Class Groups and Picard Groups of Group Rings and Orders

board conference ofthemathematical sciences inmathematics regional conference series

number 26

IRUIiIG REII{ER ANt|PICARD CLASS GR(IUPS GR(IUPS (lFGR(IUP RINGS ANII(IRIIERS

supported bythenational foundation science published bytheamerican mathematical society

Conference Board of the Mathematical Sciences

REGIONAL CONFERENCE SERIES IN MATHEMATICS

supported by the National Science Foundation

Number 26

CLASS GROUPS AND PICARD GROUPS OF GROUP RINGS AND ORDERS

by

IRVING REINER

Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society Providence, Rhode Island

Expository Lectures from the CBMS Regional Conference held at Carleton College August 11-15, 1975

AMS (MOS) 1970 subject classifications. Primary 16-02, 16A18, 16A54, 20ClO; Secondary 13D15, 18-02, 20-02

Library of Congress Cataloging in Publication Data

Reiner, Irving. Class groups and Picard groups orders.

o~

group rings and

(Regional conference series in mathematics ; no. 26) ffEx"pository lectures ~rom the CBMS regional conference held at Carleton College, August ll-15, 1975.!l Includes bibliographical references and index. 1. Class groups (Mathematics) 2. Picard groups. 3. Fields, Algebraic. 4. Ideals (Algebra) 5. Group rings. I. Conference Board of the Mathematical Sciences. rI. Title. HI. Series. QAl.R33 no. 26 [QA247] 510'.8s [512'.2] 76-10337 ISBN 0-8218-1676-4

Copyright © J 976 by the American Mathematical Society Printed in the United States of America

All rights reserved except those granted to the United States Government. This book may not be reproduced in any form without permission of the publishers.

CONTENTS 1. Introduction

1

2. Explicit formulas

6

3. Change of orders

13

4. Class groups of p-groups

18

5. Mayer-Vietoris sequences

20

6. Calculations

24

7. Survey of specific results

28

8. Induction techniques

31

9. Picard groups

35

References

41

Index

44

Other Monographs in this Series

No. l. Irving Kaplansky: Algebraic and analytic aspects of operator algebras

2. Gilbert Baumslag: Lecture notes on nilpotent groups 3. Lawrence Markus: Lectures in differentiable dynamics

4. H.S.M. Coxeter: TU/isled honeycombs 5. George W. Whitehead: Recent advances in homotopy theory

6. WaIter Rudin: Lectures on the edge-of-the-wedge theorem 7.

Y0Z0

Matsushima: Holomorphic vector fields on compact Klihlcr manifolds

8. Peter Hiltan: Lectures in homological algebra 9. I. N. Herstein: Notes from a ring theory conference

10. Branko Grunbaum: Arrangements and spreads 11. Irving Glicksberg: Recent results on function algebras 12. Barbara L. Osofsky: Homological dimensions of modules

13. Michael Rabin: Automata on infinite objects and Church's problem 14. Sigurdur Helgason: Analysis on Lie groups and homogeneous spaces 15. R. G. Douglas: Banach algebra techniques in the theory of Toeplitz operators 16. ]oseph L. Taylor: Measure algebras 17. Louis Nirenberg: Lectures on linear partial differential equations 18. Avner Friedman: Differential games 19. B~la Sz.-Nagy: Unitary dilations of Hi/bert space operators and related topics

20. 2l. 22. 23. 24.

Hyman Bass: Introduction to some methods of algebraic K-theory Wilhelm Stall: Holomorphic functions of finite order in several complex variables O. T. O'Meara: Lectures on linear groups Mary Ellen Rudin: Lectures on set theoretic topology Melvin Hochster: Topics in the homological theory o( modules over commutative rings

25. Karl. W. Gruenberg: Relation modules o( (inite groups

1. Introduction The aim of these lectures is to provide an introduction to recent developments in the theory of class groups and Picard groups. By way of orientation, let us indicate where this theory fits into the general framework of algebra. It is first of all a branch of number theory, since we will be generalizing, to the noncommutative case, the concept of the ideal class group of an algebraic number field. The techniques employed come from three main areas: algebraic number theory, representation theory of algebras and orders, and algebraic K-theory. As is common with many topics in number theory, part of the appeal of this subject is aesthetic. On the other hand, it has applications to other branches of mathematics. For example, associated with a topological space with fundamental group n, there is a SwanWall invariant which measures whether the space has the same homotopy type as a finite complex. This invariant takes values in the reduced projective class group Ko(Zn). There are similar invariants which measure obstructions to the problem of fitting a boundary onto an open manifold. For references, see Milnor [23, page x]. For an algebraic example, consider a finite galois extension E of Q, with group G, and suppose that E/Q is tamely ramified. Then the ring alg. int. {E} is a projective ZG-module, and its class in Ko(ZG) measures the obstruction to whether E has a normal integral basis over Q. As main references for these lectures we cite: Bass [I], CR (Curtis and Reiner [3]), Milnor [23], MO (Reiner [26]). Throughout, R denotes a Dedekind domain with quotient field F of characteristic O. For example, if F is an algebraic number field (= finite extension of the rational field Q), then the ring alg. int. {F} of all algebraic integers in F is a Dedekind domain with quotient field F (See CR, §18). Any nonzero prime ideal P of R gives rise to a P-adic valuation of the field F, in which an element a E F is "small" if P occurs to a high power in the factorization of Ra into powers of prime ideals. By a prime of F we mean an equivalence class of valuations of F The class of the above P-adic valuation will also be denoted by P. However, F may have other valuations as well, not arising from prime ideals of R; in particular, there are the "infinite primes" of F, corresponding to archimedean valuations of F. For each prime P of F, infinite or not, we may use the P-adic valuation to make F into a metric space, and then we may form its P-adic completion Fp relative to this metric. For P an infinite prime, the completion Fp is either the real field R or the complex field C, and the prime P is accordingly called real or complex. On the other hand, for a nonarchiCopyright © 1976, Amt'ric.1n Matht'il\;.'ILal S'J' Il'!~

2

IRVING REINER

medean or finite prime P of an algebraic number field F, the P-adic field Fp is a finite extension of the field Qp' where p is the unique rational prime in P. If M is any R-module, we may form its P-adic completion Mp = Rp OR M. Call M an R-lattice if M is finitely generated and torsion free as R-module; in this case, Mp is an Rp-lat-

tice for each P. On the other hand, if M is a finitely generated P-torsion R-module, then M ~ Mp. In general, properties of modules can be deduced from those of their completions,

as we shall see later. We recall the definition of the ideal class group Cl R of the Dedekind domain R. By an R-ideal in F (or fractional ideal) we mean an R-lattice X C F such that FX = F. Two such ideals X,

x'

are in the same ideal class if X ~ X' as R-modules, or equivalently, if X' =

Xa for some a E F' (where F" = F - {On. Let [X] denote the ideal class of X; the ideal

classes form a group Cl R, relative to the multiplication given by [X] [X'] = [XX'], where

XX' is the set of all finite sums ~XiX;'

Xi

E X,

x; E X'.

When F is an algebraic number

field, the group Cl R is finite; this is a special case of the Jordan-Zassenhaus Theorem(see (1.7) below). Our main purpose here is to study various generalizations of the ideal class group Cl R. In the noncommutative case, it turns out that there are two natural generalizations. The first of these considers certain types of left ideals, and yields the theory of class groups of orders. The second deals with certain types of two-sided ideals, and gives rise to Picard groups. For the commutative case, these two theories will coincide (but it requires a proof that they do so !). Most of these lectures will be devoted to class groups, saving Picard groups for the end. Now let A be a finite dimensional semisimple F-algebra. We are especially interested in the case where A

= FC,

the group algebra of a finite group C over the field F. By

Maschke's Theorem (CR, §1O) this group algebra is necessarily semisimple. Left FC-modules are just vector spaces over F, on which the elements of C act from the left as F-linear operators. Returning to the general case, by an R-order in A we mean a subring A of A such that (i) R is contained in the center of A, (ii) A is finitely generated as R-module, and (iii) FA = A, that is, A contains an F-basis of A.

(1.1) Examples. (1) The integral group ring RC is an R-order in the algebra FC. (2) The ring alg. into {F} is a Z-order in the algebraic number field F. (3) For any algebraic integer a, the ring Z [a] is a Z-order in the field Q(a). (4) Let Mn(F) denote the ring of all n

x

n matrices over F. Then the ring Mn(R) is

an R-order in Mn(F). Hereafter, let A be an R-order in the semisimple F-algebra A, where char F = O. We wish to define the class group Cl A. By a left A-lattice M we mean a left A-module which is an R-Iattice. Each such M is embedded in the A-module F OR M via m -+ I

°

° m, m EM.

We always identify M with its image 1 M in F OR M; then F OR M may be written as FM, the set of F-linear combinations of the elements of M.

3

CLASS GROUPS AND PICARD GROUPS

A left A-ideal in A means a left A-lattice M C A such that FM == A. Two such ideals M, N are in the same ideal class if M ~ N as A-modules, or equivalently, if M == Nx for some x E u(A). Unfortunately, these ideal classes do not form a group relative to the product [M] [N] == [MN]. To overcome this difficulty, we must restrict the type of ideals considered. Two left A-modules M. N are in the same genus (notation: M V N) if Mp ~ N p as left Ap-modules for each prime ideal P of R. A left A-ideal M in A is locally free if it is in the same genus as A, that is, Mp ~ A p for each P. Let us recall some results on genera (for

proofs, see MO, §27). (1.2) ROITER'S LEMMA. Let M, N be left A-lattices. Then M V N if and only if for each finite set S of prime ideals of R. there exists a short exact sequence of A-modules o ---+ M ---+ N ---+ T ---+ 0 such that Tp == 0 for each PES. (1.3)

PROPOSITION.

Let L, M, N be left A-lattices such that L V (M E!l N). Then

there exist A-lattices X, Y such that

L (1.4)

PROPOSITION.

~

X E!l Y,

YVN.

XVM,

Let L, M, N be A-lattices in the same genus. Then

M E!l N ~ L E!l L' for some L' V L.

Now let M and M' be a pair of locally free A-ideals in A; then M and same genus as A, so by (1.4) we may write (1.5)

M' are in the

M-+-M==A+-M"

for some locally free ideal M'. (We have used the symbol +- to denote external direct sum, in order to avoid any confusion with the concept of internal direct sum of a pair of ideals in a ring.) Let (M) temporarily denote the class of the locally free ideal M. We wish to make these classes into an additive group, by defining (M)

+ (M') ==

(M")

whenever (1.5) holds. It is clear that (A) acts as zero element. To show the existence of inverses, we note that A, A and M are in the same genus, so by (1.4) we have A + A ~ M + N for some locally free ideal N; thus (M) + (N) == (A), so (N) is an additive inverse for (M). There is, however, one difficulty which we have ignored: in order for addition of classes to be well defined, we must know that the isomorphism classes of M and M' uniquely determine the class of M" in (1.5). This is unfortunately not true in general. Swan [39] gave an example in which A +- A ~ A -+- M", with M" not isomorphic to A. In this example, A is the integral group ring ze, where e is the generalized quatemion group of order 32. We shall therefore introduce a new equivalence relation which is slightly weaker than isomorphism. Two left A-modules M, N are called stably isomorphic if M-+- A(k) ~ N -+- A(k)

for some k,

4

IRYING REINER

where A(k) is the external direct sum of k copies of A. Let [M] denote the stable isomor· phism class of M. We have thus shown (1.6) THEOREM. The stable isomorphism classes of locally free ideals form an abelian additive group Cl A, called the locally free class group of A. Addition is given by [M]

+

[M']

=

[M"]

whenever M -+- M' ='" A

-i- M".

The zero element is the class [A].

For brevity we shall call Cl A the class group of the R-order A. If F is an algebraic number field, the group Cl A is finite. This is a consequence of the following basic result (see MO, §26): (1.7) THEOREM. (JORDAN·ZASSENHAUS). Let A be an R-order in the semisimple F·algebra A, where F is an algebraic number field, and let L be any left A-lattice. Then there are only finitely many isomorphism classes of left A-lattices M such that FM ='" FL as left A·modules. In particular, there are only a finite number of isomorphism classes of left A-ideals in A. We shall see later that in the special case where A = R, the locally free class group Cl A is isomorphic to the usual ideal class group Cl R. Two other special cases should be mentioned. First, suppose that A is a maximal R·order in A (that is, an order not contained in any larger R-order in A). Then (see MO, (11.6)) every left A·ideal in A is locally free, so Cl A is the group of stable isomorphism classes of all left A-ideals in A. Second, and more important, suppose that A is the integral group ring RG of a finite group G. One is interested in the projective ideals of A, that is, those left ideals of A which are projective as A·modules. A basic result states (see CR, § 78): (1.8) THEOREM (SWAN). Suppose that no rational prime divisor of IGI is a unit in R. Then every projective ideal of RG is locally free. As we shall see later, this implies that Cl RG ='" Ko(RG) if no prime divisor of IG I is a unit in R. In particular, Cl ZG ='" Ko(ZG) for every finite group G. A left A-lattice X is called locally free of rank n if X V A(n). We claim that X is necessarily A·projective} For consider a A-exact sequence (1.9)

o~u~ V~X~O.

For each prime ideal P of R, there is then a Ap-exact sequence (1.10) But X p is free as Ap.module, so (1.10) splits. This holds for each P, whence also (1.9) splits (see MO, (3.20) and §5). Therefore X is A-projective, as claimed. The study of locally free modules reduces readily to the case of locally free ideals, by virtue of I This also folloWl; from (1.4): since X V A (n), then A (n)

-i- A (n)

~

X

-i-

Y for some Y

V A(n).

5

CLASS GROUPS AND PICARD GROUPS

(1.11)

PROPOSITION.

If X V A(n), then X ~ A(n-l)

+-

M for some locally free left

A-ideal M in A. PROOF.

This follows easily by induction on n, by repeated use of (1.3) and (l.4).

To conclude this section, let us consider the relation between stable isomorphism and isomorphism of locally free A-modules, restricting our attention to the case where F is an

.

algebraic number field. It can be shown that two locally free left A-modules M, N are stably

.

isomorphic if and only if M + A'=" N + A. This follows from Jacobinski [15], [16] (see also Frohlich [8]), by using a theorem of Eichler. It also follows from the Serre-Bass Cancellation Theorem (see Bass [1, Chapter IV, §3] or Swan [40a, Chapter 12]). As a matter of fact, in most cases which arise in practice, stable isomorphism implies isomorphism for locally free A-modules. A sufficient condition for this is that A satisfy the Eichler condition relative to R (notation: A = Eichler/R; see MO, (38.1) for the definition). (1.12) REMARKS. (i) If A is commutative, or if A is a direct sum of matrix algebras over fields, then A = Eichler/R.

(ii) If A is a simple algebra whose center F has at least one complex prime, then A Eichler /R. For the case of group algebras, the following result is useful (see MO, p. 344): (1.13)

=

Let A = FG, where G is a finite group and F is an algebraic Eichler/R if G has no homomorphic image of any of the following

PROPOSITION.

number field. Then A = types: generalized quaternion group of order 4n, n;;;' 2; binary tetrahedral group of order 24; binary octahedral group of order 48; binary icosahedral group of order 120.

2. Explicit Fonnulas We shall use the Localization Sequence of algebraic K-theory to obtain explicit formulas for the class group of an order. This approach, due to Wilson [44], avoids the use of a deep theorem of Eichler. Earlier formulas by Jacobinski [l6] and Frohlich [8] were based on Eichler's Theorem. Let us review some basic facts from K-theory. As general references, we cite Bass [1], Milnor [23], and an excellent expository article by Lam-Siu [20]. Let A denote an arbitrary ring (not necessarily an order), and let C be some category of finitely generated left A-modules. We wish to define the Grothendieck group Ko(C) of the category C. To begin with, let F denote the free abelian group generated by symbols (M), one for each isomorphism class of objects M E C. Let F 0 be the subgroup of F generated by all expressions (M) - (M') - (M"), one such for each short exact sequence, O-M'-M-M"-O

(2.1 )

which belong to the category C. We then define Ko(C) = F IF 0 and denote by [M] the image of (M) in Ko(C)' Of especial interest is the case where we start with the category peA) of all finitely generated projective left A-modules. The corresponding Grothendieck group Ko(P(A)) is called the projective class group of A, and will be denoted simply as Ko(A). Since every exact sequence (2.1) of objects in peA) must split, we sometimes say that Ko(A) is generated by symbols [M], with M E peA), and relations [M] = [M'] + [M"] whenever M ~ M' EB M". It is easily verified that for M, NE peA), we have [M] = [N] in Ko(A) if and only if M is stably isomorphic to N. If the Krull-Schmidt Theorem holds for the ring A (that is, every finitely generated A-module is expressible as a finite direct sum of indecomposable modules, with the summands uniquely determined up to isomorphism and order of occurrence), then stable isomorphism implies isomorphism. Further, suppose that A = 1:'" Pi is a decomposition of A into indecomposable left ideals, numbered so that PI' ... , Pk are a full set of nonisomorphic modules among the {P). Then [P.J, ... , [Pk ] fonn a free basis for the free abelian group Ko(A). We remark that the Krull-Schmidt Theorem holds if A is a left artinian ring; it also holds for an order Ap over a P-adic ring Rp (see MO, Exercises 6.6, 6.7). Let 'fJ: A -

r

= 1). = r ~A

be a ring homomorphism (always assumed such that '.(J(l)

'fJ determines an additive homomorphism 'fJ*: Ko(A) -

Ko(r), given by 'fJ*

other words,

[M] E Ko(A). 6

Then •.

In

CLASS GROUPS AND PICARD GROUPS

7

We also need the abelian multiplicative group K I (A), the Whitehead group2 of A, which is analogous to the group u(A) of units of A. Let CL(A) be the general linear group consisting of all n x n invertible matrices over A, for all n, and let CL' (A) be its commutator subgroup. We set KI(A) = CL(A)/CL'(A). It is easily checked that CL'(A) contains all elementary matrices (those that arise from the identity matrix by inserting one off-diagonal entry). An arbitrary ~ E K I (A) is represented by an invertible n x n matrix X with entries in A. If it happens that by elementary row and column operations X can be put into diagonal ° An' viewed as a I x I matrix. form diag(A I , . . . , An)' then ~ is also represented by Al If this procedure can be carried out for each X, then we obtain a surjection u(A) ---- K 1 (A). This occurs whenever A is a field or skewfield, or a discrete valuation ring. We shall need a more general result of this type (see Bass [I, Chapter V, (9.1)]), namely 00

(2.2)

THEOREM (BASS).

There is a surjection u(A) ---- K I (A) whenever A is a semi-

local ring.

We recall that A is semiloeal if A/rad A is a semisimple artinian ring, where rad A denotes the Jacobson radical of A. In particular, every left artinian ring is semilocal, as is each finite dimensional algebra over a field. Further, every order A over a discrete valuation ring R is semilocal, since A/rad A is a finite dimensional algebra over the residue class field of R

(see MO, (6.15)). We make several remarks about Whitehead groups. First, each ring homomorphism If!: A ---- f induces a map If!* : K I (A) ---- K I (f). Secondly, if f = Mn(A) then there is an

isomorphism K I (f) ~ K I (A), gotten by viewing each element of CL(f) as an element of CL(A). Finally, let D be a skewfield, and let DO = D - {O} be its multiplicative group. Set D# = DO/[DO, DO], the commutator factor group of DO. Each X E CL(D) may be diagonalized by elementary row operations. The product of the diagonal elements, viewed as element of D#, is called the Dieudonne determinant of X. We then obtain an isomorphism det: K I (D) ~ D#. Let us now turn to the Localization Sequence, assuming hereafter that A is an R-order in A. A left A-module M has finite homological dimension if there exists an exact sequence (2.3) in which each Pi is A-projective. We set

assuming that M and each Pi are finitely generated. By repeated use of Schanuel's Lemma, one shows that X(M) is independent of the choice of the sequence (2.3). We denote by T(A) the category of all finitely generated R-torsion 3 left A-modules of finite homological dimension. Then we form the Grothendieck group Ko(T(A)). It turns out that X induces a homomorphism Ko(T(A) ---- Ko(i\), again denoted by X. Next, we define fJ.: K I (A) ---- Ko(T(A)) as follows: given a locally free left A-ideal M in A, we may choose a nonzero r E R such that rM C A. Since there is an exact sequence 2 Some authors use this term to denote K 1(ZG)/ {±G} for the case where A = ZG. 3This means that for ME T(A), there exists a non zero a ER such that aM = O.

8

IRYING REINER

r o -----+ M -----+ A -----+ AjrM -----+ 0, it follows that AjrM E T(A). We define [AIIM] = [AlrM] - [A/rA] E Ko(T(A». It is easily shown that [A//M] is independent of the choice of r. To define /1: K I (A) -----+

Ko(T(A», we note that since A is semilocal, each element of K I (A) is of the form

a for

some a E u(A). We set /1(a) == [All Aa] E Ko(T(A»,

a E u(A).

Then /1 turns out to be a well-defined homomorphism. Finally, we observe that the inclusion A C A induces homomorphisms

'Po : Ko(A)

-----+

Ko (A ),

'PI: K I (A)

----+

K I (A).

We have now the following basic result (Bass [1, Chapter IX, 6.3]): (2.4) THEOREM (LOCALIZATION SEQUENCE). Let A be an R-order in the F-algebra A. Then there is an exact sequence of abelian groups K I (A)

~ K I (A) £..

Ko(T(A»

~

Ko(A)

~ Ko(A).

It should be stressed that the above result is a rather formal one, in that it does not

rely on deep properties of orders or algebraic number fields, but depends instead on relatively straightforward (albeit detailed) arguments based on the definitions of the groups Ko and K I ' and some of their properties. Let us make one comment about the mysterious term Ko(T(A». For each ME T(A), we may decompose the R-torsion module M into a direct

sum of its P-torsion submodules, with P ranging over all prime ideals of R. However, the = 0 a.e., 4 so we have M = 1;1%1 Mp.

P-torsion submodule of M is isomorphic to Mp, and Mp

Further, Mp E T(A p ) since ME T(A). This yields an isomorphism Ko(T(A» ~

(2.5)

L:1%1 Ko(T(Ap», p

which will be extremely useful for us. Returning to the sequence in (2.4), we set Ko(A) = ker 'Po, the reduced projective class group. Thus the sequence /1 X ~ (2.6) I ----+ K I (A)/imK I (A) ----+ Ko(T(A» -----+ Ko(A) ----+ 0 is exact, with /1, X induced from those in (2.4). Likewise, there is an exact sequence (2.7)

I

-----+

L:1%1 K I (Ap)/im K I (A p ) -----+ L:1%1

Ko(T(A p» -----+ L:1%1 Ko(A p )

----+

0,

where in the direct sums P ranges over all nonzero prime ideals of R. By (2.5), the middle terms in (2.6) and (2.7) are isomorphic. We shall construct homomorphisms Xo, A' making the following diagram commute:

(2.8)

I-

K I (A)/imK 1 (A) ~ Ko(T(A» ~ Ko(A) ~ 0

A'l I

----+

iso·l

L:1%1 K I (Ap)/imK I (Ap) -----+ L:1%1 Ko(T(Ap »

Aol -----+

L:1%1 Ko(Ap ) -----+ O.

4 "a. e." means "almost everywhere", that is, for all but a finite number of prime ideals P.

CLASS GROUPS AND PICARD GROUPS

9

To begin with, it is clear that the homomorphism Ko(A) ---+ Ko(Ap ) induces a map t/Jp: Ko(A) ---+ Ko(A p ). Now let ~ = [X] - [Y] E Ko(A), where X, YE peA); then FX == FY, so replacing X by an isomorphic copy, we may assume that in fact FX = FY. Then X p = Yp a.e. (MO, Exercise 4.6), whence t/Jp(~) = 0 a.e. Hence the map Xo = L t/Jp is a well-defined homomorphism from 1(o(A) into L EEl Ko(Ap ). Since the rows of (2.8) are exact, there is then a unique homomorphism A' making the left hand square commute. Let us show at once that there is an isomorphism {3 : Cl A == ker AO '

(2.9)

given by MM] = [A] - [M], [M] E Cl A. It is clear that (3 is well defined and monic, and that im (3 C ker Ao ' To prove that (3 is surjective, let ~ E ker Xo. We may write ~ = [A(k)]

- [X] for some k and some X E peA). Then t/J p(~) = 0 for each P, whence X p == A~k). Therefore X is locally free of rank k, so by (1.11) we have X == A(k- t) -i- M for some locally free ideal M But then ~ = [A] - [M] E im (3. We have now proved that {3 is a surjection, and it remains to prove that (3 is a homomorphism. Let M, M', M" be locally free ideals such that M -i- M' == A -1- M", so [M] + [M'] = [M"] in Cl A. Then {3 [M]

+ (3 [M']

+ [A]

=

[A] - [M]

- [M']

=

[A] .- [M"] = (3 [M"] ,

=

[A

-i- A]

- [M

-i- M']

as desired. This establishes (2.9). We are going to apply the "Snake Lemma" to (2.8), so let us recall it. (2.10) SNAKE LEMMA. Given a commutative diagram of abelian groups

L'

~

L

L

L"

---+

0

o ---+ M' ---+ M ---+ M' y

there

~

an exact sequence ker

" a e, Cl:* ----+ ker e ----+ ker e ---+

If Cl: is monic, so is Cl:*; if

°

°

cok

0." e, ----+ cok e ----+ cok e

is epic, so is 8 *.

PROOF. Standard diagram chase. The map a is given by y-I e{3-I, suitably interpreted. Applying this to (2.8), we conclude that ker we obtain exact sequences

Kt (A)

---+

A'

K 1 (A)---+

Xo

== cok A', and Ao is surjective. Hence

a Ko(A) AO EEl L EEl K, (Ap)/im K 1 (A p ) ----+ ~ L Ko (Ap ) ---+ 0

(2.11 )

a~

0/

/'

\

CIA

o with im {3

= ker Xo

and

(3a' = a.

Let us describe the map

a'

0 explicitly. By (2.2), each ele-

ment in Kt (A p ) is the image of a unit in A p . Thus each x in the domain of a' is expressible as a product x = flap with ap E u(A p ) for all P, and ap = 1 a.e. The image of x in

10

IRVING REINER

L EB Ko(T(A p » is then L[ApIIApapl, and we must find x' E Ko(T(A» which corresponds to this sum under the isomorphism (2.5). We set

(2.12)

X

=

A () {

QApap }.

By MO, (5.3), X is a locally free A-ideal in A such t~at X p = Apa p for all P. The desired x' is therefore [All Xl. Since the image of [All Xl in Ko(A) is [AJ - [XJ, it follows that a'(x) = [Xl by virtue of the definition of (3. We shall find it convenient to describe the cokernel of A' in terms of ideles. Recall that the idele group of the field F relative to R is defined by

(2.13)

J(F)

{(a p) Enu(Fp): ap

=

E

u(R p ) a.e}

where P ranges over all maximal ideals of R. Analogously, we define the idele group of K 1(A) relative to R by

(2.14)

JK(A)

=

{(X p ) E nK 1 (A p ): x p E

im K 1 (A p )

a.e}

This group is independent of the choice of the R-order A in A, since if A' is another order, then Ap = A~ a. e. We shall also need the group of unit ideles

UK(A)

=

n K (A p), 1

p

which of course depends on A. There is an obvious homomorphism UK(A) by K 1 (A p ) ~ K 1 (A p ) for each P. We have at once

JK(A)

=

whence

~

(2.15)

EB

~

JK(A), induced

{~EB K 1(A p)}. im UK(A), K 1 (A p) ~ JK(A) imK1(A p ) =imUK(A)'

Now we observe that there is a homomorphism K1(A) ~ npK1(A p ), induced by the inclusion A CAp at each P. The image of K1(A) need not lie in ~EB K1(A p), as is already clear from the case where A = Q and R = Z. On the other hand, for each X E GL(A) we observe that X p E GL(Ap ) a. e. Thus there is a well-defined homomorphism

Kl(A)~{~EB K1(Ap)}{l}imK1(Ap )} given by the diagonal map x

=JK(A),

~

(x), x E K 1 (A). It follows at once from (2.15), and the definition of the map A' in (2.8), that cok A' ==

JK(A) imK1(A). im UK(A)

Since Cl A == ker Ao == cok A', we obtain

(2.16)

THEOREM.

There is an isomorphism q; ..

JK(A) imK1(A). im UK(A)

"'" - Cl

A 1\,

given as follows: let x = (x p ) E JK(A), where x p is the image of an element ap E u(A p) for each P, and where ap = 1 a. e. Then q;(x) = [Xl, where X is the locally free left A-ideal in A given by formula (2.12).

11

CLASS GROUPS AND PICARD GROUPS The above result is due to Wilson [44], and as we have seen, is a consequence of the

Localization Sequence of K-theory. An analogous formula, proved earlier by Frob1ich [8], expresses Cl A as a quotient of the idele group J(A). Frob1ich's formula is the idele-theoretic version of an earlier result of Jacobinski [16]; in both cases, Eichler's Theorem plays a vital role. For another K-theory approach, see Wall [45, IV] . In order to eliminate the occurrences of the Whitehead groups in (2.16), we shall use the reduced norm map nr, whose properties we now review (see MO, §9): (i) If A = Mn(F), then nr: u(A) -+ F" is just the determinant map, where F" = F - {O}.

(ii) If A = Mn(D) where D is a skewfield with center F, we may write (D: F) = k 2 for some integer k. Let E be a maximal subfield of D. Then there is an embedding D -+ Mk(E), which gives rise to an embedding r : A -+ Mkn(E). Set nr a = det r(a), a E u(A). It can be shown that nr a E F", so we again obtain a multiplicative map nr: u(A) -+ F". Hereafter we use the following notation:

A =A t EB """ EBA s (2.17)

(simple components),

C = Ft EB""" EB Fs = center of A, F i = center of Ai' D == R t EB""" EB R s = integral closure of R ~ C

Dp is the integral closure of

Then for each prime ideal P of R, Cp is the center of A p , and

Rp in Cp. There is a multiplicative map nr: u(A) -+ u(C), defined componentwise, which extends to a reduced norm nr: u(Ap) -+ u(Cp ) for each P. Suppose from now on that F is an algebraic number field.

For each i, 1";; i .,;; s, let Si denote the set of all real primes P of F i such that (A;)p is not a full matrix algebra over the real field (Fi)p (one says that Ai ramifies at each such P). Defme

Ft ={aEF;:ap>O

for all PES),

and let (2.18) We quote without proof some vital facts about reduced norms, keeping the above notation. (2.19) THEOREM. The reduced norm map nr: u(A p) -+ u(Cp ) is surjective, and its kernel is the commutator subgroup of u(A p ). If A p is a maximal Rp-order in A p , then

nr u(Ap )

= u(D p ).

This result is not too hard to prove; for the image of 1Ir, see MO, Exercise 14.6 and §33. For the kernel of nr, see Nakayama and Matsushima [23a]. The global analogue of (2.19) is considerably deeper: (2.20) THEOREM. Let F be an algebraic number field. kernel of nr is the commutator subgroup of u(A).

Then nru(A) = C+, and the

The first assertion is proved in MO, (33.15). The second is a deep result due to Wang, and will not be used here. The reduced norm map nr: u(A)

-+

u(C) extends uniquely to a map GL(A)

-+

u(C),

and thence to a map nr : Kt (A) -+ u( C). This last map is consistent with the surjection u(A) -+ Kt (A). Likewise, for each P we obtain a map Kt (A p ) -+ u(Cp ). This yields a

12

IRVING REINER

homomorphism nr: JK(A) ---+ J(C), where J(C) = rr J(Fi ), using the notation of (2.13) and (2.17). If x = (tip) E JK(A), where ap E u(A p ) for all P, and ap E u(Ap ) a. e., then nr x = (nr ap). It follows at once from (2.19) that nr maps JK(A) onto J(C). Further, if nr x = 1, then by (2.19) each ap lies in the commutator subgroup of u(A p ), whence x = 1 in JK(A). We have thus shown that nr :'JK(A) ~ J(C). Using this, together with the first part of (2.20), we obtain (2.21) THEOREM.

For F an algebraic number field,

CIA ~J(C)/C+.

rr nr u(Ap ),

where C+ is given by (2.18), and where the product extends over all nonzero prime ideals PaIR.

Let us try out this result for the case where A is a simple algebra with center F, and where A is a maximal R-order in A. In this case, we know that for each P, Ap is a maximal Rp-order in A p (MO, (11.6)), so by (2.19) nru(A p ) = u(R p ). Hence we obtain (2.22)

CIA ~J(F)/P.

rr u(R p ).

The right hand expression may be interpreted in terms of ideals rather than ideles, as follows: let 0: = (O:p) E J(F), where O:p E F; for each P, and O:p E u(R p ) a. e. Set Ro: = Fn{

n RpO:p },

an R-ideal in F such that (Ro:)p = RpO:p for each P. If I(R) denotes the multiplicative group of R-ideals in F, then we obtain a surjection a: J(F) ---+ I(R). Clearly ker a = rr u(Rp), so J(F)/II u(R p ) ~ I(R). Under this isomorphism, F+ maps onto P+(R) = {Ro:: 0: E F+}, a certain group of principal R-ideals in F. Hence (2.22) gives Cl A ~ I(R)/ p+ (R).

(2.23)

If p+ (R) consists of all principal ideals, then Cl A ~ Cl R. In general, this is not the case, but there is a homomorphism of I(R)/ p+ (R) onto Cl R with kernel a finite elementary abelian 2-group (see MO, Exercise 35.2). Hence, the problem of calculating the class group of a maximal order is solved (or, at least, is reduced to a standard problem in algebraic number theory). In particular, for the special case where A = R and A = F, it is clear that P+(R) is the group of all principal ideals. Thus by (2.23), Cl A coincides with the usual ideal class group Cl R. We may finally observe that if A is a maximal R-order in the semisimple F-algebra A, then (keeping the notation in (2.17)) we may write A = At EB ... EB As, where for each i, Ai is a maximal R(order in A j (see MO, (10.5)). Therefore, Cl A ~

L Ql Cl A

j

~ TII(R)/P+(RJ i

3. Change of Orders Throughout this section, let A, A', f denote R-orders in semisimple F-algebras, where F has characteristic zero but need not be an algebraic number field. Let p : A --+ f be a homomorphism of R-orders, that is, a ring homomorphism such that per) = r, rE R. We -_ claim that p induces a map p* : Cl A --+ Cl f, given by .•. ------

[M] E Cl A, where in computing the tensor product we use p to make f into a right A-module. Indeed, for each prime ideal P of R we have

(f ®J\ M)p ::= f p ®J\p Mp ::= f p ®J\p A p ::= fp, since M is a locally free A-ideal. Thus f ®J\ M is a locally free f-modu1e, and can be identified with a f-ideal in Fr. It is clear that the stable isomorphism class of M determines that of f ®J\ M, so p* is well defined, and is obviously a homomorphism. We call p a "change of rings" homomorphism, and write p* = f ®J\ .. Keeping the above notation, let A = FA, B = Fr. Then p induces a homomorphism A --+ B of F-algebras, and also induces maps K 1 (A) --+ K 1 (B),

JK(A) --+ JK(B),

UK(A) --+ UK(f).

It is easily checked that the isomorphism in (2.16) is consistent with the change of rings

maps induced by p. Of special interest is the case where p is the inclusion map A C A', where A' is a maximal R-order in A. Since JK(A) I JK(A) (3.1) CIA::= , CIA::= " imK)(A).imUK(A) imK)(A).imUK(A) it is clear that the map Cl A --+ Cl A' is surjective. Let us prove this fact directly, without use of (2.16). If M is any locally free A'-ideal in A. then for each P we may write Mp = A~ap, with ap E u(A p ) for all P, and ap = 1 a. e. Define

M=An{r; Apa p }.

a left A-lattice in A such that Mp = Apa p for each P. Then A'M = M ' in A, since the equality holds at each P. But A' ® I\. M ~ A'M. since M is locally free. It follows that [M] E Cl A maps onto [M'] E Cl A', as desired. Hereafter let D(A) denote the kernel of the surjection Cl A --+ Cl A'. Since Cl A' is known from §2, we hope to calculate Cl A by concentrating on its subgroup D(A). By (3.1) we have 13

IRVING REINER

14

imKl(A). im UK(A')

( 32) .

D(A)

~

--=-----im K (A) • im UK(A) l

When F is an algebraic number field, from (2.21) and (2.19) we obtain a simpler formula:

(

3.3)

C+ . ITu(Op) + ' C • IT nr u(Ap )

D(A) ~

where Op is the integral closure of Rp in the center of A p , and where C+ is as in (2.18). This latter formula shows that the subgroup D(A) of Cl A does not depend on the choice of the maximal order A' containing A. It will be desirable to give another characterization of D(A), valid whether or not F is

an algebraic number field. (3.4) THEOREM. Let M be a locally free A-ideal in A. Then [M] E D(A) if and only if there exists a finitely generated left A-module X such that M EEl X ~ A EEl X. PROOF. Let A' be a maximal R-order in A containing A, and let N be any finitely generated left A'-module. Denote by teN) the R-torsion submodule of N; then teN) is the kernel of the mapping N ----+ F ®R N. so N/t(N) is a A'-lattice. Now let M be any locally free A-ideal in A. Then A' ®A M ~ A'M, where A'M is the image of A' ®A M in FM Suppose now that M EEl X ~ A EEl X for some X. Then there is a left A'-isomorphism

1/1: A'M EEl A'0X ~ A' $A'0X. where ® means ®A' Since 1/1 preserves the R-torsion module, it induces an isomorphism

A'M EEl Y ~ A' EEl Y.

where Y

= (A' 0

X)/ teA' 0 X).

But now Y is a A'-lattice, and A' is a maximal order, so Y is A'-projective (see MO, (21.5)). We may then choose a A'-lattice y' such that Y EEl y' ~ A'(k) for some k, and then

A'M $ A'(k) ~ A' $ A'(k). Thus [A'M]

=:

0 in Cl A', whence [M] E D(A) as claimed.

The converse is somewhat harder to prove. Let [M] E D(A). Consider the exact sequence of A-bimodules: s

o ----+ A - i

A, ----+ U ----+ 0,

where i is an embedding, and U is an R-torsion A-module. Let S be the finite set of prime ideals P of R for which Ap exact sequence

"* A~.

Now M V A, so by Roiter's Lemma (1.2), there is an

(3.5) of A-modules, such that Tp Ap = A~. Next we claim that

=:

0 for each PES. Hence for each p. either Tp

=:

0 or else

5 A A-bimodule is a two-sided A-module L such that (Alx)AZ = Al(xAZ) for all Al' AZ E A, x EL.

15

CLASS GROUPS AND PICARD GROUPS

A' ®A T

(3.6)

==

T

as left A-modules. Since T is an R-torsion module, it suffices to prove that A~ ®Ap Tp for each P. But this is clear, since for each P either Tp = 0 or else A~ = Ap . From (3.5) we obtain an exact sequence of left A/-modules: A(' 8, , / Tor 1 A, n~ A ®AM~A ®A A~A ®A T~O.

==

Tp

n.

Since T is an R-torsion module, so is Tor~(A/, Hence irn 8 = 0, since we have already seen that A' ®A M is R-torsionfree. Therefore the sequence of left A-modules

,

O~A'®AM~A'L T~O

(3.7)

is exact, where we have used (3.6) to replace A' ®A T by T We now set K

= {(X, X') E

A E9 A':g(A)

= g'(A/)},

the fibre product or pullback of the pair of maps g, g' occurring in (3.5) and (3.7). Then there is a commutative diagram

K

h

------*)

A

A'--,-~)

T

g

in which h, h' are induced by the projection maps of A E9 A' onto their respective components. Since g' is epic, so is h; it is easily checked that ker h == ker g'. In this manner, we obtain a commutative diagram with exact rows and columns:

o

0

tItM t 1

M

------)0

0---+ A' ®A M ~ K ~ A ---+ 0

1 1

1

1

0---+ A' ®A M ---+ A' ---+ T----l- 0

t

o

1

0

Consider its completion at an arbitrary P. When Tp and both sequences (3.8)

0 ~ Mp ~ K p ~ A~----)-O.

= 0, we

see that K p

==

Ap E9 A~,

0 ~ (A' ® M)p ~ Kp ~ Ap ~ 0

split. On the other hand, when Tp =1= 0 then Ap = A~, so both A~ and Ap are Ap-projective, and again both sequences in (3.8) split. It follows at once (see MO, (3.20) and §5) that both of the sequences O~M~K~A'~O,

are split. Therefore

o ----)- A' ® M ~ K ~ A ~ 0

16

IRYING REINER

(3.9) as left A-modules. Finally, since [MJ E D(A) we know that A' ®A M is stably isomorphic to A', so there exists an integer k such that A'®M$ A'(k) ~ A' $ A'(k).

Adding A'(k) to both sides of (3.9), we obtain M $ A'(Hl)

~

A $ A'(k+l),

which completes the proof of the theorem. A slightly stronger version of (3.4) is proved in Jacobinski [16J; see also Endo and Miyata [4, I] . From (3.4) we see at once that the subgroup D(A) of Cl A does not depend on the choice of A'. Even more important, however, is the following consequence (Reiner [27J):

(3.10) COROLLARY. Let p : A ~ r be a homomorphism of R-orders. Then the change of rings map P* : Cl A ~ Cl r carries D(A) into D(r). PROOF. Let [MJ E D(A), so M $ X ~ A $ X for some A-module X. Therefore r®M $ r®x ~ r $ r®x, where ® means ®A' But then by (3.4) we have [r ® M] E D(r), that is, p* [M] E D(r) as desired. In particular, let H be a subgroup of the fmite group G, and let A == RH, r == RG. The inclusion A c r gives a change of rings map Cl RH ------+ Cl RG, defined by [M] ~ [RG ®RH MJ,

[M] E Cl RH.

However, RG ®RH M is precisely the induced RG-module obtained from M, and is usually denoted by W. Thus, the induction map Cl RH ~ Cl RG carries D(RH) into D(RG). We shall also be interested in the restriction map Cl RG ~ Cl RH, whose existence we now demonstrate. The crucial fact is that RG is free as left RH-module. Let us discuss this situation in somewhat greater generality than the group ring case. Let p: A ~ r be a homomorphism of R-orders, and let A = FA, B = Fr. Each left r-module X may be viewed as a left A-module, denoted by p*(X), by letting A act on X via p. We call p* the restriction operator, since in the special case where p is an inclusion, p*(X) is obtained from X by restricting the operator domain from r to A. Suppose now that p*(r) ~ A(n), that is, r is a free left A·module on n generators. Then p* : per) ~ peA), where as in §2, per) is the category of all fmitely generated projective left r-modules. Hence p* induces homomorphisms Ko(r) ~ Ko(A), Ko(r) ~ Ko(A), and there is a corn· mutative diagram with exact rows: ~ 0 e ~ o --- Cl r --+ Ko(r) --+ Ko(rp) --+ 0

p*

1

L

o --+ Cl A ~ Ko(A) ~ L ~

p*

1

e~

Ko(A p ) ~ 0

There is then a well-defined homomorphism res: Cl r ~ Cl A making the left hand square commute. The restriction map res may be described explicitly as follows: if M is any locally

17

CLASS GROUPS AND PICARD GROUPS

r

free r-ideal in B, then for each P, Mp ~ p ; therefore p*(Mp ) ~ A~n), which shows that p*(M) is a locally free left A-module of rank n. By (1.11) we have p*(M) ~ A(n - 1) -+ N for some locally free A-ideal N in A. Then res [M]

=

[N] E Cl A.

We may at once prove Matchett's [22a] result that res D(r) C D(A). Keeping the preceding notation, let [M] E D(n, so there exists a r-module X for which M EB X ~ r EB X Therefore p*(M) EB p*(X) ~ p*(r) EB p*(X), that is,

NEB A(n-l) EB p*(X) ~ A EB A(n-l) EB p*(X). Hence [N] E D(A) by (3.4), and we have established that res: D(r)

---+

D(A).

In particular, let A = RH and r = RC, with H a subgroup of the finite group C. Set

C

= U?=l Hg;;

then n

r =L

e Ag;,

;=i

so r is free of rank n as left A-module. We summarize our results: (3.11) THEOREM. The inclusion RH C RC yields an induction map ind : Cl RH Cl RC and a restriction nuJp res: Cl RC ---+ Cl RH. These maps satisfy indD(RH) C D(RC),

---+

resD(RC) C D(RH).

This theorem permits one to apply the machinery of Frobenius functors to the calculation of D(RC) and Cl RC. Applications of this type may be found in [4], [5], [28], [32].

4. Class Groups of p-Groups Let G be a finite p-group, where p is a rational prime. The aim of this section is to prove that D(ZG) is also a (finite) p-group. This result, proved first by Froblich [6] for the special case where G is abelian, was established in general by Reiner-Ullom [29], [31]. For simplicity we shall consider here only the case where p is odd. Set A = ZG, A = QG, and let A' be a maximal Z-order in A. For any finite group G, it is easily shown that IGIA ' C A C A' (see MO, (41.1)). Hence in our case it follows that Aq = A~ for each rational prime q p. We shall need some facts about the decomposition of A into its simple components {Ai}' In terms of the notation in (2.17), each field F i is a cyclotomic field Q(w) for some pnth root of unity w. Further, there is exactly one simple component, say A I' for which F I = Q; and in fact, A I = F I = Q. Finally, each Ai is a full matrix algebra over F i . These results are due to Schilling (see MO, (41.9)); for another proof, see Feit [Sa, (14.5)] . From algebraic number theory, we cite

*

(4.1) PROPOSITION. Let F i = Q(w), where w is a primitive pn th root of 1, with n ~ 1. Let Ri = alg. into {F;}. Then there is a unique prime ideal Pi of Ri containing p, and Ri / Pi ~ Zjpz. Given any element Q E Ri prime to Pi' there exists a unit u E Ri such that QU == 1 (mod PJ The uniqueness of Pi' and the isomorphism R;!Pi ~ Z/pZ, both follow from the fact that p is completely ramified in F i . Now let Q E Ri be prime to Pi; then Q == m (mod P) for some m E Z with (P, m) = 1. Set v = (w m - l)j(w - 1); then v E u(R) and v == m (mod Pi)' since Pi = (1 - w)R i . We need only choose u = V-I, and then QU == 1 (mod P), as desired. PROOF.

Since A i ~ M n /F) for 1 ,,;; i ,,;; s, no real prime of Fi can ramify in Ai' Hence the group C+ of (2.18) is precisely u(C). From (3.3) we have

~ C+

D(A)

• TInr

u(A~)/C+

• TInr u(Aq )

~TIU(Dq)/{nnrU(Aq)} {c+ where q ranges over all rational primes. However, C+ ()

q

* p we have nr u(Aq )

n u(Oq) = u(O).

Further, for

= nr u(A~) = u(Oq)' Hence we obtain

D(A) where X group

() TIU(Oq)},

= nq*p u(Oq)'

~ u(Op)·

xl

u(O)· nr u(A p). X,

TIlls implies at once that D(A) is a homomorphic image of the

18

19

CLASS GROUPS AND PICARD GROUPS

H Since

0 =

= u(Dp)/u(D). nr u(Ap)'

~EIl R j , we obtain Op

,.

=

~EIl (R j )

•

p

By (4.1), the p-adic completion (R j ) '"

p

is the same as the Pradic completion R j of R j ; we shall use F j to denote the Pj-adic completion of F j , for convenience. Thus we have

u(D p )

sEll'"

=L

u(RJ

j=t

Now let et = ~ etj E u(Op), with each cr.j E u(RJ Then et j == a j (mod Pj) for some a j E R j. Choose t E Z with (t, p) = I such that tat == I (mod Pt). Viewing t as an element of u(Ap )' we note that the reduced norm nr t has component t in (A t) . Let us replace et by et • nr t, p which does not affect the coset et EH; this replacement has the effect of making at == I (mod Pt). On the other hand, for each i> I, we can multiply etj by a factor from u(O) without changing the coset By (4.1), we can then make a j == I mod P j for i > 1. We have thus shown that each element of H is of the form where et = ~ etj and where etj == I (mod Pj) for each i. Let rad Ap denote the Jacobson radical of the lp-order Ap ' Then (see MO, (6.15)) for large n we have (rad Apt cp. Ap C rad Ap ' pn A~ CAp'

ex.

a,

It follows at once that if z E A~ is such that z - 1 E rad A~, then zpn .- I E rad Ap for large n. and therefore zpn E u(Ap )' We shall use this fact in a moment. We may write

Ap ~ L

Ell

{Mnj(Fj)}p

~E9

= L.J

"-

Mn/F;),

where F j is the Pradic completion of F j • Up to an inner automorphism of M n .(F;), each A

A

I

A

maximal ,.Rrorder in M n I.(F;) is of the form M n I.(R;) (see MO, (17.4)). Hence we have Ap' ~ ~EIl M n . (R;), from which we deduce at once (see MO, (17.5)) that I

,

~E9

"-

rad Ap ~ L.J Pj • Mn/R j ). Now let

a E H,

where et

= ~ etj

and a j == 1 (mod P;) for each i. Set

x j = diag(etj , I, ... , 1) E Mn.(R j), I

1 , ;;;;; i";;;;; s,

and let x = ~ x j ' Then x-I E rad A~, and nr x = a. If I + rad A~ denotes the multiplicative group {I + z : z E rad A~}, then by the above discussion the reduced norm map gives a surjection v: I

+ rad A~

-

H. Further, for each z E 1

+ rad A~

we proved above that = 1 in

zpn E u(A p) for large n, and therefore (nr z)pn E nr u(A p)' This shows that v(z)pn

H, and proves that every element of H has order a power of p. But D(lG) is a homomor-

phic image of H, whence D(lG) has this same property. Therefore D(lG) is a p-group as claimed, since D(lG) is already known to be a finite group. The proof for the case p = 2 is slightly more complicated, and may be found in [29]. We should caution that the obvious generalizations of the above result do not hold. Thus, if R is a ring of algebraic integers and C is a p-group, then D(RG) need not be a p-group. On the other hand, if G is an arbitrary fmite group, then lD(lG) I may have prime divisors other than those occurring in ICI (see Ullom [42]).

5. Mayer-Vietoris Sequences The explicit formulas for D(A) in §3 are not well suited for calculations, although they do yield results such as that in §4, and Ullom's [43] estimates for the exponent of D(A). In practice, most computations are based on Milnor's Mayer-Vietoris sequence, or upon the modification thereof given in Reiner-Ullom [30]. We start with a fibre product diagram (or pullback diagram)

(5.1)

of rings and ring homomorphisms. This means that there is a ring isomorphism A'='" {(A., A2 ):Ai E Ai' g.(A.) =g2(A2)}' with f i the projection map A. EB A2 -+ Ai' i = 1, 2. The following basic theorem is due to Milnor [23] (see also Bass [1]):

(5.2) THEOREM. Given a fibre product diagram (5.1) with either gl or g2 surjective, there is an exact sequence of additive groups: (5.3)

K I (A) -+ K I (AI) EB K. (A 2 ) -+ K I (A) -+ Ko(i\) -+ Ko(A I

(Furthermore, if both

gl

)

EB K o(A 2 ) -+ Ko(A).

and g2 are surjective, the sequence

(5.4)

is exact.) The sequence (5.3) is called a Mayer-Vietoris sequence. We shall not make use of (5.4) here, though it does occur in Keating's [18] calculation of K I (ZG) for G metacyclic. If A is an order, then Ko(i\) is closely related to Cl A, and K I (A) to the group of units u(A), so we may expect a sequence of the type (5.3) connecting class groups and unit groups. We shall prove (see Reiner-illlom [30]):

Let A be an R-order in a semisimple F-algebra A satisfying the Eichler condition relative to R, where F is an algebraic number field, and let (5.1) be a (5.5)

THEOREM. 6

6 The purpose of this hypothesis is to guarantee that for locally free A-modules, stable isomorphism implies isomorphism.

20

21

CLASS GROUPS AND PICARD GROUPS

fibre product diagram of rings in which Al and A2 are R-orders in semisimple F-algebras. Suppose that A is a finite ring, and that either gl or g2 is surjective. Let us set u*(A;) Then there are exact

(5.6) ( 5.7)

1 ----+ u*(A I ) • u*(A2 ) 1 ----+ u*(A I )

Here, the map Cl A

= g;{u(A;)},

i

= 1,2.

sequences 7

----+

•

----+

- 0 1/1 u(A) ----+ Cl A ----+ Cl Al $ Cl A2

----+

0,

- 0 1/1 1 u*(~) ----+ u(A) ----+ D(A) D(A I ) $ D(A 2 ) ----+ O.

Cl A; is induced by f i , i

= 1,2, and o(u) =

(5.8)

[Au], where

u E u(A).

The hypotheses of (5.5) remain in force for the rest of this section, unless otherwise stated. We set A; = FA;, i = 1, 2. We begin the proof with a few simple remarks. For each prime ideal P of R, we may replace each ring in (5.1) by its P-adic completion, thereby obtaining another fibre product diagram. Thus

where now g;: (A;)p

----+

Ap . Likewise, (5.1) yields a fibre product diagram

A--A I

1

1

Az--.O

since A is an R-torsion module (because A is finite). Thus A ~ A I $ A 2 ; we hereafter identify A with A I $ A z ' which is consistent with identifying A with a subring of Al $ A2 • For i = 1,2, let A; be a maximal R-order in A; containing A. Set A' = A~ $ A~, a maximal R-order in A containing A. If X is a locally free left A-lattice of rank n, then A; 0 A X is a locally free left A;-lattice of rank_n, and may be identified with its image A;X computed inside the A-module FX Likewise, A 0 A X is locally free. The map 1/1 : Cl A ----+ Cl Al $ Cl A2 is defined by 1/1 [X] = ([AI X], [A 2 X]) for each locally free left A-ideal X in A. There is a commutative diagram o ) D(A) • Cl A ) Cl A' •0 1/1

1

iso·l

o ~ D(A I ) $ D(A2 ) ~ Cl At $ Cl A2 ~ Cl A~ $ Cl A~ ~ 0, so 1/1 induces a map 1/1 I : D(A) ----+ D(A I ) $ D(A 2 ) making the left-hand square commute. By the Snake Lemma, it follows that ker 1/1 I = ker 1/1, and cok 1/1 I ~ cok 1/1. Hence if we show that 1/1 is surjective, we may deduce that 1/1 I is also surjective. Suppose we are given a locally free Ai-ideal X; in A, i = 1, 2. Then for each P we may write (X;) p

and

a}!) = 1

a.e., i

= 1,2.

= (A;)p aJ!),

where aJ!)

= unit in (A;) p,

Let

7 The expression u*(A I ) . u*(A2) stands for {UI u2 : u; E u*(A;)}. It is a subgroup of u(i\) by vir· tue of Lemma 5.9 below.

22

IRVING REINER

X

=A n

{Q

A p (41)

+ aJ,2»}.

Then X is a locally free A-ideal in A such that AiX = Xi' i = 1, 2. This proves that '# is surjective 8 , whence so also is '# 1 . The proof of Theorem 5.5 is based on the following result, which is established in great detail in Reiner-Ullom [30, (4.20)]: (5.9)

LEMMA

For each u E u(A), define Au = {(AI' A2): Ai E Ai' gl (Al)'U = g2(A 2 )}·

Then

(i) Each Au is a locally free A-lattice such that Ai 0 A Au == Ai' Au

= Ai'

i

= 1,2,

the last equality holding true inside A.

(ii) For u, u' E u(A') we have •

Au

I ·

+ Au ==

A

+ Auu' ==

•

A

+ Au'u.

(iii) Let A' = A~ EB A~ as above. Then A'

°

A

Au == A' • Au = A',

the equality holding true inside A.

(iv) Let X be any projective A-lattice such that Ai Au for some u E u(A).

°

A

X

= Ai' i =

1, 2. Then X ==

(v) Let u*(A) be defined as in (5.5), and let u, u' E u(A). Then Au == Au' if and only if u' E u*(A l ) • U • u*(A 2 ). Assertion (iii) shows that [Au] E D(A) for each u E u(A), while (ii) gives [Au]

+

[Au'] = [Auu']. Thus the map 0, defined in (5.5) by the formula o(u) = [Au], is a homo-

'#1 = ker ,#, we need only '#1 • 0 = O. On the other hand, let X be a locally free left A-ideal in A such that '#1 [X] = O. Then [Ai 0 AXl morphism from u(A) into D(A). In view of the fact that ker

establish the exactness of sequence (5.7). By (5.9(i», we know that

0 in Cl~, i = 1, 2. Since A = EicWer/R, and A = Al EB A 2 , it follows at once that also Ai = EicWer/R. i = 1,2. Therefore Ai 0 A X == Ai' i = 1,2, whence X == Au by (5.9(iv». This shows that ker '#1 C im 0, and proves that ker '#1 = im o. It remains for us to determine ker o. If u E u(A) is such that o(u) = 0, then [Au] o in Cl A, whence Au == A. Thus by (5.9(v» we have u E u*(A l )· u*(A 2 ), as desired. The converse argument also works, and so the exactness of (5.7) is established. It should be emphasized that this proof of Theorem 5.5 is completely routine and formal, except for the fact that stable isomorphism implies isomorphism when A = Eichler/R. In the absence of the EicWer condition, we obtain (see [30, (5.3)]) an exact sequence =

(5.10)

1 ----+ GL~(Al)' GL;(A 2 )

----+

GL 2 (A)

----+

D(A)

----+

D(A l ) EB D(A ) 2

----+

0,

where GL;(A) is the image of GL 2(A) in GL 2 (A). To conclude this section, we indicate how fibre products (5.1) arise in practice. Let

I, J be a pair of two-sided ideals in the R-order A. The diagram of natural surjections 8 The fact that 1/1 is surjective also follows from § 3, since there is an inclusion A C Al $ A 2 of orders in A.

23

CLASS GROUPS AND PICARD GROUPS

-----1 1 ------A

A

-I

I nJ

A

A

J

I +J

is always a fibre product diagram. Now FI is a two-sided ideal of A, where A = FA, and F fg)R (A/I) ~ A/F!. Hence if A/I is R-torsionfree, then A/I is an R-order in a semisimple ring summand of A. If also A/J is R-torsionfree, and

F!nFJ=O,

Ff(f)FJ=A,

then the above diagram becomes A~A/I

1

AjJ

~

1 + J).

A/(I

Here, the rings A, A/I and A/J are R-orders in semisimple F-algebras, and A/(I R-torsion ring.

+ J) is an

6. Calculations Throughout this section let A == ZG. A == QG. where G is a finite group. Let A' be a maximal Z-order in A containing A, so there is an exact sequence

o--.. D(A) --.. Cl A --.. Cl A' --.. O.

(6.1)

As we showed at the end of §2, the class group Cl A' is known quite explicitly, and so we shall concentrate here on the calculation of D(A). It may be mentioned that relatively little is known about determining the additive structure of the group Cl A, once the structures of

D(A) and Cl A' are known. Let us begin with the easiest possible example:

(6.2) THEOREM.

Let G be cyclic of prime order P. and let w be a primitive p-th root

of unity over Q. Then D(A)

= 0,

Cl A

== Cl R, where R = alg. int. {Q(w)}.

PROOF. If g is a generator for G, then A

= Z EEl

Zg EEl ••• EEl ZgP-l. We may thus

identify A with the ring Z[x] /(x P - 1) of polynomials in x with integral coefficients, modulo the principal ideal (x P - I). We set p-l

I = (x - I)A,

] =