Some remarks on the local class field theory of Serre and Hazewinkel

Bulletin

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Publié avec le concours du Centre national de la recherche scientifique

de la SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Tome 141 Fascicule 1

2013

pages 1-24

SOME REMARKS

ON THE LOCAL CLASS FIELD THEORY OF SERRE AND HAZEWINKEL

Takashi Suzuki

SOME REMARKS

ON THE LOCAL CLASS FIELD THEORY OF SERRE AND HAZEWINKEL

by Takashi Suzuki

Abstract. — We give a new approach for the local class field theory of Serre and Hazewinkel. We also discuss two-dimensional local class field theory in this framework.

Résumé (Quelques remarques sur la théorie du corps de classes local de Serre et Hazewinkel)

Nous donnons une nouvelle approche de la théorie du corps de classes local de Serre et Hazewinkel. Nous discutons également la théorie du corps de classes local de dimension deux dans ce cadre.

1. Introduction

The purpose of this paper is twofold. First, we give a geometric description of the local class field theory of Serre and Hazewinkel ([13], [7, Appendice]) in the equal characteristic case. The main result is TheoremAbelow. Second, we discuss its two-dimensional analog with the aim to seek an analog of Lubin- Tate theory for two-dimensional local fields. The discussion is taken place in Section 6with a partial result.

We formulate Theorem A. The precise definitions of the terms used below are given in Section 2. Let k be a perfect field of characteristic p >0 and set K = k((T)). The group of units of K can be viewed as a proalgebraic group

Texte reçu le 17 mars 2009, révisé le 17 décembre 2012, accepté le 30 avril 2012.

Takashi Suzuki, Department of Mathematics, University of Chicago, 5734 S University Ave, Chicago, IL 60637 • E-mail :suzuki@math.uchicago.edu

2010 Mathematics Subject Classification. — 11S31.

Key words and phrases. — Local class field theory, K-theory.

BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 0037-9484/2013/1/$ 5.00

©Société Mathématique de France

(more precisely, a pro-quasi-algebraic group) overkin the sense of Serre ([12]);

we denote this proalgebraic group byUK. For each perfectk-algebraR(perfect means that the p-th power map is an isomorphism), we have the group of its R-rational points

U_K(R) = (_∞

X

i=0

a_iTⁱ a_i∈R, a₀∈R^× )

.

Likewise, the multiplicative groupK^× of K can be viewed as a perfect group scheme K^×, which is the direct product ofUK and the discrete infinite cyclic group generated by T. Let Kp be the perfect closure of K. We consider the K_p-rational point1−TT⁻¹ ofK^× and the corresponding morphism

ϕ: SpecKp→K^×. This morphism induces a homomorphism

η: Gal(K^ab/K) =π₁^et(SpecKp)^ab→^ϕ π^k₁(K^×)

on the fundamental groups. Here K^ab is the maximal abelian extension ofK and π^et₁(·)^ab denotes the maximal abelian quotient of the étale fundamental group. The groupπ^k₁(K^×)is the fundamental group ofK^× as a perfect group scheme overk, which classifies all surjective isogenies toK^×with finite constant kernels. Now we state the main theorem of this paper:

Theorem A. — 1. The above defined map η: Gal(K^ab/K)→π^k₁(K^×) is an isomorphism.

2. The inverse of η restricted toπ^k₁(UK) coincides with the reciprocity iso- morphism θ:π^k₁(UK)→^∼ I(K^ab/K) of Serre and Hazewinkel ([13],[7]), where I denotes the inertia group.

Note that Assertion1of the theorem says that we have an essentially one-to- one correspondence between surjective isogeniesAK^× with finite constant kernels and finite abelian extensionsLofK by pullback byϕ, as expressed as a cartesian diagram of the form

SpecL_p −−−−→ A



 y



 y SpecKp

−−−−→ϕ K^×.

In other words, we have Ext¹_k(K^×, N) →^∼ H¹(K, N) for any finite constant group N. Another remark is that the proof of Assertion 2 that we give in this paper does not use the fact that θ is an isomorphism. This means that Theorem A and our proof of it together give another proof of this fact, more specifically, the existence theorem of the local class field theory of Serre and

o

Hazewinkel ([13, §4], [7, 6.3]). For a generalization of Assertion 2 for the full groups π₁^k(K^×)andGal(K^ab/K), see Remark3.5.

We outline the paper. After providing some preliminaries at Section 2, we give three different proofs of TheoremA. Each of them has its own advantages and interesting points. The first proof given in Section3 may be the standard one. The method is purely local. In this proof, for Assertion1, we explicitly cal- culate the groupsGal(K^ab/K)andπ₁^k(K^×)and the homomorphismηbetween them individually. For Assertion 2, we interpret the assertion to the compat- ibility of ϕ with norm maps and prove it by analyzing the diagonal divisors.

The second proof given in Section 4 relies on Lubin-Tate theory. Hence it is applicable only for finite residue field cases. The third proof given in Section5 is a geometric proof, which was suggested by the referee to the author. In this geometric proof, for Assertion 1, we need the Albanese property of the mor- phismϕ: SpecK_p→K^×, which was established by Contou-Carrere in [6] (see also [5]). For Assertion2, we need the local-global compatibility and the global version of Assertion2, which was obtained by Serre in [13, §5].

In Section 6, we define a morphism analogous to ϕ: SpecKp → K^× for a field of the form k((S))((T)) using the second algebraic K-group instead of the multiplicative group. If k is a finite field, the fieldk((S))((T))is called a two-dimensional local field ([8]) of positive characteristic. In analogy with the second proof of Theorem A by Lubin-Tate theory, we attempt to formulate an analogous theory to Lubin-Tate theory for the two-dimensional local field k((S))((T)). Although we have not yet obtained an analog of a Lubin-Tate formal group in this paper, we did obtain a result that may be thought of as a partial result for abelian extensions havingp-torsion Galois groups (Proposi- tion6.1). In Section7, we give an analog of the result of [2, §2.6] on

D

^-modules.

Their result is for fields of the formk((T))withkcharacteristic zero while ours k((S))((T)) with k characteristic zero. This is also regarded as an analog of Proposition 6.1.

We give a couple of comments on literature. The morphismϕ: SpecKp → K^× had been introduced by Grothendieck ([4, August 9, 1960]) long before our paper. Besides Contou-Carrere as mentioned above, the morphism ϕ has been studied by many people. The

D

-module version of TheoremAin the zero- characteristic case had been established in [2, §2.6]. Deligne had given a sketch of proof of results stronger than Assertion1. This is written in Section e of his letter to Serre contained in [3]. His method is different from our method. The author did not realize these works at the time of writing this paper.

Acknowledgements. — This is an extended version of the master thesis of the author at Kyoto University. The author would like to express his deep gratitude to his advisor Kazuya Kato and Tetsushi Ito for their suggestion of the

problems, encouragement, and many helpful discussions. The author is grateful also to the referee for the suggestion of the geometric proof of TheoremA, and to Professor Takuro Mochizuki for informing the author of the work [2].

2. Preliminaries

In this section, we give precise definitions of the terms that we used in Introduction and will use in the following sections, fix notation, prove the in- dependence of the choice of the prime elementT (Proposition2.3), and reduce the proof of TheoremAto the case of algebraically closed residue fields.

2.1. Definitions and notation. — We work on the site (Perf/k)fpqc of per- fect schemes over a perfect field k of characteristic p > 0 with the fpqc topology (compare with [11, Chapter III, §0, “Duality for unipotent perfect group schemes”]; see also [17]). The category of sheaves of abelian groups on (Perf/k)fpqc contains the category of commutative affine pro(-quasi)- algebraic groups over k in the sense of Serre ([12]) as an thick abelian full subcategory. We denote by Extⁱ_k the i-th Ext functor for the category of sheaves of abelian groups on (Perf/k)fpqc. For a sheaf of abelian groups A on (Perf/k)_fpqc and a non-negative integer i, we define the fundamental group π^k₁(A) of A to be the Pontryagin dual of the torsion abelian group inj lim_n≥1Ext¹_k(A, n⁻¹Z/Z). If A is an extension of an étale group whose group of geometric points is finitely generated as an abelian group by an affine proalgebraic group, then inj lim_n≥1Ext¹_k(A, n⁻¹Z/Z) = Ext¹_k(A,Q/Z) and hence the Pontryagin dual of Ext¹_k(A,Q/Z) coincides with π₁^k(A). For a k-algebraR, we denote byR_p the perfectk-algebra given by the injective limit of p-th power mapsR→R→ · · ·, where thei-th copy ofR in this system is given the mapk→R,a7→a^pi as itsk-algebra structure map.

Let K be a complete discrete valuation field of equal characteristic with residue field k. We denote by

O

K the ring of integers of K and by p_K the maximal ideal of

O

K. We setU_K =U_K⁰ =

O

^×K andU_Kⁿ = 1 +pⁿ_K forn≥1. We fix an algebraic closureKofK. All algebraic extensions ofK are taken inside K. We denote byK^sep (⊂K) the separable closure ofK, byK^ur the maximal unramified extension of K and by K^ab the maximal abelian extension of K, respectively. Since Khas equal characteristicp >0, the rings

O

K andK have canonical structures ofk-algebras by the Teichmüller sectionk ,→

O

K. Hence, for a sheaf of abelian groups A on (Perf/k)fpqc, we can consider the groups A((

O

K)_p)(resp.A(K_p)) of(

O

K)_p-rational (resp.K_p-rational) points ofA. We denote by A((pK)p) the kernel of the natural map A((

O

K)p) → A(k). The reduction map

O

K k and the Teichmüller sectionk ,→

O

K give a canonical splittingA((

O

K)_p) =A(k)⊕A((p_K)_p).

o

We define a sheaf of ringsOK on(Perf/k)fpqcby settingOK(R) =R⊗ˆk

O

K

for each perfectk-algebraR,⁽¹⁾where⊗ˆ denotes the completed tensor product.

LetKbe the sheaf of rings on(Perf/k)_fpqcwithK(R) =O_K(R)⊗_OKK. We set U_K=O^×_K. For eachn≥0, the sheaf of ringsO_K has a subsheaf of ideals pⁿ_K with pⁿ_K(R) =OK(R)⊗_OKpⁿ_K. The presentationOK = proj lim_n→∞OK/pⁿ_K gives an affine proalgebraic ring structure forOK. Likewise,UK has a subsheaf of groups Uⁿ_K = 1 +pⁿ_K for each n ≥1 (for n= 0, we set U⁰_K =U_K). The presentation UK = proj lim_n→∞UK/Uⁿ_K gives an affine proalgebraic group structure for UK. We have a split exact sequence0 →UK →K^× →Z→0.

For a prime element T of

O

K, let T be the k-rational point of O_K defined byT= 1⊗T ∈k⊗ˆk

O

K =OK(k). If we fix a prime elementT, we may write UK(R) = R[[T]]^× and K^×(R) = R[[T]][T⁻¹]^×, which are the descriptions ofU_K andK^×given in Introduction. The rational point1−TT⁻¹∈K^×(K_p) taken in Introduction should therefore be understood to be1−(T⊗1)⊗T⁻¹∈ ((Kp⊗ˆk

O

K)⊗_OKK)^×=K^×(Kp).

For a rational point f ∈K^×(Kp), we denote byϕf the corresponding mor- phism ofk-schemesSpecK_p→K^×. An extension0→N →A→K^×→0with N finite constant pulls back byϕf: SpecKp→K^×to anN-torsor onSpecKp. This defines a homomorphism Ext¹_k(K^×, N)→H¹(K, N). By Pontryagin du- ality, we have a homomorphism Gal(K^ab/K) → π₁^k(K^×), which we denote byη_f.

2.2. A subset of K^×(Kp). — In this subsection, we define a subset ΞK

of K^×(Kp) and prove some properties of it. This subset contains 1−TT⁻¹ for all prime elements T and is convenient for proving both the independence of the prime element (Proposition2.3) and the compatibility with norm maps (Proposition 3.4).

Definition 2.1. — We define Ξ_K to be the set of elements f ∈ K^×(K_p) satisfying the following conditions:

1. f is inK((

O

K)_p)and generates as an ideal ofK((

O

K)_p)the kernel of the multiplication mapK((

O

K)p) = ((

O

K)p⊗ˆk

O

K)⊗_OKK→Kp and 2. the natural map K((

O

K)p)→K(k)sendsf to1.

Proposition 2.2. — 1. The setΞ_K contains1−TT⁻¹ for any prime T. 2. The group UK((pK)p) = Ker UK((

O

K)p)→UK(k)

acts by multiplica- tion on ΞK transitively without fixed points.

3. The valuation map K^×→Z sends every element ofΞ_K to−1.

(1)This defines a sheaf on the site of perfectk-algebras with the fpqc topology. We actually need to take its Zariski sheafification to have a sheaf on(Perf/k)fpqc. The two sites here have equivalent categories of sheaves. The same process is applied to all sheaves in this paper.

4. For everyf ∈ΞK, we have the following commutative diagram with exact rows:

0 −−−−→ I(K^ab/K) −−−−→ Gal(K^ab/K) −−−−→ Gal(k^ab/k) −−−−→ 0



 y

η_f 

 y

η_f 

 y^o

0 −−−−→ π^k₁(U_K) −−−−→ π^k₁(K^×) −−−−→ π₁^k(Z) −−−−→ 0, where the right vertical isomorphismGal(k^ab/k)→^∼ π₁^k(Z)is the Pontrya- gin dual map of the natural isomorphism H¹(k,Q/Z) ∼= Ext¹_k(Z,Q/Z) times−1.

5. For each prime elementT, the restricted homomorphismη_f: I(K^ab/K)→ π^k₁(UK) for f = 1 − TT⁻¹ ∈ ΞK factors as the composite of the inclusion I(K^ab/K) ,→ Gal(K^ab/K) and the homomorphism Gal(K^ab/K) → π₁^k(U_K) induced by the morphism SpecK_p → U_K corresponding to the rational point −T +T∈UK(Kp).

Proof. — 1. Obvious.

2. Iff1andf2are inΞK, we havef2=uf1for someu∈Ker K^×((

O

K)p)→ K^×(k)

=K^×((pK)p). Suchuis unique sinceK((

O

K)p)is a domain. We have K^×((pK)p) =UK((pK)p)sinceZ((pK)p) = 0. Thus we have Assertion2.

3. This follows from Assertions1 and2.

4. By Assertion3, we have a commutative diagram SpecK_p −−−−→ Speck



 y

ϕ_f 

 y K^× −−−−→ Z,

where the right vertical morphism corresponds to−1∈Z(k). The maps induced on the fundamental groups give the required commutative diagram. The top row of the diagram in Assertion 4 is obviously exact. The bottom row of is exact as well since the sequence 0→U_K→K^×→Z→0 is split exact.

5. The prime element T gives a splitting K^× ∼= U_K ×Z. The rational point 1−TT⁻¹∈K^×(Kp)corresponds to the rational point(−T+T,−1)∈ UK(Kp)×Z(Kp). This implies Assertion5.

2.3. Independence of the choice of the prime element. — We show that the morphisms SpecKp →K^× corresponding to the rational points1−TT⁻¹ ∈ K^×(Kp) for various primes T actually induce the same homomorphism Gal(K^ab/K)→π₁^k(K^×)on the fundamental groups. More generally, we prove the following.

Proposition 2.3. — The homomorphism ηf: Gal(K^ab/K) → π^k₁(K^×) for f ∈Ξ_K is independent of the choice off.

o

Thus we may write η instead of ηf. To prove the proposition, we need the following lemma, which will also be used in the proof of Proposition3.4and in Section 5.

Lemma 2.4. — Let 0→N →A→A⁰ →0be an exact sequence of sheaves of abelian groups on(Perf/k)fpqc withN representable by an affine scheme. Then the induced map A((pK)p)→A⁰((pK)p)is surjective.

Proof. — Taking cohomology, we have a commutative diagram with exact rows 0 −−−−→ N((

O

K)p) −−−−→ A((

O

K)p) −−−−→ A⁰((

O

K)p) −−−−→ H_fpqc¹ (

O

K, N)



 y



 y



 y



 y 0 −−−−→ N(k) −−−−→ A(k) −−−−→ A⁰(k) −−−−→ H_fpqc¹ (k, N), where Hfpqc denotes the cohomology with respect to the site(Perf/k)fpqc.

We show that the last vertical homomorphism is an isomorphism. Since N is affine, we can write N = proj limN_λ, where each N_λ is an affine algebraic group scheme over kviewed as a sheaf on (Perf/k)fpqc ([7, III, §3, 7.5, Cor.]).

We haveH_fpqc¹ (

O

K, N)∼= proj limH_fpqc¹ (

O

K, Nλ). Since any infinitesimal group becomes zero over (Perf/k)fpqc, we can takeNλ to be smooth. Then we have H_fpqc¹ (

O

K, N_λ)∼=H_et¹(

O

K, N_λ)∼=H_et¹(k, N_λ)by [10, Chapter III, Theorem 3.9 and Remark 3.11]. ThusH_fpqc¹ (

O

K, N)∼=H_fpqc¹ (k, N).

On the other hand, the horizontal homomorphisms have natural sections cor- responding to the Teichmüller sectionk ,→

O

K. These facts imply the required surjectivity.

Proof of Proposition2.3. — By Assertion2 of Proposition2.2, it is enough to show that η_f: Gal(K^ab/K) → π₁^k(K^×) is a zero map for f ∈ U_K((p_K)_p) = K^×((pK)p). Let 0 → N → A → K^× → 0 be an exact sequence of sheaves of abelian groups on (Perf/k)fpqc with N finite constant. Then the morphism ϕ_f: SpecK_p → K^× lifts to a morphism SpecK_p → A by Lemma 2.4. This implies thatηf = 0.

2.4. Reduction to the case of algebraically closed residue fields. — We assume that TheoremAis true for algebraically closedkand show that TheoremAis true for general perfect k. By Assertion 4 of Proposition 2.2, the problem is re- duced to that for the restricted part η: I(K^ab/K) → π₁^k(UK). For this, it is enough to see that the Gal(k/k)-coinvariants of Gal((K^ur)^ab/K^ur) (for the conjugation action) is I(K^ab/K) and the Gal(k/k)-coinvariants of π₁^k(U_K) is π^k₁(UK). The first assertion follows from the fact that the natural sur- jection Gal(K^sep/K) Gal(k/k) admits a section ([15, Chapter II, §4.3, Exercises]). For the second assertion, we use the natural spectral sequence

Hⁱ(k,Ext^j

k(UK,Q/Z)) ⇒ Ext^i+j_k (UK,Q/Z). We have Hom_k(UK,Q/Z) = 0 by the connectedness of UK. Thus H⁰(k,Ext¹_k(UK,Q/Z)) = Ext¹_k(UK,Q/Z).

Hence Gal(k/k)-coinvariants ofπ^k₁(U_K)isπ^k₁(U_K). Thus we get the result.

3. First proof of TheoremA

3.1. Proof of the part “ηis an isomorphism”. — In this subsection, we assume that the residue field k is algebraically closed and fix a prime element T of

O

K to prove that η: I(K^ab/K) = Gal(K^ab/K) → π^k₁(K^×) = π₁^k(U_K) is an isomorphism. By Assertion 5 of Proposition 2.2, this homomorphism is induced by the morphism ϕ⁰: SpecKp → UK given by the rational point

−T+T∈U_K(K_p). Both groupsGal(K^ab/K)andπ₁^k(U_K)are profinite abelian groups. The Pontryagin dual of Gal(K^ab/K) isH¹(K,Q/Z). The Pontryagin dual ofπ^k₁(UK)isExt¹_k(UK,Q/Z). Therefore the problem is equivalent to show- ing that the dual mapη^∨: Ext¹_k(UK,Q/Z)→H¹(K,Q/Z)is an isomorphism.

SinceU_K∼=G_m×U¹_K, we have

Ext¹_k(U_K,Q/Z)∼= Ext¹_k(G_m,Q/Z)⊕Ext¹_k(U¹_K,Q/Z).

3.1.1. The prime-to-p part. — We show that η^∨: Ext¹_k(U_K,Q/Z) → H¹(K,Q/Z)induces an isomorphism on the prime-to-pparts. It is enough to show that η^∨: Ext¹_k(UK, n⁻¹Z/Z) → H¹(K, n⁻¹Z/Z) is an isomorphism for each integernprime top. Sincekis algebraically closed,n⁻¹Z/Zis isomorphic as a Galois module overkto the groupµnof then-th roots of unity. The group Ext¹_k(U¹_K, µn) is zero since the n-th power map is an automorphism on U¹_K while µn is killed by n. The group Ext¹_k(Gm, µn) is a cyclic group of order n generated by the extension class 0 → µ_n → G_m →ⁿ G_m → 0. The group H¹(K, µn) is a cyclic group of order n generated by the Kummer character σ7→σ((−T)^1/n)/(−T)^1/n. The morphismϕ⁰: SpecKp→UK followed by the projectionU_K G_mcorresponds to the rational point−T. These generators correspond each other via the homomorphismη^∨: Ext¹_k(UK, µn)→H¹(K, µn) since we have a cartesian diagram

SpecKp((−T)^1/n) −−−−→ Gm



 y



 yⁿ SpecK_p −−−−→ G_m.

Henceη^∨: Ext¹_k(UK, µn)→H¹(K, µn)is an isomorphism for any integern≥1 prime top.

o

3.1.2. The p-primary part. — We show that the homomorphism on thep-pri- mary partsη^∨: Ext¹_k(UK,Qp/Zp)→H¹(K,Qp/Zp)is an isomorphism as well.

We need the following lemmas.

Lemma 3.1. — The Artin-Hasse exponential mapQ

p-n≥1W →U¹_K sending a= (an)_p-n≥1∈ Y

p-n≥1

W with an= (an0, an1, . . .)∈W

to

Y

p-n≥1, m≥0

F(anmT^npm)∈U¹_K

is an isomorphism of proalgebraic groups. Here we denote by W the additive group of Witt vectors and setF(t) = exp(−P

e≥0t^pe/p^e)∈Zp[[t]].

Proof. — See [14, Chapter V, §3, 16].

Lemma 3.2. — Let f:A→B be a homomorphism between abelian groups A andB. If both Aand B arep-divisible and p-power torsion, andf induces an isomorphism on the p-torsion parts, thenf itself is an isomorphism.

The group Ext¹_k(Gm,Qp/Zp) is zero since the p-th power map is an auto- morphism onGmas a perfect group scheme whileQp/Zpis a union ofp-power torsion groups. The group Ext¹_k(U¹_K,Qp/Zp) = π^k₁(U¹_K)^∨ is p-divisible since U¹_K ∼= Q

p-n≥1W by Lemma 3.1 and π₁^k(W) is p-torsion-free by [12, §8.5, Prop. 5]. The group H¹(K,Qp/Zp) is p-divisible since the largest pro-p quo- tient of Gal(K^sep/K)is pro-pfree ([15, Chapter II, §2.2, Corollary 1]). Thus, using Lemma 3.2, we are reduced to showing that η^∨: Ext¹_k(U¹_K,Z/pZ) → H¹(K,Z/pZ)is an isomorphism. We have

Ext¹_k(U¹_K,Z/pZ)∼= M

p-n≥1

Ext¹_k(W,Z/pZ)∼= M

p-n≥1

Ext¹_k(G_a,Z/pZ)∼= M

p-n≥1

k.

Here the last isomorphism comes from the isomorphism k→^∼ Ext¹_k(G_a,Z/pZ) sending an elementa∈k^× to the extension class given by

(1) 0−→Z/pZ−→G_a^a

−1℘

−→ G_a−→0,

where℘(x) =x^p−x([12, §8.3, Prop. 3]). On the other hand, the map defined by

M

p-n≥1

kT⁻ⁿ→H¹(K,Z/pZ),

aT⁻ⁿ7→ σ7→σ(℘⁻¹(aT⁻ⁿ))−℘⁻¹(aT⁻ⁿ)

is an isomorphism by Artin-Schreier theory. To show that η^∨: Ext¹_k(U_K,Z/pZ)∼= M

p-n≥1

k→ M

p-n≥1

kT⁻ⁿ∼=H¹(K,Z/pZ)

is an isomorphism, we need to calculate the following morphism:

SpecK_p→U¹_K/(U¹_K)^p∼= Y

p-n≥1

W/pW ∼= Y

p-n≥1

G_a.

The morphism SpecKp → U¹_K/(U¹_K)^p corresponds to the Kp-rational point 1−T⁻¹T of U¹_K/(U¹_K)^p. The isomorphism Q

p-n≥1Ga

→∼ U¹_K/(U¹_K)^p sends each element (an)_p-n≥1 of the left-hand side to Q

p-n≥1F(anTⁿ)of the right- hand side.

Proposition 3.3. — 1. The inverse of the isomorphism Q

p-n≥1Ga

→∼

U¹_K/(U¹_K)^p is given by the map U¹_K/(U¹_K)^pdlog→ Y

n≥1

GaTⁿdlogT→^α Y

p-n≥1

Ga,

where α(P

n≥1bnTⁿdlogT) = (−bn/n)p-n≥1.

2. The rational point 1−T⁻¹T corresponds to (1/(nTⁿ))_p-n≥1 via the iso- morphism U¹_K/(U¹_K)^p(Kp)∼=Q

p-n≥1Ga(Kp).

3. The mapSpecKp→Q

p-n≥1Gagives theKp-rational point(1/(nTⁿ))_p-n≥1 of Q

p-n≥1Ga.

Proof. — 1. Using the identitydlogF(t) =−P

e≥0t^pedlogt, we have

dlog Ñ

Y

p-n≥1

F(anTⁿ) é

=− X

e≥0 p-n≥1

(anTⁿ)^pedlog(anTⁿ)

= X

e≥0 p-n≥1

(−n)(anTⁿ)^pedlogT.

Thus the mapα◦dlogsendsQ

p-n≥1F(anTⁿ)to(an)p-n≥1, as desired.

2. A simple calculation shows that(α◦dlog)(1−T⁻¹T) = (1/(nTⁿ))_p-n≥1. 3. This follows from Assertion2.

Now we calculate η^∨. Let n≥1 be an integer prime to panda6= 0 be an element ofkregarded as an element ofL

p-n≥1kby the inclusionk ,→L

p-n≥1k

o

into then-th summand. The corresponding extension ofGa is given by (1). We have a cartesian diagram

SpecKp(℘⁻¹(a/nTⁿ)) −−−−→ Ga



 y



 y^a

−1℘

SpecKp −−−−→ Ga. Thus η^∨: L

p-n≥1k → L

p-n≥1kT⁻ⁿ preserves the direct factors and the map on the n-th factor is given by multiplication by 1/n. This shows that η^∨: Ext¹_k(U_K,Z/pZ)→H¹(K,Z/pZ)is an isomorphism.

This completes the proof of Assertion1 of TheoremA.

3.2. Proof of the part “η⁻¹=θ”. — In this section, we assume that the residue fieldkis a general perfect field of characteristicp >0(see Remark3.5below for the point this generality helps for). To prove Assertion2of TheoremA, we first quickly recall the construction of the reciprocity mapθ:π₁^k(UK)→^∼ I(K^ab/K) of the local class field theory of Serre and Hazewinkel. LetLbe a finite totally ramified Galois extension ofK. We can apply the constructions in Section2.1 to L to get sheaves UL and L^× on(Perf/k)fpqc. Let VL/K be the subgroup ofU_Lgenerated by elements of the formσ(u)/uforσ∈Gal(L/K)andu∈U_L. By [7, Appendice, 4.2], we have exact sequences

0→Gal(L/K)^ab→UL/V_L/K ^N→^L/K U^×_K →0, (2)

0→Gal(L/K)^ab→L^×/V_L/K^N→^L/K K^×→0, (3)

where the second map NL/K is the norm map and the first map sends σ to σ(π_L)/π_L for any choice of a prime element π_L of

O

L, with π_L defined as 1⊗πL∈k⊗ˆk

O

L =OL(k)andσ acting onL^× as a morphism of sheaves.

To write the isomorphism θ:π₁^k(U_K)→^∼ I(K^ab/K), note that the Pontryagin dual groups ofπ₁^k(UK),I(K^ab/K)areExt¹_k(UK,Q/Z),H¹(I(K^ab/K),Q/Z), respectively. Now the dual map θ^∨: H¹(I(K^ab/K), G) →^∼ Ext¹_k(UK, G) for a finite abelian group G is given by sending L totally ramified abelian with Gal(L/K) =Gto the extension class (2) ([7, Appendice, 7.1]).

On the other hand, the dualη^∨: Ext¹_k(K^×, G)→^∼ H¹(K, G)of our isomor- phismη: Gal(K^ab/K)→^∼ π^k₁(K^×)sends the extension class (3) to theG-torsor on SpecK_p given by pulling back (3) by the morphism ϕ_f: SpecK_p → K^×, where f ∈ ΞK. Therefore our claim η⁻¹ = θ is interpreted as the claim that the G-torsor on SpecKp thus obtained is SpecLp. Hence Assertion 3 of the following proposition proves Assertion2 of TheoremA:

Proposition 3.4. — Assume that the residue field k is perfect of character- istic p >0. LetL/K be a finite totally ramified Galois extension.

1. For any f ∈ΞK and any g∈ΞK, the diagram SpecLp −−−−→

ϕ_g L^×



 y



 y^NL/K SpecK_p −−−−→^ϕf K^×

is commutative “up to a (p_L)_p-rational point”, namely the ratio of the two elements of K^×(Lp) corresponding to the two morphisms SpecLp⇒K^× coming from the above possibly non-commutative diagram is in K^×((p_L)_p) = Ker(K^×(

O

L)_p)K^×(k)).

2. For any f ∈ ΞK, there exists g ∈ ΞL such that the above diagram is indeed commutative.

3. Assume thatL/K is abelian. For anyf ∈Ξ_K and anyg∈Ξ_Lthat makes the diagram commutative, the induced diagram

SpecLp −−−−→

ϕ_g L^×/V_L/K



 y



 y^NL/K

SpecKp ϕ_f

−−−−→ K^×

is cartesian, and is equivariant under the action ofGal(L/K)on the right vertical arrow coming from the exact sequence (3)and the natural action of Gal(L/K) on the left vertical arrow.

Proof. — 1. The two elements ofK^×(L_p)mentioned in Assertion1 are given byf andN_L/Kg. It is enough to show thatf /N_L/Kg∈K^×((

O

L)p)since then we immediately havef /N_L/Kg∈K^×((pL)p)by Condition2 of Definition2.1.

The claimf /N_L/Kg∈K^×((

O

L)_p)is equivalent to the equality(N_L/Kg) = (f) of ideals of the ring K((

O

L)_p). Let ∆_K ⊂ SpecK((

O

K)_p) be the diagonal divisor defined by the kernel of the multiplication map

K((

O

K)_p) = ((

O

K)_p⊗ˆ_k

O

K)⊗_OKK→K_p.

Then we have ∆K = (f) as divisors on SpecK((

O

K)p) by Condition 1 of Definition2.1. Similarly, forσ∈Gal(L/K), letσ(∆_L)⊂SpecL((

O

L)_p)be the σ-twist of the diagonal divisor defined by the kernel of the map

L((

O

L)_p) = ((

O

L)_p⊗ˆk

O

L)⊗_OLL^{(1⊗σ)⊗σ}−→ L_p.

Then we have p_∗∆L = (N_L/Kg) as divisors on SpecK((

O

L)p), where p: SpecL((

O

L)p) → SpecK((

O

L)p) is the natural map. Therefore what we have to show is the equality p_∗∆_L = ∆_K of divisors on SpecK((

O

L)_p).

o

This equality follows from the equalities p_∗∆L= X

σ∈Gal(L/K)

σ(∆L) = ∆K

as divisors onSpecL((

O

L)p). This proves Assertion1.

2. First take g ∈ ΞL to be arbitrary. Then by Assertion 1, we have f /NL/Kg ∈ K^×((pL)p). The homomorphism NL/K: L^× → K^× and its re- striction U_L →U_K are surjective and have the same kernel; see (2) and (3).

SinceUL andUK are affine, so isKer(N_L/K). Thus we can choose an element u∈L^×((pL)p)so thatN_L/Ku=f /N_L/Kg by Lemma 2.4. Replacingg byug, we have N_L/Kg=f. This choice ofgmakes the diagram commutative.

3. Before the proof, note that the field L(Lp) has two different actions ofGal(L/K). One comes from the action on the sheafL, which we have already denoted byh7→σ(h),σ∈Gal(L/K). The other comes from the action on the coefficient fieldLp, which we denote byh7→[σ](h),σ∈Gal(L/K).

Now we show the equivariance first. This is the same, in terms of the rational pointg, as the equality[σ](g) = (σ(πL)/πL)·g in(L^×/VL/K)(Lp). Below we use the exponential notation, so the equality to be shown can be written as g^[σ]−1=π^σ−1_L . We have

g^[σ]−1= (g^−[σ])^σ−1g^σ[σ]−1.

By Assertion 3 of Proposition 2.2, the element g^−[σ] is a prime element of the complete discrete valuation ring O_L(L_p). Hence (g^−[σ])^σ−1 = π^σ−1_L in (L^×/V_L/K)(L_p). Therefore it is enough to show thatg^σ[σ]−1∈V_L/K(L_p).

By Condition 1 of Definition 2.1, the element g^σ[σ] generates as an ideal ofL((

O

L)p)the kernel of the map

L((

O

L)p) = ((

O

L)p⊗ˆk

O

L)⊗_OLL^{(σ⊗σ)⊗σ}−→ Lp,

which is the same as the kernel of the usual multiplication map L((

O

L)_p) = ((

O

L)_p ⊗ˆ_k

O

L) ⊗_OL L → L_p. Therefore we have g^σ[σ]−1 ∈ L^×((

O

L)_p).

Since g ∈ L^×(Lp), we have g^σ[σ]−1 ∈ UL((

O

L)p). Moreover, we have g^σ[σ]−1 ∈ UL((pL)p) by Condition 2 of Definition 2.1. Furthermore, we have g^σ[σ]−1 ∈ (KerNL/K)((pL)p) since f is K-rational and there- fore N_L/K(g^σ[σ]−1) = f^[σ]−1 = 1. The exact sequence (2) leads an ex- act sequence 0 → V_L/K → KerN_L/K → Gal(L/K)^ab → 0, hence (KerN_L/K)((p_L)_p) = V_L/K((p_L)_p). Thus we have g^σ[σ]−1 ∈ V_L/K((p_L)_p).

This proves the equivariance of the diagram.

Finally we show that the diagram is cartesian. The pullback of the exact sequence (3) by ϕf: SpecKp →K^× gives a Gal(L/K)-torsor Z on SpecKp. The diagram factors through a morphism from SpecLp to Z by universality.

The equivariance just proved says that this is a morphism ofGal(L/K)-torsors

onSpecKp. Such a morphism is automatically an isomorphism. ThereforeZ= SpecLp and the diagram is cartesian.

Remark 3.5. — In [16, §4], the reciprocity isomorphism θ: I(K^ab/K) →^∼ π^k₁(UK) of Serre and Hazewinkel was extended to an isomorphism Gal(K^ab/K) →^∼ π^k₁(K^×). A full discussion about this result was taken place in [17]. The dual map θ^∨: H¹(K, G) →^∼ Ext¹_k(K^×, G) sends a finite totally ramified abelian extension L of Galois group Gto the extension class (3). The above proposition also proves the equalityη⁻¹ =θ as isomorphisms between fullGal(K^ab/K)andπ₁^k(K^×).

4. Second proof of TheoremA: Lubin-Tate theory

In this section, we assume that the residue field kis finite and give another proof of Theorem A for finite k using Lubin-Tate theory. Assume k is the finite field with q elements. We fix a prime element T of

O

K. This defines a splittingK^×∼=UK×Z. By Assertion5of Proposition2.2, the homomorphism Gal(K^ab/K) →^η K^× UK comes from the morphism ϕ⁰: SpecKp → UK

corresponding the rational point −T +T ∈ UK(Kp). We consider the short exact sequence 0 → UK → UK

F−1→ UK → 0, where F is the q-th power Frobenius morphism. The isogeny F−1, called the Lang isogeny, is universal among isogenies onto UK with (pro-) finite constant kernels ([14, Chap. VI,

§1, Prop. 6]), hence it identifiesUK asπ^k₁(UK).

Proposition 4.1. — There is a cartesian diagram Spec(K_T^ram)p −−−−→ UK



 y



 y^F−1

SpecKp

ϕ⁰

−−−−→ UK.

Here K_T^ram is the field K adjoining all the T^m-torsion points (where m runs through the integers≥1) of the Lubin-Tate formal groupF_f ([9]) whose equa- tion of formal multiplication by T is equal to f(X) = T X+X^q. The mor- phism Spec(K_T^ram)p →UK corresponds to the rational point P∞

m=0αm+1T^m, where αm is a generator of the module of T^m-torsion points of Ff such that f(α_m+1) =α_m. The induced isomorphism Gal(K_T^ram/K)∼=U_K coincides with the one given by Lubin-Tate theory.

Proof. — We calculate the geometric fiber of F −1 over −T +T. Let g = PamT^mbe an element ofUK(K). The equationF(g)/g=−T +Tis equiv- alent to the system of equations f(a0) = 0,f(am+1) =am, m≥0. Thus, for eachm≥0,a_mis a generator of the module ofT^m+1-torsion points ofF_f. This

o

proves the existence of the above cartesian diagram. Next we calculate the ac- tion ofGal(K_T^ram/K)on the fiber ofF−1over−T+T. The Lubin-Tate group F_f forf(X) =T X+X^qis the formal completionG“_aof the additive group with formal multiplication of each elementPbmT^mof

O

K being given by the power seriesPb_mf^◦m(X)∈EndG“_a, where f^◦mis them-th iteration off. Hence, if σ corresponds to u(T) = PbmT^m via the isomorphism Gal(K_T^ram/K) ∼=UK

of Lubin-Tate theory, we have σ

Ñ X

m≥0

αm+1T^m é

= X

m≥0

σ(αm+1)T^m= X

0≤k≤m<∞

bkαm+1−kT^m

=u(T)X

m≥0

α_m+1T^m.

Thus the action ofσon the fiber ofF−1over−T+Tis given by multiplication byu(T), as required.

Therefore the mapη: Gal(K^ab/K)→π₁^k(K^×) factors through the restric- tion map Gal(K^ab/K) Gal(K_T^ramK^ur/K), the isomorphism of Lubin-Tate theory Gal(K_T^ramK^ur/K)→^∼ (K^×)^∧ (the profinite completion ofK^×) and the natural isomorphism (K^×)^∧ ∼= π₁^k(K^×). Thus the assertion that η is an iso- morphism is equivalent to the local Kronecker-Weber theorem for Lubin-Tate extensions:K^ab=K_T^ramK^ur. Since the isomorphismθof Serre and Hazewinkel for finite k coincides with the one given by Lubin-Tate theory, the equality η⁻¹=θ for suchkfollows.

Remark 4.2. — This proof also shows that the proof for η being an isomor- phism given in Section 3.1 gives another proof of the local Kronecker-Weber theorem for Lubin-Tate extensions in the equal characteristic case.

5. Third proof of TheoremA: geometry

In this section, we give a geometric proof of Theorem A. This method was suggested by the referee to the author. As in Section3, we assume the residue field kis algebraically closed and fix a prime elementT of

O

K. Both the mor- phism ϕ: SpecKp →K^×, which corresponds to the rational point1−TT⁻¹, and the morphismϕ⁰: SpecKp→UK, which corresponds to the rational point

−T+T, give the same homomorphismη: Gal(K^ab/K)→π^k₁(K^×) =π₁^k(U_K).

We use the same letter to express morphisms and corresponding rational points.

It is enough to study the part η^∨: Ext¹_k(UK, N)→H¹(K, N) for each finite constant groupN.

For the proof of Assertion1of TheoremA, we need the following result due to Contou-Carrere ([6]). LetAbe a commutative quasi-algebraic group overk.

Then the homomorphism

Hom(UK, A)→A(Kp)/A((

O

K)p)

induced by the morphismϕ⁰: SpecK_p →U_K is an isomorphism ([6, Théorèm (1.5)]). If A is a commutative proalgebraic group over k, the homomorphism Hom(UK, A) →A(Kp)/A((

O

K)p) induced by the morphism ϕ⁰: SpecKp → U_K is still an isomorphism.

First we show that η^∨: Ext¹_k(UK, N) → H¹(K, N) is surjective, that is, every finite abelian extension L of K with Galois group N is obtained by the pullback of an extension 0 → N → A → U_K → 0 by the morphism ϕ⁰: SpecK→UK. Sincekis algebraically closed, the abelian extensionL/K is a composite of a Kummer extension of degree prime topand Artin-Schreier- Witt extensions. A Kummer extension ofK of degreen,p-n, is the pullback of G_m →ⁿ G_m by some morphism SpecK_p → G_m. An Artin-Schreier-Witt extension of K of degree pⁿ is the pullback of Wn

→℘ Wn by some morphism SpecKp → Wn, where Wn is the additive group of Witt vectors of lengthn,

℘(x) =F x−xandF is the Frobenius. Combining them, we can find an exact sequence 0→N →A⁰ →^α A⁰⁰→0 of commutative quasi-algebraic groups and a cartesian diagram

SpecLp −−−−→

β A⁰





y ^α



 y SpecKp

−−−−→γ A⁰⁰.

Since ϕ⁰: SpecKp → UK induces an isomorphism Hom(UK, A⁰⁰) →^∼ A⁰⁰(Kp)/A⁰⁰((

O

K)p) by the result of Contou-Carrere quoted above, we have γ +ε = ϕ⁰ ◦ δ for some δ: U_K → A⁰⁰ and ε ∈ A⁰⁰((

O

K)_p). Since α: A⁰ → A⁰⁰ is surjective and k is algebraically closed, Lemma 2.4 shows that α:A⁰((

O

K)p) →A⁰⁰((

O

K)p) is also surjective. Hence ε= α(ζ)for some ζ ∈ A⁰((

O

K)_p). The above diagram is still commutative and cartesian if we addζtoβandεtoγ. The resulting cartesian diagram splits into the following commutative diagram with cartesian squares:

SpecL_p −−−−→ A −−−−→ A⁰



 y





y ^α



 y

SpecK_p ^ϕ

0

−−−−→ U_K −−−−→^γ A⁰⁰

Thus we get the surjectivity ofη^∨: Ext¹_k(U_K, N)→H¹(K, N).

o

Next we show that η^∨ is injective. Let 0 → N → A →^α UK → 0 be an extension whose image in H¹(K, N)byη^∨ is trivial, i.e.α◦β =ϕ⁰ for some β ∈A(K_p). The diagram

Hom(UK, A) A(Kp)/A((

O

K)p)





y^{α α}



 y

Hom(UK,UK) UK(Kp)/UK((

O

K)p)

is commutative. Let γ ∈ Hom(UK, A) correspond β ∈ A(Kp) via the top horizontal map. Then we have α◦γ= idby the equality α◦β =ϕ⁰ and this diagram. Hence the extension 0→N →A→^α UK →0 is trivial. Thus we get the injectivity of η^∨: Ext¹_k(U_K, N)→H¹(K, N).

Finally we show that η⁻¹ = θ. For this, we need generalized Jacobians ([14]). We denote byP¹ the projective line overkwith coordinateT. For each effective divisor m on P¹ with support Supp(m) ⊂P¹, let J_P1,m be the mod- uli of line bundles on X with level m structure, which is isomorphic to the direct product of Z and the generalized Jacobian of P¹ with modulus m. Let Φm:P¹−Supp(m) → J_P1,m be the Abel-Jacobi map times −1. Since P¹ has trivial Jacobian, J_P1,m is an affine algebraic group times Z. The groupsJ_P1,m, the schemesP¹−Supp(m)and the morphismsΦ_m:P¹−Supp(m)→J_P1,mform projective systems with respect to m. LetJ_P1 (resp.Φ : Speck(T)→ J_P1) be the projective limit of theJ_P1,m(resp. theΦm), the limit being taken over all ef- fective divisorsmonP¹. They are naturally regarded as sheaves on(Perf/k)_fpqc and morphisms of sheaves on(Perf/k)fpqc. The inclusionk(T)⊂k((T)) =K identifiesKas the fraction field of the completion of the local ring ofP¹atT = 0. This identification gives a natural morphismSpecK_p → Speck(T)_p and a natural homomorphismK^× →J_P1. Consider the following (non-commutative) diagram:

SpecKp −−−−→

ϕ K^×



 y



 y

Speck(T)_p −−−−→^Φ J_P1.

This diagram gives two morphismsSpecK_p →J_P1, or equivalently, twoK_p-ra- tional points ofJ_P1. Under the natural identificationJ_P1 = (Q

a∈P¹(k)−{0}Ua× K^×)/Gm with Ua defined by Ua(R) = R[[T⁻¹ −a⁻¹]]^×, the difference of these two rational points is given by ((1−TT⁻¹)_a∈P¹_(k)−{0},1), which is in J_P1((p_K)_p). Hence this diagram induces, on fundamental groups, a

commutative diagram

Gal(K^ab/K) −−−−→

η π^k₁(K^×)



 y



 y Gal(k(T)^ab/k(T)) −−−−→ π^k₁(J(P¹))

by the same argument as in the proof of Proposition 2.3. The top horizontal mapηis an isomorphism by Assertion1of TheoremA. The bottom horizontal map is an isomorphism as well by [14]. Regarding this diagram, Serre proved the global version of the equality η⁻¹ = θ and the local-global compatibility ([13, §5.2, Prop. 2 and §5.3, Lem. 2]; actually Serre’s Proposition 2, which says ψ⁻¹ =θin his notation, should be corrected as −ψ⁻¹ =θ). This implies the local equalityη⁻¹=θ.

6. A two-dimensional analog

In this section, we discuss an analog of the above theory for a field of the form K=k((S))((T)). We denote byK₂the functor of the second algebraicK-group ([1]). For each perfect k-algebra R, we have the abelian group K2(R[[S,T]]).

This gives a group functor that we denote byK2[[S,T]]. TheKp-rational point {−S+S,−T+T} ∈K2[[S,T]](Kp) =K2 k((S))((T))p[[S,T]]

gives a morphism ϕ⁰: SpecKp → K2[[S,T]], where {·,·}denotes the symbol map. This is an analog of the mapSpeck((T))_p→U_k((T))we have defined and studied before this section.

Now assume kis the finite field Fq withq elements. Define a functorX by the following cartesian diagram:

X −−−−→ K2[[S,T]]



 y



 y^F−1

SpecKp ϕ⁰

−−−−→ K2[[S,T]],

where F is theq-th power Frobenius morphism overk. Then we expect that K2[[S,T]] can be viewed as a sort of an “algebraic group over k” and the equation (F −1)x = {−S +S,−T +T} gives a two-dimensional analog of Lubin-Tate theory so thatX is theSpecof the perfect closure of a large totally ramified abelian extension ofK (compare with Proposition4.1).

Although this hope is hard to be made precise, we can formulate and prove a rigorous partial result (Proposition6.1below). For each perfectk-algebraR,

o