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Graduate Texts in Mathematics 67

Editorial Board S. Axler F.W. Gehring K.A. Ribet

Springer Science+Business Media, LLC

Jean-Pierre Serre

Local Fields

Translated from the French by

Marvin Jay Greenberg

Springer

Jean-Pierre Serre CoHege de France 3 rue d'Ulm

Marvin Jay Greenberg University of California at Santa Cruz Mathematics Department 75005 Paris, France Santa Cruz, CA 95064

Editorial Board S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA

F. W. Gehring Mathematics Department East HaU University of Michigan Ann Arbor, MI 48109 USA

K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720- USA

Mathematics Subject Classifications (1991): llR37, llR34, 12G05, 20106

With 1 Figure

Library of Congress Cataloging-in-Publication Data

Serre, Jean-Pierre. Local fields.

(Graduate texts in mathematics; 67) Translation of Corps Locaux. Bibliography: p. lncludes index.

1. Class field theory. 2. Homology theory. I. Title. II. Series. QA247.S4613 512'.74 79-

L' edition originale a ete publiee en France sous le titre Corps locaux par HERMANN, editeurs des sciences et des arts, Paris.

AH rights reserved.

No part of this book may be translated or reproduced in any form without written permission from Springer Science+Business Media, LLC.

© 1979 by Springer Science+Business Media New York Originally published by Springer-Verlag New York Berlin Heidelberg in 1979 Softcover reprint of the hardcover 1st edition 1979

9 8 765 4 3

ISBN 978-1-4757-5675-3 ISBN 978-1-4757-5673-9 (eBook) DOI 10.1007/978-1-4757-5673-9 SPIN 10761519

Vl

Part Two RAMIFICATION

Chapter III

Discriminant and Different

§1. Lattices §2. Discriminant of a Lattice with Respect to a BilinearForm §3. Discriminant and Differentofa Separable Extension §4. Elementary Properties of the Different and Discriminant §5. Unramified Extensions §6. Computation of Different and Discriminant §7. A Differential Characterisationofthe Different

Chapter IV

Ramification Groups

§1.Definition of the Ramification Groups; First Properties §2.The Quotients GjGi+ ~> i 2 0 §3.The Functions^ rjJ^ and^ ljJ;^ Herbrand's Theorem §4. Example: Cyclotomic Extensionsofthe Field QP

Chapter^ V

The Norm

§1. Lemmas §2.The U nramified Case §3.The Cyclic^ of^ Prime^ Order^ Totally Ramified Case §4.Extension of the Residue Field in a Totally Ramified Extension §5. Multiplicative Polynomials and Additive Polynomials §6. The Galois Totally Ramified Case §7. Application: Proof ofthe Hasse-Arf Theorem

Chapter VI

Artin Representation

§1.Representations and Characters §2.Artin Representation §3.Globalisation §4.Artin Representation and Homology (for Algebraic Curves)

Part Three GROUP COHOMOLOGY

Chapter VII

Basic Facts

§1.G-Modules §2.^ Cohomology §3. Computing the Cohomology via Cochains

Contents

48 50 51 53 55 59

61 65 73 77

81 83 87 90 91 93

97 99 103 105

109 Ill 112

Contents Vll

Introduction

The goal of this book is to present local class field theory from the cohomo- logical point of view, following the method inaugurated by Hochschild and developed by Artin-Tate. This theoryis about extensions-primarily abelian-of "local" (i.e., complete for a discrete valuation) fields with finite residue field.Forexample, such fields are obtained by completing an algebraic number field; thatisone of the aspects of "localisation". The chapters are grouped in "parts". There are three preliminary parts: the first two on the general theory of local fields, the third on group coho- mology. Local class field theory, strictly speaking, does not appear until the fourth part. Here is a more precise outline of the contents of these four parts: The first contains basic definitions and results on discrete valuation rings, Dedekind domains (which are their "globalisation") and the completion process. The prerequisite for this part is a knowledge of elementary notions of algebra and topology, which may be found for instance in Bourbaki. The second part is concerned with ramification phenomena (different, discriminant, ramification groups, Artin representation). Just as in the first part, no assumptions are made here about the residue fields. It is in this setting that the "norm" map is (^) studied; I have expressed the results in terms of "additive polynomials" and of "multiplicative polynomials", (^) since using the language of algebraic geometry would have led me too far astray. The third part (group cohomology) is more of asummary-andan incom- plete one atthat-thana systematic presentation, which would have filled an entire volume by itself. In the two first chapters, I do not give complete proofs, but refer the reader to the work ofCartan-Eilenberg [13] as well as to Grothendieck's 'Tohoku" [26]. The next two chapters (theorem of Tate- Nakayama, Galois cohomology) are developed specifically for arithmetic

2 Introduction

applications, and there the proofs are essentially complete. The last chapter

(class formations) is drawn with little change from the Artin-Tate seminar

[8]-a seminar which I have also used in many other places.

The last part (local class field theory) is devoted to the case of a finite or,

more generally, quasi-finite residue field; it combines the results of the three

first parts. (The logical relations among the different chapters are made more

precise in the Leitfaden below.) Besides standard results, this part includes a

theorem of Dwork [21] as well as several computations of "local symbols".

This book would not have been written without the assistance of Michel

Demazure, who drafted a first version with me in the form of lecture notes

("Homologie des groupes-Applications arithmetiques", College de France,

1958-1959). I thank him most heartily.

Leitfaden

I

j

2

j

3 7

I

10 8 4

j l

9

5 l

6/ ~12 II

13

14 15

CHAPTER I

Discrete Valuation Rings

and Dedekind Domains

§1. DefinitionofDiscrete Valuation Ring

A ring Aiscalled a discrete valuation ring^ if it is a principal ideal domain (Bourbaki,^ Alg.,^ Chap. VII)^ that^ has a unique non-zero prime ideal m(A). [Recall that an ideal:pofa commutative ring Aiscalled^ prime^ ifthe^ quotient ringA/:pisan integral domain.] The field A/m(A) is called the residue field ofA.The invertible elements of A are those elements that do not^ belong to m(A); they form a multiplicative group and are^ often called the^ units^ of^ A (or^ of^ the field^ of^ fractions^ ofA). In a principal ideal domain, the non-zero prime ideals are the ideals^ of^ the form nA, where n is an irreducible element. The definition above comes down to saying^ that^ A has one^ and^ only one irreducible element,^ up^ to^ multiplica- tion by an invertible element; such^ an^ element is called a^ uniformizing element of A (or uniformizer;^ Weil [123] calls it a^ "prime^ element"). The non-zero ideals of A are of the form m(A)^ =^ nnA,^ where^ n^ is a uniform- izing element. If^ x^ =/:.^0 is^ any element^ ofA,one can write^ x^ =^ nnu,^ with^ n^ E^ N and u invertible; the integer n iscalled the valuation^ (or the^ order)^ of^ x^ and is denoted v(x); it does not depend^ on^ the choice^ of^ n. Let K be the field^ of^ fractions^ ofA,^ K^ *^ the multiplicative^ group^ of^ non-zero

elements of K. If x = ajb is any element^ of^ K^ *,^ one can again write^ x^ in the

form nnu, with n E Z this time, and set v(x) = n. The following properties are easily verified:

a) The map v: K*-+^ Z^ is a surjective homomorphism.

b) One has v(x + y)^ ~^ Inf(v(x),^ v(y)^ ).

(We make the convention^ that^ v(O)^ =^ +^ oo.)

5

(^6) I Discrete Valuation Rings and Dedekind Domains

The knowledge of the function v determines the ring A: it is the setof those x E K such that v(x) ~ 0; similarly, m(A) is theset ofthose x E K such

that v(x) > 0. One could therefore have begun with v. More precisely:

Proposition 1. Let K be a field, and let v: K * ~ Z (^) be a homomorphism having properties a) and b) above. Then the set A ofx E K such that v(x) ~ 0 isa discrete valuation ring having v as its associated valuation.

Indeed, let n be an element suchthat v(n) = 1.Every x E A can be written in the form x = n"u, with n = v(x), and v(u) = 0, i.e., u invertible. Every non- zero ideal of Aisthereforeofthe form nnA, with n ~ 0, which showsthatA isindeed a discrete valuation ring. 0

EXAMPLES OF DISCRETE VALUATION RINGS 1)Let p be a prime number,andlet Z(p) be the subsetofthe field Q of rationals consistingofthe fractions r/s, wheresisnotdivisible by p; this is a discrete valuation ring with residue field the field F P of p elements. If vP denotes the associated valuation, vp(x) isnone other than the exponentof p in the decomposition of x into prime factors. An analogous procedure applies to any principal ideal domain (and even to any Dedekind domain,cf.§3). 2)Let k be a field, and let k( (T)) be the fieldof formal power series in one variable over k. Forevery non-zero formal series

n ;,no one defines the orderv(f)off to be the integer n (^0) (cf.Bourbaki, Alg., Chap. IV). One obtains thereby a discrete valuation of k( (T)),whose valuation ring is k[[T]], the setofformal series with non-negative exponents; its residue field is k. 3)Let V be a normal algebraic variety, of dimension n, and let W be an irreducible subvariety ofV,of dimension n - 1.Let Av 1 w be the local ringof V along W (i.e., the set of rational functions f on V which are defined at least at one point of W). The normality hypothesis shows that Av 1 w isinte- grally closed; the dimension hypothesis shows that it is a one-dimensional local ring; therefore it is a discrete valuation ring(cf. §2,prop. 3); its residue field is the field of rational functions on W. If vw denotes the associated valuation, and iff is a rational function onV, the integer vw(f) is called the "order" off along W; it is the multiplicity of W in the divisor of zeros and poles of f.

4. Let S be a Riemann surface (i.e., a one-dimensional complex manifold), and let PES. The ring ~P of functions holomorphic in a neighborhood (unspecified) of P is a discrete valuation ring, isomorphic to the subring of convergent power series in C[[T]]; its residue field is C.

8 I Discrete Valuation Rings^ and^ Dedekind Domains

[Recall that an element x of a ring containing Aiscalled^ integral^ over A

if it satisfies^ an^ equation^ "of^ integral^ dependence":

a; EA.

One says that Ais integrally closed in a ring B containing it if every element

of B integral over A belongs toA.^ One^ says^ that^ Ais^ integrally closed^ if it

isanintegral domain integrally closed in its fieldoffractions.Cf.Bourbaki,

Alg. comm., Chap.V,§1.]

It isclear that a discrete valuation ring satisfies (ii). Let us show that^ it

satisfies (i). Let K be the field of fractionsofA,and let x be an elementofK

satisfying an equation of type (*),and suppose x were not inA.That means

v(x) = -m, with m > 0.^ In equation(*), the first term has valuation^ -nm,

while the valuation^ of^ the othersis^ 2:::^ -(n-^ 1)m,whichis >^ -nm;^ thatis

a contradiction, according to the following lemma:

Lemma 1.^ Let^ A^ be a discrete valuation ring, and let^ X;^ be elementsofthe

fieldoffractions of A such that v(x;) > v(x1) for i 2::: 2. One then has

x 1 + x 2 + · · · + Xn i= 0.

One can assume x^1 =^ 1 (dividing by x^1 if necessary), whence^ v(x;)^ 2:::^1

for i 2::: 2,i.e., X; E m(A); as x 1 ¢; m(A}, it follows that x 1 + · · · + xn ¢; m(A),

which proves the lemma.

[This proof also shows that^ x^1 +^ ·^ ·^ ·^ +^ xn^ has the same valuation as x^1 .]

Let us now show that a Noetherian integral domain satisfying (i) and^ (ii)

isa discrete valuation ring. Condition (ii) shows that^ Aisa local ring whose

maximal ideal m is =1= 0. Let m' be the set of x E K such that xm c: A (i.e.,

xy E A for every y Em); it is a sub-A-module of K containingA.^ If^ y^ is^ a

nonzero element of m, it is clear that m' c: y-^1 A, and as A is Noetherian,

this shows that m' isa finitely generated A-module (thatiswhat one calls

a "fractional ideal" of K with respect toA).Let m. m' be the product of m

and m', i.e., the set of all LX;Y;,^ X;^ Em,^ Y;^ Em';by definition^ of^ m', one has

m. m' c: A; on the other hand, since A c: m', one has m. m'^ ::::>^ m; since

m. m' is an ideal, one has either m. m' = m or m.m'=A.We will successively

show:

I. If m.m'= A, the ideal m is principal. II. If m. m' = m, and if (i) is satisfied, then m' =^ A.

III. If (ii) is satisfied, then m' i= A.

By combining II and III, one sees that m. m' = mis impossible, whence,

by I, m must be principal, therefore A is a discrete valuation ring (prop.2).

It remains to prove assertions I, II, III.

PRooF OF I.If m. m'=A,one has a relation^ LX;Y;^ = 1,with^ X;^ Em,^ Y;^ Em'.

The products X;Y; all belong to A; at least one of them-say^ xy-does^ not

§3. Dedekind Domains 9

belong to m, therefore is an invertible element u. Replacing x by xu-^1 , one

obtains a relation xy = 1, with x Em and y Em'. If z Em, one has z =

x(yz), with yz E A since y Em'; therefore z is a multiple of x, which shows

that m is indeed a principal ideal, generated by x.

PRooF OF II. Suppose m. m' = m, and let x Em'. Then xm c m, whence, by

iteration, xnm c m for all n, i.e., xn E m'. Let an be the sub-A-module of K

generated by the powers {1, x, .. .. , x"} of x; one has an can+ t> and all the

an are contained in the finitely generated A-module m'. Since A is Noetherian,

one gets an- 1 = an for n large, i.e., xn E an- 1• One can then write xn = bo +

b 1 x + · · · + bn- 1 xn- 1 , b; E A, which shows that x is integral over A. Con-

dition (i) then implies x E A, hence m' = A.

PRooF OF III. Let x be a non-zero element of m, and form the ring Ax of

fractions of the type yjxn, with y E A, and n ~ 0 arbitrary. Condition (ii)

implies Ax = K: indeed, if not, Ax would not be a field, and would contain

a non-zero maximal ideal p; as x is invertible in Ax, one would have x ¢: p,

which shows that p n A =F m. On the other hand, if yjxn is a non-zero

element of p, one has y E p n A, so that p n A =F 0. But since p is prime,

so is p n A, which contradicts (ii).

Thus every element of K can be written in the form yjxn; let us apply

this to 1/z, with z =F 0 in A. We get 1/z = y/xn, whence xn = yz E zA. There-

fore every element of m has a power belonging to the ideal zA. Let x t> ... , xk

generate m, and let n be large enough so that x? E zA for all i; if one chooses

N > k(n - 1), all the monomials in the x; of total degree N contain an x?

as factor, therefore belong to zA; as the ideal mN is generated by these mono-

mials, one has mN c zA. Apply this with z Em: one concludes that there is

a smallest integer N ~ 1 such that mN c zA; choose y E mN- 1 , y ¢ zA

(putting m 0 =A by convention). One then has my c zA, whence yjz Em',

and yjz ¢A, which indeed proves that m' =FA. D

Remark. The construction of m' does not use the hypotheses made on A

and m; for every non-zero ideal a of an integral domain A, one can define

a' as the set of x E K such that xa c A; if A is Noetherian, this is a fractional

ideal. When aa' = A, one says that a is invertible. The proof of I shows that

every invertible ideal of a local ring is principal.

§3. Dedekind Domains

Reminder. Let A be an integral domain, K its field of fractions, and let S

be a subset of A that is multiplicatively stable and contains 1 (such a set

will be called multiplicative); suppose also that 0 does not belong to S. The

§3. Dedekind Domains 11

If V is an affine algebraic variety, defined over an algebraically closed field k, the coordinate ring k[VJ of V is a Dedekind domain if and only if V is non-singular, irreducible and of dimension .:::;; 1.

Proposition 5. In a Dedekind domain, every non-zero fractional ideal is invertible. [If K is the field of fractions of A, a fractional ideal a of A is a sub-A- module of K finitely generated over A. One says a is invertible if there exists a' c K with a. a'= A.]

In a discrete valuation ring, a fractional ideal has the form n"A, where

n E Z, and is therefore invertible. The proposition follows from this by

localisation, taking into account that:

if b is finitely generated. D

[(a: b) denotes the ideal of those x E K such that xb ca. If a' =(A: a), to say that a is invertible amounts to saying that a. a' = A.]

Corollary. The non-zero fractional ideals of a Dedekind domain form a group under multiplication.

This group is called the ideal group of the ring.

Proposition 6. If x E A, x =I= 0, then only finitely many prime ideals contain x.

Indeed, the ideals containing x satisfy the descending chain condition: if Ax c a c a' c A, one has Ax - 1 :::J a-^1 :::> a'-^1 => A, and A is Noetherian. It follows that if x E P1> p (^) 2 , ... , Pb ... , the sequence P1 => P1 n Pz => · · · => P1 n Pz n · · · n Pk => · · · is stationary, which means that from some point onward, one has

Pi=>P1 npz···npk=>P1P2···pk which, as the pi are prime, shows that Pi is one of the p (^) 1, ... , Pk· D

Corollary. If one denotes by vv the valuation of K defined by Av, then for every x E K*, the numbers vv(x) are almost all zero (i.e., zero except for a finite number).

Now let a be an arbitrary fractional ideal of A; it is contained in only

finitely many prime ideals p. The image av of a in Av has the form av = (pAv)"v(al, where the vv(a) are rational integers, almost all zero.

12 I Discrete Valuation Rings and Dedekind Domains

If One COnsiders the ideal al = nil pvv(a) and the ideal a2 Of those X SUCh that vll(x) ~ vll(a) for all p, the three ideals a, ab and a2 are equal locally (i.e., have the same images in all the All). An elementary argument shows that they must then be equal, whence:

Proposition 7. Every fractional ideal a of A can be written uniquely in the form:

where the vll(a) are integers almost all zero.

The following formulas are immediate:

Furthermore:

vll(a. b)= vll(a) + v"(b) vll((b:a)) = vll(b.a- 1 ) = vll(b)- vll(a) vll(a +b)= Inf(vll(a), vll(b)) vll(xA) = vll(x).

Approximation Lemma. Let k be a positive integer. For every i, 1 :::;; i :::;; k, let

Pi be distinct prime ideals of A, xi elements of K, and ni integers. Then there

exists an x E K such that vlli(x - xJ ~ ni for all i, and vq(x) ~ 0 for q =I=

P1> · · ·, Pk·

Suppose first that the xi belong to A, and let us seek a solution x belonging

to A. By linearity, one may assume that x 2 = · · · = xk = 0. Increasing the ni

if necessary, one may also assume ni ~ 0. Put

a = p~~ + p~z ... PZk.

One has vll(a) = 0 for all p, whence a =A. It follows that

and the element x has the desired properties. In the general case, one writes xi = a)s, with ai E A, sEA, s =1= 0, and x = ajs. The element a must fulfill the conditions:

vlli(a - aJ ~ ni + vlli(s), 1 :::;; i :::;; k, vq(a) ~ vq(s) for q =I= P1, ... , Pk·

These conditions are of the type envisaged above (if one adds to the family

{Pi} the prime ideals q for which vq(s) > 0); the existence of a then follows

from the previous case. 0

Corollary. A Dedekind domain with only finitely many prime ideals is principal.

It sufficies to show that all its prime ideals are principal. Now if p is one