = Z/n on X . Hence we are left with the exact sequence

0→ H1K(X ,Z/n)→ H1

et(X ,Z/n)→ lim←−Y⊂X

dimY=0

H1et(Y,Z/n), (86)

which shows

H1K(X ,Z/n) = ker

H1et(X ,Z/n)→ lim←−

Y⊂XdimY=0

H1et(Y,Z/n)

= H1c.s.(X ,Z/n). (87)

16

Of course, the inverse limit is in this case just a product. H1c.s.(X ,Z/n) is the completely split cohomology

as defined in [6, Definition 3]. Hence H1K(X ,Z/n) classifies the abelian finite etale coverings of exponent

n which split completely.

5 Singular Class Field Theory

In this chapter we will only consider the constant sheaves F = Z/n and F = Q/Z.

Definition 33 We denote the Pontryagin-dual of the compact group HqK(X ,Z/n) by HK

q (X ,Z/n).

5.1 Class Field Theory for Curves

We will use Corollary 2 of [6]:

Proposition 34 Let C be an arithmetic scheme of dimension 1. Then there is an exact sequence

κ(C)×/n −→⊕x∈C

Z/n⊕⊕

x∈C\C

κ(C)×x /n⊕⊕v|∞

κ(C)×v /n

−→ πab1 (C)/n −→ πc.s.

1 (C)/n −→ 0.

(88)

In the first sum, x runs over all closed points of C. In the second sum, C is the normalization of C, andC is the normalization of a completion C of C. We write κ(C)x for the completion of κ(C) at the placecorresponding to a closed point x ∈ C \ C. Finally, πc.s.

1 (C)/n is the dual of H1c.s.(C,Z/n).

Proposition 35 Let Y be an arithmetic scheme of dimension 1, proper over S. Then there is an exactsequence

0→ CY/n→ πab1 (Y)/n→ HK

1 (Y,Z/n)→ 0. (89)

Proof. We adapt Proposition 34 to our terminology. Note that the second term of (88) is just the idelegroup IC/n. Hence we can replace the first two terms of (88) by the cokernel of the map between them,which is CY/n. The equality H1

K(Y,Z/n) = H1c.s.(Y,Z/n) holds by Remark 32, as we are assuming that

Y is proper over S.

Example 36 Note that the etale fundamental group πet1 (X ) only describes unramified coverings of X .

We can, however, allow ramification in certain points by applying the results to an open subscheme withthose points removed where we want to allow ramification. (But note that to use the theory developedherein after, we have to retain properness over S.) Hence the idele class group CX considered in thispaper corresponds to an S-class group in the classical theory, where S is a finite set of places where weallow ramification. For example, let X = P1

Z and consider U = Spec(Z[X,X−1, 12 ]). This is an open

subscheme of X with the horizontal divisors∞, (0) and the vertical divisor (2) removed. There is an etalecovering V of U given by V = Spec(Z[Y, Y −1, 1

2 ]) with Y 2 = X. While U is not proper over Spec(Z[ 12 ]),

it is regular and its class field theory is described by [24].We want to illustrate the statement of Proposition 34 by an example. Consider curves C on U . If C

is normal, then by Cebotarev’s Theorem every connected covering of C has an inert point. Hence thereare no completely split coverings and we have an isomorphism CC/n ∼= πab

1 (C)/n. On the other hand,consider the curve C given by the equation (X − 1)2 − 9X = 0. The curve C is singular as the pointgiven by the maximal ideal (3, X−1) is a node. Now there are more coverings of C than those describedby the class group alone. Indeed, consider the covering of C induced by V → U . It is given by

(Y 2 − 1)2 − 9Y 2 = (Y 2 + 3Y − 1)(Y 2 − 3Y − 1) = 0. (90)

It is connected because the points (3, Y ± 1) are on both irreducible components Y 2 + 3Y − 1 = 0 andY 2 − 3Y − 1 = 0. Hence we have found a connected covering that is split over each closed point x ∈ C.These coverings are accounted for by πc.s.

1 (C)/n.

17

5.2 The Theory modulo n

Proposition 37 Let X be an arithmetic scheme, proper over S, with dimX ≥ 2. The sequence

0→ H1K(X ,Z/n)→ H1

et(X ,Z/n)⊕ lim←−Y⊂X

dimY=1

H1K(Y,Z/n)→ lim←−

Y⊂XdimY=1

H1et(Y,Z/n)→

→ H2K(X ,Z/n)→ H2

et(X ,Z/n)→ lim←−Y⊂X

dimY=1

H2et(Y,Z/n)

(91)

is exact.

Proof. Let q = 1,F = Z/n in Corollary 31.

Theorem 38 (Class Field Theory modulo n) Let X be an arithmetic scheme, proper over S, withdimX ≥ 2. The following sequence is exact.

lim−→Y⊂X

dimY=1

H2et(Y,Z/n)∗ → H2

et(X ,Z/n)∗ → HK2 (X ,Z/n)→

→ CX /nρ→ πab

1 (X )/n→ HK1 (X ,Z/n)→ 0

(92)

Proof. We dualize Proposition 37. The first three terms of the dual are

lim−→Y⊂X

dimY=1

H2et(Y,Z/n)∗ → H2

et(X ,Z/n)∗ → HK2 (X ,Z/n)→ . . . (93)

and give the first three terms of sequence (92). The remaining terms of the dual sequence, together withthe injective limit of sequence (89) in Proposition 35 over all closed, reduced and connected Y ⊆ X ,dimY = 1, form a commutative diagram.

. . . // HK2 (X ,Z/n) // lim−→

Y⊂XdimY=1

πab1 (Y)/n // lim−→

Y⊂XdimY=1

HK1 (Y,Z/n)⊕ πab

1 (X )/n //

pr1

HK1 (X ,Z/n) // 0

0 // lim−→Y⊂X

dimY=1

CY/n // lim−→Y⊂X

dimY=1

πab1 (Y)/n // lim−→

Y⊂XdimY=1

HK1 (Y,Z/n) // 0.

(94)

Note that lim←−CY/n = CX /n by the exactness of injective limits. The remaining maps of sequence (92)are constructed as follows. Starting at HK

2 (X ,Z/n), we use the maps of diagram (94) to go right-down-right. Commutativity of (94) and the exactness of the first row show that we end up with zero. Hencethe image in lim←−π

ab1 (Y)/n has a preimage in lim←−CY/n, which clearly is well-defined. From there we

use the maps of the diagram to go to the sum lim←−HK1 (Y,Z/n) ⊕ πab

1 (X )/n, but only use the entryin the second summand. This map is induced by Wiesend’s reciprocity map ρ, because ρ is definedby maps which factorize over Y ⊂ X ,dimY = 1 (cf. part c of the proof of [24, Lemma 3]). Fi-nally, the map πab

1 (X )/n → HK1 (X ,Z/n) is given by mapping πab

1 (X )/n on the second summand oflim←−H

K1 (Y,Z/n)⊕ πab

1 (X )/n, and then taking the respective map of sequence (94). This explains all themaps needed for the theorem. The exactness of the sequence follows from further diagram chases.

Conjecture 39 HKq (X ,Z/n) is finite. If X is regular and q > 0, HK

q (X ,Z/n) = 0.

Remark 40 These conjectures are inspired by Conjecture 0.5 of [8]. For dimX = 1 and q = 1, theconjecture follows from Proposition 35 and Theorem 18. For dimX ≥ 2 and q = 1, 2, note first thatCX /n is finite as CX ⊂ nCX ⊂ CX , nCX is open and CX /CX is profinite by Proposition 43 below. Thenthe finiteness of HK

q (X ,Z/n) follows from Theorem 38 and Proposition 8. For q = 1, the second part(X regular) of the conjecture follows from Theorems 38 and 18. For q = 2, we know by Theorems 38and 18 that HK

2 (X ,Z/n)→ CX /n is the zero map. Hence the conjecture is equivalent to an inclusion

H2et(X ,Z/n) ⊆ lim←−

Y⊂XdimY=1

H2et(Y,Z/n). (95)

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5.3 The General Theory

Definition 41 Let X be an arithmetic scheme. Define Kato cohomology groups with values in Q/Z by

HqK(X ,Q/Z) = lim−→

n

HqK(X ,Z/n), (96)

and let HKq (X , Z) be the dual of Hq

K(X ,Q/Z).

Let CX be the connected component of zero of the idele class group CX . We will start by using deJong’s alterations and the fact that CX /CX is profinite for regular X (Main Theorem of [24]) to provethat CX /CX is profinite in general. To recover the general theory (without modulo n) we will then takethe projective limit over n of sequence (92) of Theorem 38.

We need the following proposition before we can proceed.

Proposition 42 Let π : Y → X be a proper and surjective morphism of reduced, separated schemes offinite type over S. Then the image of CY in CX is open and of finite index.

Proof. The proof is by induction over the dimension of X . If dimX = 0, CX is a finite sum of copiesof Z. The image of the summand Z of CY corresponding to a closed point y ∈ Y in the summand Z ofCX for x = π(y) is nonzero. As π is surjective, this shows the claim for dimX = 0.

Assume now that X is of dimension at least one. Let U ⊆ X be an open dense subscheme such thatU is regular (so connected components of U are irreducible), and let V := π−1(U).

From the Stein factorization [4, III.4.3.1 and 4.3.4] of π : V → U we get a scheme V ′ and a factorizationπ = γ β, where β : V → V ′ is a proper and surjective morphism with geometrically connected fibersand γ : V ′ → U is a finite surjective morphism.

We start by discussing β : V → V ′. By [24, Lemma 2] it induces an homomorphism Cβ : CV → CV′ ofthe class groups. We will show that Cβ is surjective by checking the terms in the definition of the classgroup one by one. For a closed point x ∈ V ′, consider the fiber Vx = π−1(x). If Vx has dimension zero,then it is just a single point as β has connected fibers. There is no residue field extension as the fibersof β are geometrically connected and the residue field is perfect. Hence Cβ maps CV′ = Z isomorphicallyto the factor Z of CU corresponding to x.

If Vx has dimension at least one, then choose a closed point y0 ∈ Vx. The image of the group Zcorresponding to y0 in the group Z corresponding to x is generated by a positive integer d0 = [κ(y0) :κ(x)]. Tensoring Vx → x with the field extension κ(y0)/κ(x) gives a cyclic covering Vx × κ(y0)→ Vx ofdegree d0. By Cebotarev’s density theorem [21, Theorem 7] there exists a closed point y1 ∈ Vx which isinert in this covering. If we let d1 = [κ(y1) : κ(x)], then d0 and d1 are relative prime. Thus the image ofthe factors Z corresponding to these two points generate the full group Z corresponding to x.

Finally, consider the terms κ(C)×v for an irreducible curve C ⊆ V ′. Clearly C ×V′ V is proper overC. There is a curve D ⊆ C×V′ V which maps surjectively onto C. Given an element l ∈

⊕v∈C∞ κ(C)×v ,

we show that its class is in the image of Cβ . For each valuation v ∈ C∞, choose a valuation wv ∈ D∞of κ(D) extending v. The image of the norm map κ(D)×wv → κ(C)×v is open and of finite index. Denoteit by Uv. Given an element l ∈

⊕v∈C∞ κ(C)×v , we can use the approximation theorem to write it as

l = l0 · f , with l0 ∈⊕

v∈C∞ Uv and f ∈ κ(C)×. As the class of l0 is in the image of Cβ , so is the classof l. (The image of f may also give another contribution on factors Z corresponding to closed points.)This completes the proof of the surjectivity of Cβ .

Consider the finite surjective morphism γ : V ′ → U . Note first that if V ′ and V are varieties over afinite field of characteristic p, and γ is a purely inseparable morphism of degree p, then Cπ : CV′ → CU issurjective. This can already be seen on the ideles IU : There is a bijection of closed reduced subschemesof V ′ and U . The map on the factor Z for a closed point x ∈ U is multiplication by the degree of theresidue field extension. But the residue fields are finite and thus perfect, hence there is no extension.Let C be an irreducible curve on U , and D the curve of V ′ lying above C. κ(C) and κ(D) are functionfields of transcendence degree 1. For each pair of places v ∈ C∞, w ∈ D∞ with w extending v, the mapκ(D)×w → κ(C)×v is a norm map. If κ(D) = κ(C) there is nothing to show. Otherwise κ(D)v = κ(C)1/p

v

is the unique inseparable extension of degree p of κ(C)v. Hence the norm map is surjective. Thisshows that Cγ is surjective in this case. For the general case, note that we constructed U such that allconnected components are irreducible. On each component of U , γ factorizes into a separable morphismand a sequence of purely inseparable morphisms of degree p as discussed above. Hence we can assumethat γ is separable.

By shrinking U (and V ′) we can also assume that γ is flat and unramified and therefore etale (thenew maps β and γ are the just restrictions of the previous maps). We replace V ′ by its Galois hull so

19

that γ becomes a finite, etale and Galois morphism. This will only increase the index of the image of CV′in CU . Then by Corollary 1 of [24], CU/γ(CV′) is isomorphic to the abelianized Galois group Gal(V ′/U)ab

and thus finite. Also by (the proof of) this corollary, γ(CV′) is open in CU .We have now shown that A = Cπ(CV) is open and of finite index in CU .Let Z be the closed complement of U in X , and define W = π−1(Z). As the dimension of Z is

strictly less than the dimension of X , the induction hypothesis shows that B = Cπ(CW) is open and offinite index in CZ . Let A0 ≤ IU and B0 ≤ IZ be the preimages of A and B in the corresponding idelegroups. There is a surjective homomorphism

ϕ : IU ⊕ UZ → IX (97)

mapping the summands of IU and IZ isomorphically to the corresponding summands of IX . By Defini-tion 16, open subsets of IU , IZ and IX are characterized by having an open intersection in each finitepartial sum. Therefore ϕ(A0 ⊕ B0) is open in IX . Since A0 and B0 have finite index in IU and IZ ,respectively, also ϕ(A0 ⊕ B0) has finite index in IX . On the level of idele classes we have an inducedcommutative diagram

CV

⊕ CW

// // CY

Cπ

CU ⊕ CZ // // CX

(98)

which by the discussion on the idele level gives the claim that Cπ(CY) is open and of finite index in CX .

Proposition 43 Let X be an arithmetic scheme, flat over S. Then CX /CX is profinite.

Proof. Let X be integral. By Theorem 15, there is a regular and integral arithmetic scheme Y and asurjective and proper map ϕ : Y → X . The scheme S is an open part of the spectrum of a Dedekind ring.Modules over Dedekind rings are flat if and only if they are torsion-free. As Y is integral and dominantover S, Y is flat over S.

The associated homomorphism Cϕ : CY → CX maps CY into CX . By Theorem 18, the group CY/CY isprofinite and hence compact. By [5, Theorem II.1], CX /CX is a (Hausdorff) totally disconnected group.This shows that the image of CY/CY in CX /CX is compact and totally disconnected, and thus profiniteas well. By Proposition 42, the image of CY in CX is open and of finite index. Hence CX /CX is theextension of a finite group by a profinite group, and thus profinite itself.

If X has several irreducible components Xi, then there is a surjective homomorphism

ϕ :⊕i

IXi → IX (99)

mapping the summands of IXi isomorphically to the corresponding summands of IX . It induces asurjective homomorphism on the idele classes, which, in turn, induces a surjective homomorphism

ϕ :⊕i

CXi/CXi → CX /CX . (100)

Since by Definition 16 the kernel of this homomorphism is clearly closed, this shows that CX /CX isprofinite.

Remark 44 Note that we have shown that CX /CX is profinite without using that X is proper over S.

Corollary 45 Let X be an arithmetic scheme, proper and flat over S. Then

lim←−n

CX /n = CX /CX . (101)

20

Proof. We first claim that nIX is an open subgroup of IX for all n ∈ N. This can be checked separatelyfor all factors in IX . For a closed point x ∈ X , nZ is open in Z. Consider an irreducible curve C ⊂ X .Since X is proper over S, we have C∞ 6= ∅ only for horizontal curves, and therefore κ(C)×nv is alwaysopen in κ(C)×v . This shows that nIX is an open subgroup of IX , and consequently nCX is an opensubgroup of CX . Hence we have nCX ⊇ CX for all n ∈ N. Using this and Proposition 43 completes theproof:

lim←−n

CX /n = lim←−n

(CX /CX )/n = CX /CX . (102)

Theorem 46 (Class Field Theory) Let X be an arithmetic scheme with dimX ≥ 2, proper and flatover S. Then the following sequence is exact.

H2et(X ,Q/Z)∗ → HK

2 (X , Z)→ CX /CX → πab1 (X )→ HK

1 (X , Z)→ 0 (103)

For dimX = 1, one has the exact sequence

0→ CX /CX → πab1 (X )→ HK

1 (X , Z)→ 0. (104)

Proof. We take the projective limit of the sequence in Theorem 38 over n, leaving off the first term.Note that the groups occuring here are finite by the part of Conjecture 39 that we proved. Hencethe projective limit is exact by [13, Satz 1.9]. The dual will transform the projective limit to a directlimit. For the first term, note that etale cohomology is compatible with direct limits. For the Katocohomology groups, note how Hq

K with values in Q/Z was defined at the beginning of this chapter. Theequality lim←−

n

CX /n = CX /CX is by Corollary 45. The etale fundamental group is profinite, so we have

lim←−n

πab1 (X )/n = πab

1 (X ). The last part of the theorem follows in the same way from Proposition 35.

6 Acknowledgements

My thanks go to my late friend and colleague Gotz Wiesend, without whose helpful remarks this workwould not have been possible and to my advisor, Prof. W.-D. Geyer, for his uninterrupted supportthroughout all the years that I was allowed to stay in Erlangen. I would also like to thank Ralf Gerkmannand Uwe Jannsen for many helpful remarks.

7 References

[1] S. Bloch. Algebraic K-theory and class field theory for arithmetic surfaces. Ann. Math., 114(2):229–265, 1981.

[2] S. Bloch and A. Ogus. Gersten’s conjecture and the homology of schemes. 1974.

[3] A. de Jong. Smoothness, semi-stability and alterations. Publ. Math., Inst. Hautes Etud. Sci.,83:51–93, 1996.

[4] A. Grothendieck. Elements de geometrie algebrique. I: Le langage des schemas. II: Etude globaleelementaire de quelques classes de morphismes. III: Etude cohomologique des faisceaux coherents(premiere partie). Publ. Math., Inst. Hautes Etud. Sci., No. 4 (1960); ibid. 8 (1961); ibid. 11 (1962).

[5] P. Higgins. Introduction to topological groups. London Mathematical Society Lecture Note Series.15. Cambridge: At the University Press., 1974.

[6] W. Hofmann and G. Wiesend. Non-abelian class field theory for arithmetic surfaces. Math. Z.,250(1):203–224, 2005.

[7] U. Jannsen and S. Saito. Kato homology of arithmetic schemes and higher class field theory overlocal fields. Doc. Math., J. DMV Extra Vol., pages 479–538, 2003.

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[8] K. Kato. A Hasse principle for two dimensional global fields. Appendix by Jean- Louis Colliot-Thelene. J. Reine Angew. Math., 366:142–183, 1986.

[9] K. Kato. Generalized class field theory. Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 419-428(1991), 1991.

[10] K. Kato and S. Saito. Two dimensional class field theory. In Galois groups and their representations,Proc. Symp., Nagoya/Jap. 1981, Adv. Stud. Pure Math. 2, 103-152. 1983.

[11] K. Kato and S. Saito. Unramified class field theory of arithmetical surfaces. Annals of Mathematics,118:241–275, 1983.

[12] K. Kato and S. Saito. Global class field theory of arithmetic schemes. Contemporary Mathematics,55:255–331, 1986.

[13] H. Koch. Galoissche Theorie der p-Erweiterungen. VEB Deutscher Verlag der Wissenschaften,Berlin., 1970.

[14] K. Matsumi, K. Sato, and M. Asukura. On the kernel of the reciprocity map of normal surfacesover finite fields. K-Theory, 18:203–234, 1999.

[15] J. S. Milne. Etale Cohomology. Princeton University Press, Princeton, 1980.

[16] J. S. Milne. Arithmetic duality theorems. Perspectives in Mathematics, Vol. 1. Boston etc.: AcademicPress. Inc. Harcourt Brace Jovanovich, Publishers., 1986.

[17] M. Nagata. Imbedding of an abstract variety in a complete variety. J. Math. Kyoto Univ., 2:1–10,1962.

[18] M. Nagata. A generalization of the imbedding problem of an abstract variety in a complete variety.J. Math. Kyoto Univ., 3:89–102, 1963.

[19] S. Saito. Class field theory for curves over local fields. J. Number Theory, 21:44–80, 1985.

[20] A. Schmidt and M. Spieß. Singular homology and class field theory of varieties over finite fields. J.Reine Angew. Math., 527:13–36, 2000.

[21] J.-P. Serre. Zeta and L functions. Arithmetical Algebraic Geometry, Proc. Conf. Purdue Univ.,1963:82–92, 1965.

[22] P. Stevenhagen. Unramified class field theory for orders. Trans. Am. Math. Soc., 311(2):483–500,1989.

[23] G. Wiesend. A construction of covers of arithmetic schemes. Journal of Number Theory, 121:118–131, 2006.

[24] G. Wiesend. Class field theory for arithmetic schemes. Math. Z., 256(4):717–729, 2007.

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Lebenslauf

Geboren am 13. Februar 1976 in Nurnberg.

Ausbildung

1982–1986 Grundschule Nurnberg1986–1995 Mathematisch-naturwissenschaftliches Gymnasium Rothenbach/Peg.1995 Abitur (Note 1.5), Leistungskursfacher: Mathematik und Physik1995 Sommerakademie der Studienstiftung uber Elementarteilchenseit 1995 Mathematik- und Physikstudium an der Universitat Erlangen Nurnberg

als Stipendiat der Studienstiftung des deutschen Volkes1996 Sommerakademie der Studienstiftung uber randomisierte Algorithmen1997 Vordiplome in Mathematik (Note 1.075) und Physik (1.25)1997 Sommerakademie der Studienstiftung uber ”Frustration in komplexen Systemen“

Sommerakademie uber Rechnerarchitektur1997–1998 einjahriger Auslandsaufenthalt am Fitzwilliam College in Cambridge/UK1998 Sommerakademie der Studienstiftung uber Seiberg-Witten InvariantenSept. 2001 Diplom Mathematik (mit Auszeichnung)seit 2001 Mitarbeiter am Mathematischen Institut der Universitat Erlangen2002 Tagung uber Korperarithmetik, Oberwolfach, Deutschland

Vortrag uber Einbettungsprobleme von quadratischen Zahlkorpern2003 Konferenz uber nicht-abelsche Zahlentheorie, Durham, UK

Vortrag uber nicht-abelsche Klassenkorpertheorie bei arithmetischen Flachen2004 Einmonatiger Forschungsaufenthalt in Belo Horizonte, Brasilien als Mitglied

des DAAD-Projekts ”Vector Bundles on Algebraic Curves“2006 Tagung uber Korperarithmetik, Oberwolfach, Deutschland

Einmonatiger Forschungsaufenthalt in Belo Horizonte, Brasilien als Mitglieddes DAAD-Projekts ”Vector Bundles on Algebraic Curves“

Veroffentlichungen

2001 Zyklische Einbettungsprobleme bei reell-quadratischen ZahlkorpernDiplomarbeit

2005 Non-abelian Class Field Theory for Arithmetic Surfaces (with Gotz Wiesend),Math. Z. 250, 203–224 (2005)

Auszeichnungen

1992 Einladung zur Endrunde des 10. Bundeswettbewerbs Informatik1993 Bundessieger des 11. Bundeswettbewerbs Informatik

Silbermedaille bei der Internationalen Olympiade in Informatik in Mendoza/ArgentinienAufnahme als Stipendiat in die Studienstiftung des Deutschen Volkes

1994 Goldmedaille bei der Internationalen Olympiade in Informatik in Stockholm/Schweden1. Preis beim Regionalwettbewerb Jugend forscht, Sonderpreis beimLandeswettbewerb Jugend forscht

1995 Teilnahme am Bundeswettbewerb Mathematik (zweiter Preis in der zweiten Runde)sowie am Auswahlverfahren zur Internationalen Physik-Olympiade

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