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This paper concerns a unified theory of the Galois module structure of the integers O N and the S-units U(S) in a finite Galois extension N/K of number fields ...

Invent. math. 74, 321-349 (1983)

I~ve?ltlO~eS mathematicae $) Springer-Verlag 1983

On the Galois structure of algebraic integers and S-units T. Chinburg* Department of Mathematics, University of Pennsylvania Philadelphia, PA 19104, USA

I. Introduction

This paper concerns a unified theory of the Galois module structure of the integers O N and the S-units U(S) in a finite Galois extension N/K of number fields, where S is a sufficiently large finite set of places of N stable under the action of G = Gal(N/K). Our main interest is an invariant ~?m in the class group CI(Z[G]) of G, which is associated to U(S). This invariant measures the difference of U(S) from a simpler module X(S), in a manner similar to that in which the stable isomorphism class f2~=(Ou)-[K:Q](Z[G]) measures the difference of O N from a free module in case O u is a projective G-module. Our main result, Theorem 3.1, is that for sufficiently large S, (2m is independent of S and of certain other choices made in the course of its definition, and consequently is an invariant of the Galois extension N/K. Let V be a representation of G, and let L(s, V) be its Artin L-function. In Sect. II of this paper we develop a parallel between the leading coefficient c(V) in the expansion at s = 0 of L(s,V) and the Galois Gauss sum T(V) of V. In Sect. III we define f2,, and discuss its relation to c(V) as V runs over the representations of G. We conjecture, and very partially prove, a description of (2,, in terms of the c(V) which is analogous to A. Fr6hlich's description of ~ , in terms of the z(V). By developing simultaneously the theories of O,,, (2,, c(V) and z(V), one is led to questions concerning them which correspond to known or conjectured results for their additive or multiplicative counterparts. The proofs of the results stated in Sects. II and III are given in Sects. IV-IX. This paper is based on the work of J. Tate in [19] and of A. Fr6hlich in [43. It was motivated by the '+norm conjecture" of [2]. It is a pleasure to thank S. Chase, S. Lichtenbaum, H. Stark and J. Tate for their inspiration and for many helpful conversations.

Notations. The symbols Z, Q, ~ and IE will denote the ring of integers and the field of rational, of real, and of complex numbers, respectively. By a number *

Supported in part by NSF fellowship MCS80-17198

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field N we will m e a n a finite extension of Q c o n t a i n e d in the a l g e b r a i c closure (~ of Q in 112. The n o r m a l i z e d a b s o l u t e value at a place v of N will be d e n o t e d H I[~. If v is real (resp. complex) then ]1 I1~ results from the o r d i n a r y a b s o l u t e value on IR (resp. the square of the o r d i n a r y a b s o l u t e value on (;). If v is nona r c h i m e d e a n and rt is a local uniformizer at v, then ll~l[,7 x is the o r d e r of the residue field of N at v. T h e tensor p r o d u c t A | of two a b e l i a n g r o u p s will d e n o t e the tensor A | B over Z. A l l G - m o d u l e s will be left G-modules. Let cg, be an a d d i t i v e c a t e g o r y of G-modules. T h e G r o t h e n d i e c k g r o u p of m o d u l e s in is the g r o u p h a v i n g g e n e r a t o r s the i s o m o r p h i s m classes {M} of m o d u l e s M in ~ , a n d relations { M } - { M ' } - { M ' } = 0 whenever O ~ M ' - ~ M - . M " - ~ , O is an exact sequence of m o d u l e s in ~. T h e class in this G r o t h e n d i e c k g r o u p of a m o d u l e M in (g will be written (M). If G is a s u b g r o u p of a g r o u p T, and M is a G - m o d u l e , then Ind~.M will d e n o t e the T - m o d u l e Z [ T ] | M.

II. A correspondence

a. Definitions N / K be a finite G a l o i s extension of n u m b e r fields with g r o u p G =GaI(N/K). W e distinguish two cases; an a d d i t i v e one, d e n o t e d G o, a n d a

Let

m u l t i p l i c a t i v e one, d e n o t e d G,,. Let V be a finite d i m e n s i o n a l c o m p l e x repres e n t a t i o n of G, a n d let V be the dual representation. Let N c~ be a fixed finite G a l o i s extension of Q c o n t a i n i n g N. Define Ind~/QV to be the i n d u c t i o n to GaI(NC~/Q) of the inflation of V to Gal(NCl/K). Let t(V) be the G a l o i s G a u s s sum of V (cf. [12, p. 48]), a n d let L(s, V) be the A r t i n L-function of V. If S is a finite set of places of N, let ~3(S,K) be the set of p r i m e ideals of K which d e t e r m i n e places of K lying u n d e r places in S. Define

Ls(s,V)= t~ det(1-Fr~ ~r

-~

K)

to be L(s, V) with the Euler factors at p r i m e s ~3~3(S,K) removed. Define rs(V ) to be the o r d e r of vanishing of Ls(s, V) at s = 0 . T h e b a s i c terms S, U, v(a) a n d c v in the cases G a a n d G,, are as defined in the following table. Table 2.1. Basic terms Go

Gm

S

The set S~ of embeddings of N into

A finite set of places of N, stable under G, which contains the set S~ of infinite places of N

U

The group ON of integers of N

The group U(S) of S-units of N

v(a) for v~S The embedding of a into • by v and a~U Cv t(lndr/QV)

log I]aLl~,,where I] I1ois the normalized absolute value at v lim s-'~(v)Ls (s, V) s~0

On the Galois structure of algebraic integers and S-units

323

Our discussion of the G~ and the G~ cases now proceeds in parallel. Let Y be the free abelian group on S. Define 2:t12 | U -+ IE | Y to be the G-homomorphism induced by 2(a)= y v(a)v for aeU. In the G, case, let X=Y,, while in the yes

G,, case let X be the kernel of the homomorphism Y--~Z sending each yeS to l. (In both cases, X is the intersection of Y with the image of 117| U under 2). Let q0: X--+ U be an injective G-homomorphism inducing an isomorphism Q|174 We define the regulator R(V,q~) (after Tate [19], in the Gm case) to be the determinant of the endomorphism (2 o ~P)v of Homa(V,,~E | X) induced by 2 o q0.

b. Stark's conjecture and Galois Gauss sums We may now give a unified statement of Tate's form of Stark's conjecture ([16]) and results of Fr6hlich on Galois Gauss sums.

Conjecture 2.1.

Let

A(V,~p)=R(V,~p)/c v .

Then

A(V~,q~)= A(V,~p) ~

for

c~Aut(IE/Q) This conjecture is proved in the Ga case in Sect. IV. We now consider formulas for A(V,q)). Let E be a number field that is large enough so that there is a finitely generated torsion free O [ G ] module M such that M | where O = O ~ denotes the ring of integers of E. Suppose that B is a finite O-module. Define the order ideal a(B) of B to be H ~ lengthlBp), where the product is over the prime ideals p of O, and length(Bp) la

is the length of the localized module B~. Suppose that C is a G-module. Let C 6 and C~ be, respectively, the largest submodule and the largest quotient module of C on which G acts trivially. Let CM=Homo(M, C| Define n M" (CM)G~(CM) ~ to be the homomorphism induced by left multiplication by the norm element h a = ~ a of Z[G]. Suppose o'~G

that D is a finitely generated G-module, and that f : C--*D is a G-homomorphism with finite kernel and cokernel. Let fM: (CM)G ~(DM) a be the homomorphism induced by.s The kernel and cokernel offMo n M" (CM)c,--*(OM)~ --*(DM)G are finite O-modules. Define qM(f) to be the O-ideal o'(coker(fM o n~))/a(ker ( f Mo riM)). We will say that the pair (N/K,S) is tame in the Gm case if (i) S contains the ramified places of N/K, and (ii) when F is a subfield of N containing K, the ideal classes of primes of F determined by places in S generate the ideal class group of F. We will say that (N/K,S) is tame in the G, case if N / K is at most tamely ramified.

Theorem 2.1 (Fr6hlich-Tate). Suppose that (N/K,S) is tame, and in the Gm case also that O = Z . Then conjecture 2.1 is true, and the number A(V,~o) lies in the field Q(Zv)~_E generated by the values of the character Zv of V. The ideal A(V, q))O equals qM(~O), where M |

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Corollary 2.1 (Fr/Shlich-Tate). Suppose that (N/K,S) is tame and that V is an absolutely irreducible representation of G. Let W be a representation of G whose character Zw is the sum of the conjugates of the character Zv. Then the number A(W,~p) is rational. In the G, case, A(W, qo) is the norm .from Q(zv) to Q of A(V, ~o)eQ(zv). In the G,~ case, at least the fractional ideal A(W, ~p) Z is the norm to Q of a fractional ideal of Q(zv). The Gm case of Theorem 2.1 and Corollary 2.1 are shown by Tate in [19]. (See also [2].) In Sect. IV we deduce the G, case from formulas of Fr6hlich. Conjecture 2.1 implies that A(V, cp)~Q(Zv ). Theorem 2.1 suggests

Conjecture 2.2. If (N/K, S) is tame, the ideal A(V, q~)O equals qM(Cp). Remark 2.1. A similar conjecture has been made by S. Lichtenbaum in [10]. It will be shown in Sect. V that Conjecture 2.2 is independent of the choice of M. In the next two paragraphs we show that it is independent of the choice of q~. Suppose ~p': X ~ U is another injective G-homomorphism inducing an isomorphism Q | 1 7 4 Let ((p'-loq))v be the endomorphism of Homa(F,,C| induced by ~0'-~o~o: ff2|174 and let d v =det(q ~'-1 ~ ~P)v- Then R(V,q))=R(V,~o')d v, so A(V,(p)=A(V, cp')dv. As is shown in [-19, Chapt. 1.6], we have dw=(dvY for eeAut(02/Q), so Conjecture 2.1 is independent of the choice of q~. Now ~PM: Xu--" UM is injective because q) is injective, and similarly for ~o~u. Hence ker(q~Mo riM)= ker(~o~ onm) = ker(nM) and coker(q)M)/coker(~oM onM) _~coker(~o~t)/coker(~p~ t onM)

~_X~/nM(X~) ~. Thus from qM(~0)= a(coker(~o Mo nM))/a(ker(q~M o riM)) we find that

qM(~O)/qM(~O')= a (coker (~pM))/a(coker (~oM)). Now X~ is a torsion free O-submodule of Homc, (F,, q? | X), and X~| = Homc,(V,, r | X). We have (q~)(~o '-1 o q))vX~ = q)MX~. Therefore

qM(~O)/qM(~O')= a(coker(cpM))/a(coker(~o'M)) r

= a(coker(q~ o (~o'-1 o (p)v)))/a(coker(~oM)) = d v O = (A(V, ~o)/A(V, ~o'))O.

Hence Conjecture 2.2 is independent of the choice of q~.

III. Galois module structure We now define the invariants O. and 0,~ and discuss their relation to the numbers A(V, ~p) as V runs over the representations of G.

On the Galois structure of algebraic integers and S-units

325

Because X is torsion free, the spectral sequence HP(G, Ext}(X, U)) ~ Extf~+~

U)

degenerates. Definition3.1. Suppose that (N/K,S) is tame. Define the canonical class ~6Ext~(X, U)=H2(G, H o m z ( X , U)) to be the trivial class in the G, case, and to be the canonical class of [20, p. 711] i~1 the G,, case. The next lemma is proved by Tate in [19] in the G,, case. In the G, case, a theorem of Noether states that N / K is tamely ramified if and only if U = O N is a projective G-module. Therefore the lemma follows in the Ga case on letting A = U, B = X , and on defining U - ~ A and B - . X to be the identity homomorphisms. L e m m a 3.1 (Tate, in the G m case). Suppose that (N/K,S) is tame. Then there exist finitely generated cohomologically trivial modules .4 and B and an exact sequence O ~ U--, A - * B--, X--~O (.)

whose class ill ExtZ(x, U) equals the canonical class. Let ~ be an order in Q[G]. An ~-module M is said to be locally free of rank 12 if for every rational prime p, the completion Mp of M at p is free of rank , over the completed ring ~p. Define Ko(,~) to be the Grothendieck group of finitely generated locally free ~-modules. The rank function rk defined by r k ( M ) = , if M is locally free of rank , extends to a homomorphism rk: K0(s~)-~Z. The kernel of rk is called the class group CI(~) of ~r When : ~ = Z [ G ] , we will make use of the facts (cf. [17] and [15]) that every projective Z[G]-module is locally free, and that every cohomologically trivial Z[G]-module has a resolution of length at most two by locally free Z[G]modules. By Schanuel's Lemma ([1, p. 36-39]), this induces an identification of Ko(Z[G]) with the Grothendieck group of finitely generated cohomologically trivial Z [G]-modules. We may now state the main result of this paper. Theorem 3.1. In both the G, alld the G~ cases, !/" (N/K, S) is tame, the class (A) - ( B ) ~ K o ( Z [ G ] ) .for A and B as in Lemma 3.1 lies in CI(Z[G]) alld is il~dependent of the choice of S and of the sequence (.). Definition 3.2. Let f2=(A)-(B), and let C2~ (resp. Qm) be f2 in the G, (resp. the G,.) case. Remark 3.1. In each of the G~ and G,, cases, the invariant f2 depends only on N/K. The invariant f2,, is defined for every N/K, but the invariant ~, is defined only for N/K which are at most tamely ramified, and is then the class (ON) - [K:Q](Z[G]). We now study s along lines suggested by Fr~ihlich's analysis of ~,. Let R G be the ring of virtual characters of G. Let E be a number field that is large enough so that every representation of G may be realized over the ring of integers of E. Then A = G a l ( @ Q ) acts on R~, E and the group Id(E) of

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T. Chinburg

fractional ideals of E. Let H(G) be the group of b~HOmA(RG, Id(E)) such that for all x e R G, b(x) is an ideal of the field Q(X) generated by the values of Z. Let P(G) be the subgroup of beH(G) such that b(z ) is a principal ideal of Q(x) for all X, with a totally positive generator in Q(x) if x is symplectic. Suppose that R is a subring of Ra. Define H~(G) (resp. P~(G)) to be the subgroup of HomA(R, Id(E)) formed by the restrictions to R of elements of H(G) (resp.

p(c)).

Let Jg be a maximal order of Q [G] which contains Z [ G ] . Let I ~_.//r be a projective left Cg-ideal. Define n~: ~ - - ~ Z ( ~ # ) to be the reduced norm from/r to the center Z ( ~ ) of .#. If )~ is an irreducible character of G, let e~ be the idempotent of Z in (12[G]. The next lemma is shown in [21, p. 504-507] and [4, p. 140-143]. L e m m a 3.2. There is a unique isomorphism t u: CI(~/g)--~ H(G)/P(G) such that if

11 ~_12 are projective left ~l-ideals contained in ~ t u((It)-(12) ) is represented by b~H(G), where b(x)=ez(n~(l O/n e(I2) ) if ~ is an irreducible character oJ" G. Tensoring on the left with d// over Z[G] induces a homomorphism s~u: CI(Z[G])--~CI(J). The composite t~osju: CI(Z[G])--~H(G)/P(G) does not depend on ~r (cf. [4, p. 140-143]), and will be denoted los. The kernel D(Z[G]) of los is called the kernel group of G, and is independent of Jg. We will assume that the field E is chosen large enough so that there exist modules M z as in the following definition. Definition 3.3. If )~ is an irreducible character of G, let M z be a finitely generated torsion-free 0 E module such that M x | has character Z. Let q(~o) be the element of Hom(R~,Id(E)) such that q(cp)()0=qM~(q~) fi~r all irreducible

characters )~ of G. The following algebraic result is proved in Sect. V. Proposition 3.1. The homomorphism q(cp) lies in H(G), and does not depend on the

choice of the M z. The image of q(q~) in H(G)/P(G) equals (to s(~2))-1. Corresponding to the involution on Z[G] we have an involution ~ on CI(Z[G]). If M is a locally free Z[G] module of rank n, then c = ( M ) - n ( Z [ G ] ) is in CI(Z[G]), and ? = ( ( M * ) - n ( Z [ G ] ) ) -1, where M * = H o m z ( M , Z ) is the contragredient of M. (See [4, p. 154]). We define the contragredient class c* to be ~-~. Suppose that the class of b6H(G) in H(G)/P(G) equals los(c). Then on defining /~()0=b(~, the class of /) in H(G) equals tos(~). Proposition 3.1 and the Fr6hlich-Tate Theorem and Corollary2.1 give the following arithmetic information on the class (2~CI(Z[G]). Proposition 3.2. Let R = R ~ in the G~ case, and let R be the subring of rational characters of G in the G,, case. Define A(cp)6Hom(R, Id(E)) by letting A(q~)()~v) =A(V, cp)O if V is a representation of G whose character )~v lies in R. Then

A(cp) lies in Ht~(G), and the image of A((p) in HR(G)/Pe(G) equals the restriction of tos((2*) to R. This proposition proves the following conjecture in the G~ case; in the Gm case, the conjecture is a consequence of Proposition 3.1 and Conjecture 2.2.

On the Galois structure of algebraic integers and S-units

327

Conjecture3.1. The function A(q~) defined by A(q))(Xv)=A(V, qo)O is an element of H(G) whose image in H(G)/P(G) equals to s(O*). Clearly A(V,,~o)O is a principal ideal if this conjecture is true. Therefore tos(O*) is conjecturally determined by the signs of the A(V,~o) for symplectic V. These signs are determined in

Proposition 3.3. Suppose that V is symplectic and that (N/K,S) is tame. Then the sign of A(V,,q)) equals the constant W(V)= ++_1 in the functional equation of L(s, V).

Corollary 3.1. If Conjecture 3.1 is true, then tos(~2*)=0 if and only if for each symplectic representation V of G, there is a unit ev6Q(zv ) such that W(V~)e~v>O for all c~Aut(ll?/Q). Fr6hlich [4] has proved the following result on the action of Galois on the signs in the functional equations of symplectic representations. Theorem 3.2 (Fr/Shlich). Let V be a symplectic representation of G=Gal(N/K), and let N f ( V ) be the absolute norm of the conductor of V. Then W ( V ~) (Nf(V)l!2y W(V) N f ( V ) ';2 for all ~ A u t ( ~ / Q ) . If N / K is at most tamely ramified, then W(V~)= W(V) fisr all c~ Aut(Ir/Q). Corollary 3.2 (Frtihlich). In the G~ case, if N / K is at most tamely ramified, we may let ev= W(V) in Corollary3.1. Hence tos((2*)=0, or equivalently, f2, is in the kernel group D(Z [G]).

Corollary 3.3. Suppose we are in the tame G,, case, and that Conjecture 3.1 is true. Then tos(~2")2=0, and tos((2*)=0 if N / K is at most tamely ramified. There exist examples in which there do not exist units e v as in Corollary 3.1; in these examples, Conjecture 3.1 implies t o s (f2*) 4=O. Let Go(Z[G]) be the Grothendieck group of finitely generated ZIG] modules. The Cartan map h: K o ( Z [ G ] ) ~ G o ( Z [ G ] ) is induced by letting h((A)) =(A) if A is a finitely generated locally free Z I G ] module. One can prove an amusing formula for h(O,,):

Proposition 3.4. Let X (S ~ ) be X in the Gm case when S is the set S o~ of infinite places of N. Let Cl(N) be the ideal class group of N. Then h (~2m)= (U (S ~)) - (X (S ~)) - (C1 (N)) where U(So~) is the group qf units of N, and where (M) is the class in Go(ZIG]) of the module M. Endo and Miyata have shown (cfi [21, p. 501]) that h(D(Z[G]))=O. Therefore we have the following corollary:

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T. Chinburg

Corollary 3.4. I f Conjecture 3.1 is true, the class h(~2m) = ( U (S m)) - ( X (S~)) - (CI(N))

in Go(Z[G]) has order 1 or 2.1 Remark module We abelian

3.2. Corollary 3.4 may be interpreted as a conjectural analog for Galois structure of classical analytic class number formulas. conclude this section by discussing more refined results when N is over Q.

Question 3.1. I f K =Q, N is abelian over Q, and (N/K,S) is tame, is f2=07 Is this the case if N is also assumed to be at most tamely ramified over Q?

Theorem 3.3. (i) (Hilbert) I f K = Q and N/Q is abeliaft and at most tamely ramified, then f2a =0. (ii) Suppose that N/Q is an abelian extension of prime degree. Then f2 m= 0 ![ the "Gras conjecture" is true. Remark 3.2. The in [8] (resp. by (resp. at p = 2 ) is Iwasawa theory. of B. Mazur and

Gras conjecture is stated in [7]. It is shown by R. Greenberg R. Gillard in [-6]) that the Gras conjecture at primes p + 2 a consequence of a suitable version of the Main Conjecture of All of these conjectures will surely be proved by the methods A. Wiles in [13].

IV. The Friihlieh-Tate Theorem The G,, case of the FriShlich-Tate Theorem and its corollary are shown Throughout this section we will suppose that we are in the G, case. Let a be an element of N which generates a normal basis {ag}g~ Let T: G---,GL(n,Q.) be a matrix representation of the representation which has coefficients in an algebraic closure (~ of Q containing resolvent of a with V is defined by Fr6hlich ([4, p. 143]) to be

in [19]. of N/K. V of G N. The

(a IV) = det (~ a g T(g- ')). g

This depends only on a and the isomorphism class of V, and is a nonzero element of Q. Suppose that k is a subfield of K. Let A be a set of automorphisms of over k which is a right transversal of Aut(C/K) in Aut(C/k). One defines

N~/k(alV)= I-I (alVe-1)7

'

Then NK/k(alV) depends on the choice of A, but only up to a root of unity in the field of values of the character of V. Lemma 4.1. Let {mr}aEzI be a basis for the integers of K. L e t {W),}7~A be a subset of S such that {w.]K}~A is the set of embeddings of K into IlL Define d K to be the discriminant of K, and let r2(K) be the number of complex places of K. 1 J. Queyrut has shown (c.f. 1-14])that Conjecture 3.1 implies h(f2,,)=0

On the Galois structure of algebraic integers and S-units

329

Let as ldKlU2, and order the w.t so that det(w.,(m~))=d Let aeO N generate a normal basis .(or N/K. Define q%: X--~ U by qoa(gw~.)=agm~ for g~G and 7~A. Then -1/'2 d i m V R(V,~G)-(dK ) Nt;/a(aW) and A(V, ~oa)= NK/e(a IV)/~(V). Proof If V1 and V2 are two representations of G, and the lemma holds for two of the three representations V~, V2 and [email protected], then it also holds for the third. Hence by Brauer's theorem, we may reduce to the case in which V=Ind~)~ for some one-dimensional character )~ of a subgroup H of G. Let Y be a left transversal of H in G. For g e g and a s A , let f(g,a) be the element of HomG(V, I I ? | 1 6 2 1 7 4 ) for which f(g,a)(1)= ~)~(h)hg-lw~. Beh~fl

cause G = ~ H g -1, these f ( g , a ) form a basis of H o m H ( ~ , ~ |

). The matrix

of (2o~0)v: H o m H ( Z , , ~ | 1 7 4 ) relative to this basis is a block matrix, the blocks indexed by pairs (7, a) of elements of A. The (?, a) block is a [G : H] • [G : H] matrix equal to

w~(mo)( ~ w,~(at) T(t- 1))T. . . . pose t~G

where T is a matrix representing I n d ~ , . Thus R(V, qG)= det ((2 o qO)v)=det(w~,(m))fG:mNr/e(alV ). Because [G: H] = dim V and det(w~,(m~)) = d~/2, the formula R (V, qG) = (d ~"2)dim V NK/Q(a IV) is proved. Martinet shows in Theorem 8.1(iii) of [12, p. 54] that z (IndK/Q V) = (d~/2)dim

i7 Z"( g ) .

Therefore A(K q~,) = R(V, qG)/z(IndK/oV) = N~/o(alF)/z(V ).

Pro~( of the Fr6hlich-Tate Theorem in the G, case. Let V be a representation of G with character Z, and let M be a finitely generated O [ G ] module which is torsion free and has character Z- Remark 2.1 shows that to prove Conjecture 2.1, it will suffice to consider the case in which ~0=~G is as in L e m m a 4.1. Conjecture2.1 then follows from the expression A(Kq~)=Nr./e(alF)/r(V) and formula 3.6 of [5, p. 205]. Hence A(Kcp) for an arbitrary q~ lies in the field Q(z). Suppose now that N/K is at most tamely ramified. We are to show that A(V,~0)O=q~a(q)). It will be enough to show that if p is a prime of O, then A(V,q~)O~=qM(q~)p. By Remark 2.1, if this is true for one ~o, it is true for all ~0, so it is enough to show it for one ~0. Because N/K is at most tamely ramified, a theorem of Noether states that if p is the rational prime beneath p, there is a b~U so that (OK[G ] b)p= Uv. For this b we have (q~bX)p= Uv.

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Consider qM(~pb)= a(coker((~pb)u o nM))/a(ker((%)M onM)), where nM: (XM)a--~X ~ is induced by the norm, and (%)M: X ~ - * U~ is induced by q0b. Because X is a free ZIG] module, X M = H o m o ( M , X | ) is an induced Gmodule. Therefore nM is an isomorphism. Since (~pb)Xp=Up, we have (q~b)M(X~)~=(U~)p. Thus qM(q~b)~=Op. TO prove the theorem, it will hence suffice to show that A(V,~pb)O~=O p. By Lemma4.1, this is equivalent to z(V)O~=NK/eR(blV)Op, where the two sides of this equality are regarded as free rank one O~ modules inside some algebraic closure of the quotient field of O~. This last equality is shown in [4, p. 160], so the proof is complete.

V. Projective Euler characteristics The following proposition is proved by C.T.C. Wall in [22, Lemma 1.3] when the A i are projective G-modules. The proof for cohomologically trivial Ai is similar.

Proposition 5.1. Let C and D be finitely generated G-modules, and suppose that ~ E x t ~ ( C , D ) . Suppose also that there exist finitely generated cohomologically trivial G-modules A 1,..., A,, such that 0-~ D--~ An-~ . . . - ~ A 1--, C ~ O is exact and has extension class o:. Then the class ~ ( - 1 ) i ( A i ) E K o ( Z [ G ] ) depends only on C, D and ~, not on the choice of the A i. Remark 5.1. If A is a cohomologically trivial G-module, then A | is a free Q[G] module of rank over Q[G] equal to the rank of A. It follows that ( - l ) i (Ai)~CI(Z [G]) if n is even and C | Q and D | are isomorphic. In the remainder of this section we prove a special result concerning the case n=2. As in Sect. III, let E be a number field with ring of integers O that is sufficiently large so that for every irreducible character Z of G, there is a finitely generated torsion free O [ G ] module M x such that V x = M x | has character ZLet H (G), P (G) and the homomorphism t o s: CI(Z[G])--~H(G)/P(G) be as defined in Sect. III. We will make the following hypothesis on the finitely generated G-modules C and D. (5.1)

There is an exact sequence O-~ D --~ A 2 --~ A1--~ C - , O

in which the A i are cohomologically trivial, and there is a homomorphism f : C -* O including an isomorphism Q | C ~ Q | D.

Proposition 5.2. The function q ( f ) 6 H o m ( R ~ , I d ( E ) ) defined by q ( f ) ( z ) = q ~ ( f ) if Z is an irreducible character of G does not depend on the choice of the M x. Suppose that the character of C | Q is an integral combination of characters of representations of the form Ind~ 1H, where H runs over the subgroups of G, and where ltt is the trivial representation of H. Then q ( f ) lies in H(G) and has image t o s((A 1)- (A2)) in H(G)/P(G). Remark 5.2. It seems likely that the proposition is true without the hypothesis on the character of C | The result above is sufficient to prove Proposition 3.1 of Sect. III.

On the Galois structure of algebraic integers and S-units

331

The proof that the function q ( f ) does not depend on the M z is a straightforward extension of [19, L e m m a 6.7], so we will indicate only what changes are necessary in the proof. Let Z be an irreducible character of G, and let M = M z. It will be enough to show that if M' is an O [ G ] submodule of finite index in M, then qM(f) =q~r(f)Suppose that A is a finitely generated G-module. Let j: A M - - , A M, be the h o m o m o r p h i s m induced by M ' c M . Let .j~ 9 A M ~ ~ A M, and JG: (AM)G---*(AM')G be the homomorphisms induced by j. Then the kernel and cokernel of j~ and j~ are finite O-modules. Define qG(A)=a(coker(jG))/a(ker(ja)) and qG(A) = a(coker (jG))/a (ker (j~)). As in [19, L e m m a 6.7 and Theorem 6.4], one has _

qM(f)/qM'(f) = qa( C)/q~ (D) = qG( A O/qG (A 2)9 Since A 1 and A 2 are cohomologically trivial modules of equal rank, to prove q M ( f ) = q M , ( f ) , it will suffice to show that if A is a cohomologically trivial module of rank n, then q~ (A) = qG (A) = cr( M / M ' ) n This may be proved by reducing to the case A = Z [ G ] as in [19, L e m m a 6.5]. We now show that the assertion of Proposition 5.2 is independent of the choice of f. Let f ' : C--~D be a G-homomorphism ~ , Q| Then q ( f ) / q ( J ' ) e P ( G ) .

Lemma5.1.

Q| C

inducing an isomorphism

Proof Let Z be an irreducible character of G, and let M = M X. To simplify notation, we will use throughout the proof of this lemma C, D and f to denote (CM) ~, D~ and the h o m o m o r p h i s m fM ~ nM: (Cm)a---~ D~t, respectively. We have an exact diagram 0

1

kf

O-

, C,o ~ - ,

C --

0

, Dto r ---~-

D ~

0

, C/C,o ~ - - ,

0

l D/Dto r -

1

cf

c'f

0

0

>0

332

T. Chinburg

By the Snake Lemma,

q M(f ) = a( cf )/ ff (k f ) = a ( cf ) o"(Dtor)/o'(Ctor). Hence

q M(f)/qM (f') = a (c'f)/a (c'f,)= a (de t(t) O), where t is the endomorphism of ( C / C t o r ) | | ~ induced by the isomorphism f - 1 o f ' of Q | C. We thus find q(f)/q(f') from the following lemma. Lemma 5.2. Let h: Q | C--~Q| C be an isomorphism. Let b~Hom(RG, Id(E)) be

the homomorphism for which b(z)=det(tx)O if Z is an irreducible character of G, where t x is the endornorphisrn of C Mx 6 | C induced by h. Then b~P(G). Proof. It is shown in [19, Chapt. I.6] that det(t~)=det(t S for fl~Aut(r Suppose now that Z is an irreducible symplectic character of G. Then det(tz) is the reduced norm of an invertible element of a simple algebra component A of HomG( Q | C , Q | C), where ~ | is a matrix algebra over the quaternions. Hence det(tz) is totally positive, so b~P(G). To prove Proposition 5.2 when C and D are cohomologically trivial, we will need the following lemma, which is proved as in [15, Chapt. IX, Theorem 9]. Lemma 5.3. Let Z be a character of G, and let M = M x. I f A is a cohornologi-

cally trivial G-module, then A M is cohornologically trivial. Lemma 5.4. I f C and D are cohornologically trivial, then q(f)~H(G) has class

tos((AO-(A2))=tos((C)-(D))

in

H(G)/P(G).

Proof Suppose first that C and D are projective left ideals in ZIG], and that f : C--~D is the inclusion of C into D. Because C M is cohomologically trivial, riM: (CM)G--*(CM)6 is an isomorphism. Therefore qM(f)=a(coker(fM)), where fM: CM-~D G G M is induced by f From the definition of the reduced norm n Q[G] -~ Z(Q [G]), one finds that a( cok er( f M)) = 0 |

O (z) e z (n( C)/n( D )),

where e z is the idempotent corresponding to Z, and where O(z) is the ring of integers of the field generated by the values of Z- This proves the lemma in this case. Now suppose that C and D are projective or rank rn + 1 for some rn > 0. By a theorem of Swan (cf. [17]), there are projective left ideals 1 I~_I2c_Z[G ] so that C and D are isomorphic to Z[G][email protected] and Z[G][email protected], respectively. As we have shown that the truth of Lemma 5.4 does not depend on the choice of f : C--,D, we may assume that f is the identity (resp. inclusion) on the Z[G]" summand (resp. the I~ summand) of C. The desired result now follows from the rn = 0 case. Assume now only that C and D are cohomologically trivial. We may find projectives Po, Po, P~ and P~' so that we have an exact diagram

On the Galois structure of algebraic integers and S-units

0--

0

,E

'Po

- - ~ P~'

333

~c

'Pd . . . .

,0

D -------~ 0

By redefining the P/, P/' a n d f/ a p p r o p r i a t e l y , we m a y assume that f, Jo and j ] have finite kernels and cokernels. The S n a k e L e m m a now shows t h a t q(f) = q(fo)/q(fl). Since

( C ) - (D) = (Po)- (Po)- ((P,)- (Pl')), the assertion of the l e m m a now follows from the case of projective C a n d D. T h e following l e m m a is shown as in [ t 9 , Sect. II.6.1~. L e m m a 5.5. Under hypothesis (5.1), we have an exact diagram

0

,P~u

, (A2)~4

---, (A,)~

((A 2)M)G - - - - - ~

((A 1)M)G . . . .

--~ ( C M ) G

~.0

in which the two vertical arrows are isomorphisms induced by the norm. L e m m a 5.6. With the notations of hypothesis (5.1), whether q(f) is in H(G) and

has class t os((A1)-(A2) ) in H(G)/P(G) depends only on the character of Q | C. Suppose that q(f) has these properties, and that G is a subgroup of almther group T. Then 0 - + Ind r D ~ I n d r A2--~ I n d r A , -~ Ind r C -+ 0

is an exact sequence of T-modules, in which the Ind~A~ are cohomologically trivial. Let IndT: Indcr, C - - , I n d ~ . D be induced by f. The,7 q ( I n d ~ f ) i s in H(T)and has class tos((Ind~A~)--(Ind~A2) ) in H(T)/P(T). Proof. Let 0 -~ D ' - ~ A'2 -.-~A'1 -~ C' --* 0 be an exact sequence in which the A'~ are c o h o m o l o g i c a l l y trivial, and suppose that C'| D'| C| a n d D| are all i s o m o r p h i c . By a d d i n g free m o d u l e s o n t o A , , A 2, A', and A 2, we m a y assume that Q| Q| Q| and Q| are all isomorphic. W e m a y then construct an exact d i a g r a m t

!

0--~D'--

,A 2-

~A I . . . . .

0

~A2--

~A t

-~D

~C'--~0

--~ C - - - - - ~ 0

in which all the vertical a r r o w s have finite kernels a n d cokernels.

334

T. Chinburg

F r o m L e m m a 5.5 we now get a diagram 0

,o '(AgM

*(DM)a -

0~

D~, - - - - - ,

(A ~)~, ~

, '((A1)M)G

, -'(CM),~

((A ~)~)~ ~

(CM)~ - - - - ~

-,0

0

in which the vertical homomorphisms have finite kernels and cokernels. Let h: C ~ D' be a G-homomorphism inducing an isomorphism Q | C ~- Q | D', and define f ' = h onc: C'--, D' and f = ~o ~ h: C - ~ D. Then we have a diagram

,(C'~)G ----~(D'~)~----~c•

O ~ k f , -

1

0------~ k s --

, (CM) G ~

(DM) a - - - ~

CI ---------* 0

Applying the Snake L e m m a to these last two diagrams shows that

q(f')/q(f)=q(rcO/q(r~2). By L e m m a 5.4, q(nl)/q(n2) is in H(G) and has image tos((A'O-(AO+(A2)-(A'2) ) in

H(G)/P(G).

Thus q(f) is in H(G) and has class tos((AO-(A2) ) in H(G)/P(G) if and only if the same holds for f ' , A'1 and A~. Because of L e m m a 5.1, this shows that whether q(f) has these properties depends only on the character of Q | C. Consider now q(Ind~rf). Because Z[T] is flat over Z[G] we have the exact sequence of induced modules stated in L e m m a 5.6. Suppose that M = M z for some character )~ of T. Then (Ind T C)M=IndGr(C)res~ M, where resGr is restriction. F r o m this one finds that q(Indrf)OO=q(f)(resrD. Now by [4, Sect. 16] there is an exact diagram CI(Z [G]) , H(G)/P(G) Ind r

CI(Z I T ] ) in

which

the

two

horizontal

, H(T)/P(T) arrows

denote

tos,

and

in

which

H(G)/P(G)-,H(T)/P(T) is induced by resT: RT-*R a. Hence q ( I n d r f ) has the required properties for T if q(f) does for G. Proof of Proposition 5.2. Let C, D, As, A 2 and f be as in the statement of the proposition. By adding a free module F of sufficiently large rank to each of C, D, A 1 and A:, and by extending f to f : C O F - * D @ F by letting f be the identity on F, we m a y assume that the character of Q | C is a positive integral combination of characters of modules of the form I n d ~ l n. Because we have shown the truth of the proposition depends only on the character of Q | C, and because the proposition in clearly compatible with direct sums, it will suffice to show that when C = I n d ~ l n there exist D, A1, A 2 and f for which the proposition holds. Since L e m m a 5.6 shows the proposition is compatible with induction, it will suffice to consider the case H=G, i.e. where C = Z with trivial action.

O n the G a l o i s s t r u c t u r e of a l g e b r a i c i n t e g e r s a n d S - u n i t s

Let g be the order of G, and let e = -

335

~7+g.

1. Then e/g is the sum of the

yeG

idempotents in (r [G] of the non-trivial irreducible representations of G. Define ml: Z[G]-+Z by m1(7)=1 for 7eG. Let T be the free Z[G] module on elements x 0 and x~ for 7eG. Define m2: T--,Z[G] by m2(xo)=e and mz(X~,)=7 - 1 for 7EG. Let T~ be the free ZIG] module on elements y~. for 7eG. Define m3: T 1 ---~T by m3(y~)=( 7 - l)x o -gx~.. Let F = T/m3(TO. Now T

m2 , Z I G ]

"'

,Z--

,0

is exact, and m 2 o m 3 ( T 0 = 0 . Therefore we get an exact sequence

0

m2

>D--

,F---+Z[G]--

ml

,Z------+O

in which m2: F--~Z[G] denotes the map induced by m2: T-~Z[G], and D =ker(m2). Clearly F is cohomologically trivial, of rank 1, and (F)-(Z[G])=O in Ko(Z[G]). Therefore D| and what we wish to show is that if f : Z - + D induces an isomorphism Q ~-D| then q(f)eP(G). Let Z be an irreducible character of G, and let M = M z. If Z is trivial, we may take M=Z, and there is nothing to be shown regarding qM(f). So suppose Z is non-trivial. Because m 3(T 1)c~Z[G]x 0=0, the quotient homomorphism T-+F = T/m3(T0 induces an injection Z[G]xo~F. Therefore we get a diagram 0----+

0

D

>

F

----+

T

,Z--:--*Z[G]xo-

z[o]-

+z-

,ziG]e--

,0

,o

T

0

0

By L e m m a 5.5 this induces a diagram

o

0

0

C1

C2

- - +

,Df, m

T

0--

,z[e]~

0

((z [6])M)~

--,((z [e] e)~,)~--

0

, (ZM) ~

i

> 0

+ o

336

T. Chinburg

Here D~ and (Zu)r are finite because Z is non-trivial. We know that Z [ G ] C ~ F ~ is injective with finite cokernel because Z[G]xo-~F is. We have Z[G]~--~((Z[G]e)M)a because (ZM)c' is 0. Now Z[G]~ is torsion free, so ((Z [G] e)M)G is torsion free. Hence the homomorphism ((Z[G]e)M)~-~((Z[G])M)G must be injective, since otherwise the rank of the image would be too small for D~ to be finite. We have by the Snake Lemma that qM(f)=a(D~)/a((ZM)~;)=a(Cl)//a(C2). Now a(Cx)=qM(i ) where i: Z[G]xo--~F is the vertical homomorphism of the top diagram. By Lemma5.4, q(i)6P(G). Consider a(c2). Because e = - ~ 7 yeG

+ g . 1 and Ma=O, a(c2) equals a(coker(h'M)), where h: Z[G]--~Z[G] is multiplication by g and h]~: (Z[G]M)G---~(Z[G]M)G is induced by h. Because the norm nM: (Z[G]ta)G--,(Z[G~M) a is an isomorphism, we find that a(cz)=qM(h ). Now q(h)eP(G) by Lemma 5.4, so we conclude that q(f)EP(G). This completes the proof.

VI. Invariance of ~ , . with S We will suppose in this section that we are in the G,, case and that the set of places S of N is large enough so that (N/K,S) is tame. By Lemma 3.1, there are cohomologically trivial modules A and B for which there is an exact sequence

O-+ U - , A--~ B--. X-+ O whose extension class a in Ext,(X, U) is the canonical class. By Proposition 5.1, the class (A)-(B) in CI(Z[G]) does not depend on the choice of A and B. Let P be a finite place of N not in S, and let H be the decomposition group of P in G. Define S' to be the union of S with the conjugates a P of P for aeG/H. Let U' and X' equal U and X, respectively, when S is replaced by S'. Clearly (N/K,S') is tame because (N/K,S) is. We will construct cohomologically trivial modules A' and B' and an exact sequence

0-~ U' -~ A' ~ B' ~ X'-~ O whose extension class a' in Ext2(X ', U') is the canonical class of S', and for which (A')-(B')=(A)-(B). This and Proposition 5.1 wilt show that if (N/K,S) is tame, the class (A)-(B) does not change when one enlarges S, and so this class is independent of S. This will complete the proof of Theorem 3.1. Lemma 6.1. The group H is cyclic. There is a place vowS fixed by H, and a 7 in U' fixed by H which has valuation 1 at P and which has valuation 0 at all of the other conjugates of P. The modules W= ~ a ( P - v o ) Z and T= ~ aTZ are r

a~G/H

isomorphic to Ind~ 1u as G-modules, and we have X' = X • W and U' = [email protected] T. Proof Because S contains all the ramified primes of N/K, P is unramified, and so H is cyclic. As in [19, Theorem II.6.8], we have that H - t ( H , X ) = O because

On the Galois structure of algebraic integers and S-units

337

the S-class number of N n is 1. Hence H I ( H , X ) = O because H is cyclic. From the H-cohomology of the sequence O - - ~ X - ~ Y - . Z - - ~ O we have Y n - + Z H = Z - ~ H I ( H , X ) = O . Therefore there must exist a v o c Y u whose image under Y--, Z equals 1, and we may take this v0 to be a place in S. Because the S-class number of N u is 1, there is a 7 ~ N n ~ U ' with the claimed properties. It is clear that W = ~ a ( P - v o ) Z and T = ~ crTZ are cr~G/tl

~G/H

isomorphic to Z [ G / H ] = I n d ~ I n as G-modules. Since P is not in S, X ~ W ={0} and U ~ T = { 1 } . Because X + W = X ' and U + T = U ' , we have direct sum decompositions X' = X | W and U' = U | T. Corollary6.1. An element of EXtG(X,U' 2 , ) may be represented by a matrix

L e m m a 6.2. Let Np be the completion of N at P,, and let c~e be the local canonical class in HE(H,N*). Let re: N * - + Z be the valuation homomorphism. Let v: N ~ - ~ H o m z ( ( P - v o ) Z , T ) be the homomorphism which sends t~N* to the homomorphism induced by ( P - Vo)--~ (7)~'~"). Define 6~ Ext~ (W, T ) = H 2 (G, Homz(W, T))= H2(G, H o m z ( I n d ~ ( P - vo)Z, T)) = HZ(H, H o m z ( ( P - vo)Z, T)) to be the image of 7p under v. Then the canonical class ~' in ExtZ(X ', U') has the form

relative to the representation qf ExtG(X 2 , ,U') in Corollary6.1, where ~ is the canonical class in ExtZ(X, U), a~d fl is some element of ExtZ(W, U). Proof Let C be the idele class group of N. Recall that Y denotes the free abelian group on the places in S. Let J be the group of S-ideles of N. Let Y' and J ' be Y and J, respectively, when S is replaced by S'. Define exact sequences (X), (W), (T) and (U) as follows: (X): O ~ X ~ Y ~ Z ~ O (W): 0 - ~ W ~ W--~0 --,0 (T): O ~ T---. T - ~ O --,0 (U): 0-~ U ~ J

~C--,O

Define (X') and (U') to be (X) and (U), respectively, when S is replaced by S'. If (A) and (B) are any two of the sequences above, we define Hom((A),(B)) to be the group of compatible triples of homomorphisms (fl,f2,J3) between the terms of (A) and (B): O--

,A 1 - -

~A 2

~A 3 - -

0 - - - - - - * B 1- -

~B 2

~B 3- - - - ~

~0

0

338

T. Chinburg

We have X ' = X O W and U ' = U O T Further, Y ' = Y O W and J ' = J O T . It follows that the additive group Hom((X'),(U')) is isomorphic to the additive group of matrices

(~) (6)1 in which (~)eHom((X),(U)), (fl)eHom((W),(U)), (rc)eHom((X),(T)) and (6)eHom((W), (T)). We will now recall from [19] the definition of the canonical class (a)eH2(G, Hom((X),(U))), and how it gives rise to the canonical class oceH2(G, Horn(X, U)). Let it 1, n2, n3, a and b denote the homomorphisms indicated in the following diagram: Hom((X), (U))

, Horn(Z, C)

~'

H o m ( Y , J ) - - a , Horn(Y, C)

~r3

Horn(X, U)

,

Hom(X,J)

)Hom(X, C)

We have an exact sequence 0

, Horn ((X), (U)) =2•

a--b * Hom(Y, C ) - -

Hom(Y, J) x Hom(Z, C)

,0

In the cohomology of this sequence, we have H t (G, Horn( Y, C)) = ( ~ H I (G, Hom(Ind~o 16~, C)) veSo

= @H'(G~, C)=0, veSo

where S O is a set of places in S which are representatives for the distinct orbits in S under the action of G. Therefore we have an exact sequence (6.1)

O-

, H2(G, Hom((X),(U))) ~2 • ~Zl

,H2(G, Hom(Y,J))xH2(G, Hom(Z,C))

_ a-b ~HZ(G, Hom(Y, C)). Let %eH2(Gv, N *) be the local canonical class at yeS. Let it,: N * - * J be the natural inclusion. The class ~2eH2(G, Horn(Y, S)) = @ H2(G, H o m ( I n d ~ 1Gv,J)) veSo

= (~H2(Gv, J) veSo

On the Galois structure of algebraic integers and S-units

339

:~leH2(G, C)=H2(G, Hom(Z, C)) is

is defined to be the sum @ i~(%). The class veSo

defined to be the canonical class of global class field theory. From the relation between local and global class field theory, one has a(0r = b(0r Therefore the sequence (6.1) above shows that there is a unique class (oOeH2(G,Hom((X),(U))) such that 7tl((~))=a 1 and ~2((~))=a2. This (~) is the canonical class of H2(G,Horn ((X), (U))). The canonical class of HZ(G,Horn(X, U)) is defined to be a = % =~3((a)). The canonical class (~')eH2(G,Hom((X'),(U'))) is constructed in the same way as (~) on replacing S by S'. From the sequence (W): 0-~ W-~ W - * 0 - ~ 0 we have isomorphisms iw, r: Hom(W,T)-+Hom((W),(T)) Horn(W, U)--* Hom((W), (U)). Let (3) = iw,r(5). Consider

and

iw,v:

for

some

((~) (O)]6H2(G,Hom((X,),(U,))).

(b)=(~')- \(0) (6)f We wish to show that

((o) (b)= \(0) for

some

(fi)EH2(G,Hom((W), (U))).

(0)] Writing

(fl) =

iw.v(fl)

fleH2(G, Hom(W, U)), we will then have that a'=rc3((a')) equals (~ \t/ will be finished. We have as in the unprimed case a diagram Hom((X'), (U'))

~2

~) and we

Horn(Z, C)

"4 ~

Hom(Y', J ' ) - "' , Hom(Y', C)

Hom(X', U ' ) - -

,

Hom(X',J')--~

Hom(X', C).

From the definitions of (~'), (e) and (b) we have ~'1 ((b)) = ~'a ((~')) - 7r, ((~)) = 0

where ~'1 ((~')) and ~za ((u)) equal the global canonical class in Ha(G,Horn(Z, C)) = H a ( G , C). We now compute ~((b)). Recall that S o is a set of representatives of the orbits under the action of G in the set of places S. Since Y' = @ Ind~vZOInd~uPZ, we have v~So

340

T. Chinburg

HZ(G, Hom(Y',J')) = @ H2(G~,,J')G HZ(H,J ')

(6.2)

wSo

where H2(G,Hom(Ind~vZ, J'))=H2(G~,J ') by Shapiro's lemma. Relative to this isomorphism, we have 7r~((~')) = @ k.(~,)G ip(~g) v~So

(,o) Oo)

~z2

= @ i~,(%)@0. veSo

We claim that (6.3)

7Z'2

(~)

wSo

where kv: N ~ J ' sends teN~ to (?)v'(t)EJ'. To show this, we may use the isomorphisms Y'= Y| and J'=JQT to represent an element of HZ(G, Hom(Y',J')) by a matrix "r

in which ~HZ(G, Hom(Y,J)), z6HZ(G, Hom(W,J)), O~H2(G,Hom(Y,T)) and (~HZ(G, Hom(W, T)). Relative to this representation, we have

o)):(o ~ o). C o n s i d e r t h e m a t r i x (~

~) for [email protected](~p)eHZ(G, Hom(Y',J')).Since the v~So

local components of h at w S o are trivial, 3 = ~=0. The image of kp: N ~ - , J ' is contained in T = I n d ~ ? Z , so z=O. We have an isomorphism (6.6)

I n d ~ ~voZO I n d ~ P Z = Ind~v~voZ| I n d , ( P - vo)Z.

Since the vo component of h is trivial, the projection of h into

HZ(G, Horn (W, T)) = H 2(G, Hom(Indg (P - vo)Z, T)) : H2(H, Hom((P - vo)Z, T)) is jp(ap), where jp: N~,-~ H o m ( ( P - vo)Z, T) sends teN~ to the homomorphism ft for which f ( P - v o ) = ( ? ) v'('). This je(~p) is just b~H2(G, Hom(W,T)), so Eq. (6.3) follows from (6.5). We now have that

,

, (

((~) (0)]] = |174 (6)11 wSo

n2((b))=~2 ( a ' ) - \(0)

We claim that the matrix (6.4) of ~2((b)) has the form

,67)

On the Galois structure of algebraic integers and S-units

341

for some rleHZ(G, Horn(W, J)). To see this, note that the first column of rCz((b)) is trivial because the components of 7r'2((b)) at yeS o are trivial. Since 7 has valuation l at P, ip-kp: N ~ - , J ' has image in J. Therefore (ip --

kp)(O~v)GH 2(H, Hom (PZ, J)) = H 2(G, Horn (Indg PZ, J)).

Now using the isomorphism in Eq. (6.6) and the fact that the vo component of rr2((b)) is trivial, we find that the projection of rrz((b)) into HZ(G, H o m ( I n d g ( P - vo)Z, J')) = H2(G, Horn(W, J')) lies in H2(G, Hom(W,J)), which shows (6.7). By a construction analogous to that giving the sequence (6.1), we have an exact sequence O _ _ _ ~ H2(G,Hom((W),(U)))

,~2 , H2(G, Hom(W,j))

" ,H2(G, Hom(W, C)). Now r(2((b))eH2(G, Hom(W,J)) and arc'2((b))=b'rc'l((b))=O. Hence there is a (b')eH2(G, Hom((W),(U))) so ~2((b't)=~c~((b)). Clearly rrl((b'))=0, and we have shown ~'l((b))=0. We regard H2(G, Hom((W),(U))) as a summand of H2(G, Hom((Y'),(J'))). The exactness of 0 --

, , ,'z • ~'l , H2(G, Hom((Y),(J)))------~HZ(G, Hom(Y',J')) ,-b , H2(G, Hom(Z, C))

shows that we must have (b)=(b')~H2(G, Hom((W),(U))). As we have shown, this proves Lemma 6.2. We will make use of the following result in constructing sequences with prescribed extension class. Let (6.8)

O ~ D--~ A 1~ A2 ~

C--->O

be an exact sequence of finitely generated G-modules. Suppose that f : D-+D' is a G-isomorphism, and that (6.9)

O - , D'--~ A 1 --+ A 2 -+ C - + O

is the sequence which results from (6.8) on identifying D with D' via f The cokernel K 1 of the homomorphism D - , A 1 in is isomorphic to the kernel of the homomorphism A 2-~ C. We have connecting homomorphisms 1 Hom6(D, D )--, Ext~(K 1, D') and Ext~(K 1, O') ~ Ext,(C, O') whose composition is the double connecting homomorphism d2: Hom~s(D,D')-* ExtZ(C,D'). Lemma6.3. The extension -dz(f)eExtZ(C,D'). Proof See [11, p. 97, 73, 66].

class

of

the

sequence

(6.9)

is

equal

to

342

T. Chinburg

Lemma 6.4. There is an exact sequence

0--~ T-~Z[G] --,ZIG] - . W ~ O whose extension class in Ext2(W, T ) = H2(G, Hom(W,, T)) is the g~of Lemma 6.2. Proof With the notations of Lemma 6.2, let lp: Nff,~Hom((P-Vo)Z, TZ ) be the homomorphism which sends t~Nff, to the homomorphism induced by (P - V o ) ~ ( j ~P(~ Let i : 7Z--~T be inclusion. Then the homomorphism v of Lemma 6.2 is i~ o le. By definition, 6 = i~o Ip(c~p)~H2(H, H o m ( ( P - - vo)Z, T)) = H 2 (G, Horn(W, W)) = Ext,(W, T). By local class field theory, % is a generator of H2(H, N*), and lp induces an isomorphism

lv: H z (H, N*)--~ H 2(H, Hom ((P - v0)Z, ? Z), where Hom((P-vo)Z, yZ ) is isomorphic to Z as an H module. Let h o be a generator of H, and let e~ be the identity element of H. Consider the sequence (6.10)

O-~ Z - ~ Z E H ] -. Z [ H ] -~ Z - . O

in which the non-trivial homomorphisms are induced by 1--* ~ h, eH - , ( h o hen

-ell) and en---*1, extension class of choose h o so that with ?,Z (resp. ( P -

respectively. We may choose the generator h o so that the this sequence is any prescribed generator of H2(H,Z). We on identifying the leftmost Z (resp. rightmost Z) in (6.10) vo)Z), the extension class of the resulting sequence

O-~ ~Z--, Z[H]--~ Z [ H ] - ~ ( P - vo)Z--~O is the generator Ip([email protected] of Ext~((P-vo)Z, TZ ). The induction to G of this last sequence is of the form required by Lemma 6.4.

Completion of the proof of invariance of f2,n with S As in the beginning of this section, we suppose that

O-* U-~ A --~B-~ X--~ O is exact with extension class the canonical class eeExt2(X, U), and that A and B are cohomologically trivial. To this sequence we sum the sequence of Lemma 6.4, giving a sequence (6.11)

0--, U|

A G Z [ G ] ~ B|

X|

On the Galois structure of algebraic integers and S-units

343

In the notation of Corollary 6.1, the extension class of this sequence is ( ; The sequence

0). 7

O--~ T--, Z[G]-+ Z[G]-~ W ~ O gives rise to a double connecting homomorphism d2: Home(T, U)--, Ext,(W, U) which is surjective because Ext~ (Z [G], U) = Ext~(Z [G], U) = 0. Hence we may find a G-homomorphism r T-*U so that - d 2 ( 0 ) is the fleExt~(W,U) of Lemma6.2. We now define the G-homomorphism f : U @ T--* U @ T by the matrix

where Idv: U--~ U and IdT: T-*T are the identity. This f is an isomorphism. We construct a new sequence (6.12)

0--, U @ T-~ [email protected][G] --, B|

~ X | W-.O

from (6.11) by identifying U O T with [email protected] via f Lemma6.3 shows that the extension class of (6.12) is

which by Lemma 6.2 is the canonical class e ' e E x t 2 ( X Q W , U•T)=Ext(X', U'). Thus (6.12) is an exact sequence

O--~ U'-~ A'--~ B ' ~ X ' - . O with canonical extension class, and clearly (A')-(B')=(A)-(B) in K o(Z [G]), so the proof is complete.

VII. Symplectic representations In this section we prove the results on symplectic representations of G stated in Proposition 3.3 and Corollaries 3.1, 3.2 and 3.3 of Sect. 3. Suppose first that we are in the G a case, and that N / K is at most tamely ramified. Lemma 5.2 shows that the assertion of Proposition 3.3 is independent of the choice of ~o. Proposition 3.3 now follows from the last formula of Lemma 4.1 and [4, p. 145]. Corollary 3.1 of Sect. 3 now follows from Proposition 3.1, 3.2 and 3.3. Corollary 3.2 of Sect. 3 results from Corollary 3.1 and Theorem 3.2. This completes the discussion of the G a case. In the remainder of this section, we will suppose we are in the Gm case, and that (N/K,S) is tame. Because A(V, cp)=R(V,q~)/Cv, Proposition3.3 is a consequence of the following lemma.

344

T. Chinburg

Lemma 7.1. Let V be a symplectic representation of G. 111 the G,, case, R(V, q) is

a positive real number, and cv=lims-~s(V>Ls(s, V) is a real number of sign equal to w(v). Proof Because we are in the G,, case, 2: C | 1 7 4 sends IR| to R| Thus R(V,q)=det((2oq))v), where (2oq~)v is the endomorphism of H o m e ( V, ~ | X) induced by the G-homomorphism 2 o q~: lR | X --~ ~ @ X. To prove the lemma, it will suffice to consider the case in which V is irreducible. Because V is symplectic, det((2o~p)v) is then the reduced norm of an invertible element of a simple algebra component A of Hom~tG~(IR| IR| such that A is a matrix algebra over the quaternions. Hence R(V,q)=det((2 o q))v) is real and positive. Consider c v =lims-r~ Ls(s, V). Here s~0

Ls(s , V) = L(s, V) [I det(1 - N(p)-S Frob(p) i VI) vLs

is L(s, V) with the Euler factors at primes p o f / s determined by finite places in S removed. The eigenvalues of Frob(p) of V t~p> are either -t-1 or occur in complex conjugate pairs. A computation now shows that c v has the same sign as the leading term c v in the expansion of L(s, V) at s =0. The functional equation of L(s, V) has the form

A(1 -s, V)= W(V) A(s, F) where A(s,V)=A(V)~/2)~v(s)L(s,V ), A(V) is a positive real number, )~v(S) is a product of functions of the form n-~/2F(s/2) and n- 0, we find that (7.1)

sign (Cv)= sign (Cv) = sign (W(V))

which completes the proof of Lemma 7.1 and Proposition 3.1. Corollary 3.1 of Chapt. 3 in the G,, case now results from Proposition 3.3. To complete the proof of Corollary 3.3, we must produce an example in which there is a symplectic V for which there is no unit ev~Q()~v) such that W(V')e~>O for all a~Aut(tE/Q). The generalized quaternion group H4, of order 4n is the group on 2 generators a and t with the relations a " = t 2, 14= 1, and t a t -~ = a - ~ . Proposition 7.1. Suppose that G = G a l ( N / K ) is isomorphic to H24. Let V be one

of the two faithful irreducible symplectic representations of G. Let N f ( V ) be the absolute norm of the conductor f ( V ) of V, and let )~v be the character of V. Then Q(Zv)=Q(]/3), and Q ( ~ ) - - Q or Q ( ~ ) = Q ( F / 3 ) . If Q ( ~ ) =Q(]f3), there is no unit e in Q(]/3) such that e:A(V:,~o)>O fi)r all a in

On the Galois structure of algebraic integers and S-units

345

Aut(~2/Q). Furthermore, there exists atz example in which Q ( ~ ) = Q ( ] / 3 )

and K = Q. Proof. The representation V has the form IndcG27` for a faithful one-dimensional character Z of the unique cyclic subgroup C~2 of order 12 in G. Therefore Q(z)=Q(exp(2~zi/12)), and Q(Zv) is the real subfield Q(I/3) of Q(z). If eeAut(C/Q) fixes Q(I~3), then V~ and V are isomorphic. Then W(V~)= W(V), so Theorem3.2 of Sect. 3 implies that (Nf(V)I/2)~=Nf(V)I/2. Therefore Q(Nf(V) 1/2) is either Q or Q(I/3). If Q(Nf(V)~/2)=Q(]~3), Proposition 3.3 and Theorem 3.2 of Sect. 3 show that A(V,~o) and A(VL~o) have opposite sign if c~eAut(Ir/Q) induces the non-trivial automorphism of Q(I/3). Because the fundamental unit 2 + 1 ~ of QQ/3) is totally positive, there can be no unit e in Q(]fl3) such that eA(V,q~)>0 and e~A(V~,~o)>O. To complete the proof of Proposition 7.1, we must construct an example of the required kind when K =Q.

Let F=Q(]/21-). The integers O of F equal Z+Zw, where w=(1 +1/~2i)/2. The class number of F is 1, and e = 2 + w is a fundamental unit. The norm form of F is Nv/a(a+bw)=aZ+ab-5b2. Define # = ( 1 - ] f f l - ) / 2 , P 3 = I + w , p 7 = 3 +w, /57 = 3 + # , P37 = 6 + w , and /~3v = 6 + # ; these have norm - 5 , - 3 , 7, 7, 37 and 37, respectively. Let m be the O ideal 3 . 5 . 3 7 . p 7 0 =p2. w- # "P7 "P37 "/~37 O. We will construct a ray class character 7` of F of conductor m such that the field N corresponding to the kernel of 7. via class field theory is a H24 extension of K=Q. This 7` will then have order 12, and V=Indv/eZ will be a faithful irreducible symplectic representation of G = Gal(N/Q). The formula for the conductor of an induced representation gives

f(V) = Ne/q()c(Z))dv/Q = NF:r (m) 21 =32523727.21=335272372. Since Nf(V)=f(V), we will have Q(Nf(V)~/Z)=Q(I/3 ), so N and V provide the example sought. Let Clm(F ) be the ray class group m o d m of F. Then Gal(F/Q)= {1, 7} acts on CI,,,(F) and on the characters of CI,,(F). Let M=NF/Qm= 3 z 5 z 7372, and let CIM(Q) be the ray class group of Q modM. Then we have a natural homomorphism i: CIM(Q)-~Clm(F ) induced by pZ-*pO if pZ is a rational fractional ideal prime to M. The conditions that the character Z of Clm(F) must satisfy are

(7.2)

7` has order 12 and conductor m.

(7.3)

)(=){-~, where Z~ is Z acted upon by the nontrivial element 7 of

Gal(F/Q). (7.4)

the restriction of Z to i(C1M(Q)) is non-trivial.

346

T. Chinburg

The first two conditions insure that the field N corresponding to the kernel of Z via class field theory is a C12 extension of F which is Galois over Q such that Gal(F/Q) acts by inversion on Gal(N/F). There are two extensions of Gal(F/Q)= C 2 by Gal(N/Q)= Ctz with this action, namely the dihedral group D24 and the quaternion group H24. Those may be distinguished by the fact ,",ab = C12 is trivial, while ver: H ~ - , C ] ~ = C12 is that the transfer ver: rnab , 2 4 ~ t~12 nontrivial. By class field theory, the non-triviality of the transfer amounts to condition (7.4) above. Because F has class number 1, to give a character of CIm(F) is the same as giving a character of (O/m)* which is trivial on the image of O* in (O/m)*. We specify such a character as follows. Each ideal f of O in the table below is a power of a prime ideal exactly dividing m. The integer g(f)eO is a generator of the group (O/f)*, which is cyclic for these f. The entry gr(g) is the value of Z on the unique residue class g m o d m for which g - g ( f ) m o d ( f ) and g = 1 rood(m/f). This data is enough to specify a character of (O/m)*. Table 7.1. Construction of Z

f

p~O

wO

ff~O

P70

P37 O

/~37 O

g(f)

1-w

2

2

3

2

2

Zf(g)

enl/3

eni/2

e -~i/2

-- 1

e 2~i/3

e 4~I/3

It is straightforward to verify that Z is trivial on units, so that it gives a character of Clm(F ), and that this character satisfies conditions (7.2) and (7.3). To check (7.4), note that the value of g on the residue class of 2 m o d m equals - 1. This completes the proofs of Proposition 7.1 and Corollary 3.3.

VIII. The i m a g e of 12 m in Go(ZIG]) In the next two sections we prove Proposition 3.4 and Theorem 3.3(ii) of Sect. 3. (Theorem 3.3(i) is shown by Hilbert in [9, Satz 136]). Let Go(ZIG]) be the Grothendieck group of finitely generated ZIG] modules, and let h: Ko(Z[G])~Go(Z[G]) be the Cartan homomorphism. Let X(S~) be X for the set S~ of infinite places of N. Let CI(N) be the ideal class group of N, and let U(S~) be the unit group of N. To prove Proposition 3.4, we are to show (8.1)

h (Y2~,)= ( U (S 0o))- (X (S ~o))- (C1 (N)).

Let S be a set of places of N which is large enough so that (N/K,S) in the G,. case is tame. Then there are cohomologically trivial modules A and B and an exact sequence

O-~ U--~A--~ B-+ X-~ O whose extension class is the canonical class. From this we have (8.2)

h ((2m) = h ((A) - (B)) = (U) - (X).

On the Galois structure of algebraic integers and S-units

347

Let S s be the set of finite places in S, and let I(SI) be the group of fractional ideals of N with support in S I. Because (N/K, S) is tame, the S-class number of N is 1. Therefore we have an exact sequence

0 - , U (S ~) ~ U - , I (S I)--, CI( N)--~O in which the h o m o m o r p h i s m I(SI)--,CI(N ) sends an ideal p~S s to its ideal class. From (8.2) we now have (8.3)

h ((2,,) = (U(S ~)) + (I (Sy)) - (C1 (N)) - (X).

Let Y(So~ ) be the free abelian group on S o. Then the free abelian group Y on the places in S is isomorphic to Y(S~)O1(Sy), and we have exact sequences

O ~ X ( S ~,)~ V ( S ~ ) - . Z - ~ O O~

X

~

Y

-~ Z -~ O

Therefore in Go(ZIG]) we have

(I (S s)) - (X) = ( Y) - ( Y (S ~,)) - (Y) + (Z) = - ( X (S 6,)). Combining this with (8.3) proves (8.1), so Proposition 3.4 is proved.

IX. The prime order case over Q

In this section we will assume that K = Q , that G has prime order q, and that the " G r a s conjecture", which will be recalled below, is true. Theorem 3.3(ii) of Sect. 3 then asserts that ~?m=0. We will make use of the following result of Reiner (cf. [18, p. 74]). L e m m a 9.1 (Reiner). I f G is a cyclic group of prime order, the Cartan homomor-

phism h: Ko(Z[G])--~Go(Z[G]) is injective. In view of this lemma and Eq. (8.1), it will be enough to show that (9.1)

(U(S~))-(X(S~))-(CI(N))=O

in

Go(Z[G]).

Suppose first that N is a complex abelian extension of Q. Since I N : Q ] = q is prime, we must then have q = 2 . Because C I ( Z [ G ] ) = 0 if @ G = 2 , we must have that ~2,,= 0. In what follows, we will assume N is a real abelian field. Because N is real, the torsion in U(S~) is { +_ 1}. Define I U(S~)l= U(Soo)/ { + 1}. In [7] there is defined a G-submodule F~ of finite index in ]U(S~)I which is generated over ZIG] be a single element f~EF~. (This f

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