On multiplicative Galois structure invariants.

Introduction. Let N/K be a finite Galois extension of number fields of group G. Let [K.sub.0](Z[G]) denote the Grothendieck group of finitely generated projective (left) Z[G]-modules, with Cl(Z[G]) the subgroup of elements which have rank 0. In [C1] Chinburg used the four term exact sequences of prescribed extension class introduced by Tate in ([T2], pp. 56-57) to define an invariant [[omega].sup.m](N/K) [element of] Cl(Z[G]) of the multiplicative Galois structure of N. (The class [[omega].sup.m](N/K) is also often denoted [omega](N/K, 3).) This invariant can be described in terms of local and global Weil groups, and has been conjectured by Chinburg to equal the Cassou-Nogues-Frohlich root number class [C2]. There is by now a considerable amount of evidence to support this conjecture (cf. [D] [sections]4). The conjecture is in part motivated by Tate's re-formulation of the Stark conjecture and, specifically, his construction of a natural generalization of the classical Dirichlet unit regulator [T2]. More recently, Frohlich has defined an integral version of the Stark-Tate regulator, and by integrally reinterpreting much of the Tate approach has studied elements in natural quotients of Cl(Z[G]) which explicitly relate the Galois structure of unit lattices and (the regular part of) ideal class groups (cf. [F3], [F4], ([D], [sections]3)). In this paper we shall show that the results of Frohlich represent approximations to the conjecture of Chinburg. For example, we show that the `fine structure' results of [F4] are for the special classes of abelian extensions considered there the best approximations to a proof of Chinburg's conjecture which can be obtained without analysis of extension class data. Moreover, for certain classes of field extensions the relevant cohomological questions can be resolved and so we shall obtain a full proof of the Chinburg conjecture for these extensions. In this way the dependence of the conjecture on the role of the canonical cohomology classes of Tate in defining [[omega].sup.m(N//K) is made very clear. Also, taken in conjunction with recent work of Snaith [Sn] on the additive Chinburg invariant [omega](N/K,2), a special case of our results provides an infinite family of (wildly ramified) abelian extensions for which all of the conjectures of Chinburg are now completely verified (cf. Remark 1.11(11)).

Acknowledgements. I am very grateful to Ted Chinburg and David Holland for illuminating conversations.

1. Statement of main results. Let N/K be a finite abelian extension of number fields of group G. For any Z order R (of a semisimple commutative Q-algebra) we let [K.sub.0](R) denote the Grothendieck group of finitely generated projective R-modules, with Cl(R) the subgroup of elements which have rank 0. For each subgroup H [less than or equal to] G we let [T.sub.H](G) denote the generalized Swan subgroup of Cl(Z[G]) with respect to H as defined by Matchett (cf. [sections]5). With [e.sub.H] denoting the idempotent (1/#H) [mathematical expression omitted] we let [B.sub.H,G] denote the Z-order in Q[G] obtained by ring adjunction of [e.sub.H] to Z[G] (and occasionally also the image of this order in Q[G] modulo (e.sub.G)). With [M.sub.G] denoting the maximal Z-order of Q[G] one has a commuting diagram in which all rows and columns are exact

[Mathematical Expression Omitted]

Here the surjection [[pi].sub.G], respectively [[rho].sub.H,G], is induced by the functor [[cross product].sub.z][G][M.sub.G], respectively [[cross product].sub.z][G][B.sub.H,G]. It is useful to recall here that

(1.2) [T.sub.G](G) = 0 if G, is cyclic

([Ta], Chapter 3, Corollary 1.5).

Motivated by work of Tate concerning Stark's conjecture Chinburg has defined an invariant [[omega].sup.m](N/K) of the multiplicative Galois structure of N which belongs to Cl(Z[G]) and is conjecturally equal to the Cassou-Nogues-Frohlich root number class (cf. [sections]2). By integrally re-interpreting much of the Tate approach, Frohlich has more recently studied elements in Cl([M.sub.G]), and in exceptional cases in groups of type Cl([B.sub.H,G]) also, which explicitly measure the Galois structure of unit lattices together with (the regular part of) ideal class groups (cf. [F3], [F4], ([D], [sections]3)). In this paper we shall compare these different approaches via the labelled homomorphisms in (1.1), and by so doing we shall obtain new and explicit results concerning these multiplicative structure invariants.

To proceed we need some notation. With S a finite G-stable set of places of N, we write [Y.sub.s] for the G-module afforded by the free Z-module on the set S, [epsilon] : [Y.sub.s] [right arrow] Z for the G-homomorphism induced by sending s [right arrow] 1 for each s [elements of] S, and we define the G-submodule [X.sub.s] of [Y.sub.s] by the exactness of the sequence

[Mathematical Expression Omitted]

We write [U.sub.n,s] for the group of S-units of N, [[mu].sub.n] for the torsion subgroup of [U.sub.n,s], and [U.sub.n,s] for the torsion free quotient [U.sub.n,s]. If S consists solely of the archimedean places of N then we write [U.sub.n] and [[U.sub.n] in place of [U.sub.n,s] and [U.sub.n,s]. We set [w.sub.n] = #[[mu].sub.n] and let [w.sub.n,g] denote the greatest common divisor of [w.sub.n] and #G.

For any Z[G]-lattice X we write [X.sub.MG] for the associated [M.sub.g]-lattice [Hom.sub.G] ([M.sub.g], X). For any nonzero integer n we let [Z.sub.(n)] denote the semi-localization of Z obtained by inverting the prime divisors of n. We let Cl([M.sub.g]; n) denote the kernel of the surjection Cl([M.sub.g]) [right arrow] Cl([M.sub.g] [cross product].sub.z] [Z.sub.(n)]) which is induced by the functor [[cross product].sub.z][Z.sub.(n)].

Definition 1.1 A finite G-stable set of places S of N will be said to be multiplicatively tame for N/K if it contains all archimedean places and all places which ramify in N/K, and the order of the S-ideal class group [Cl.sub.s](N) of N is coprime to #G. (For what follows it would in fact suffice to require only that [[Cl.sub.s](N) has finite projective dimension as a G-module but, for simplicity, we shall use the stronger condition.)

In Section 3 we shall define an element [[mu].sub.n/k] of Cl([M.sub.g) which arises from the torsion group [[mu].sub.n], and we then prove

Theorem 1.2. Let S be a finite G-stable set of places of N which is multiplicatively tame for N/K. If for each intermediate field L of N/K the order of the S-ideal class group [Cl.sub.s](L) is coprime to #G then one has

[Mathematical Expression Omitted]

Since G is abelian it possesses no irreducible symplectic characters and so the Cassou-Nogues-Frohlich root number class is trivial. In this case therefore the conjecture of Chinburg is that [[omega].sub.m](N/K) = 0 (cf. (2.4). In this direction the result of Theorem 1.2 in conjunction with results of Frohlich gives the following

Corollary 1.3. Let N be a real abelian extension of Q of group G. Then [mathematical expression omitted].

Proof. If N is real then [mathematical expression omitted] and so Theorem 1.2 implies that

[Mathematical Expression Omitted]

On the other hand, Frohlich has used the known validity of the Iwasawa main conjecture over Q to deduce the equality

[Mathematical Expression Omitted]

(cf. ([D], Theorem 7) and [F4], Theorem 4)).

The group Cl([M.sub.g];#G) can be large and in general any refinement of Corollary 1.3 will require a rather subtle analysis of unit structures. In one special case this has recently been achieved by D. Holland using the approach of canonical factorizations, and we record this result as

Proposition 1.4 ([H2]), (6.8) and (6.10) If N/Q is a real abelian (of group G) extension of prime power conductor [mathematical expression omitted].

If G is an abelian p-group then Cl([M.sub.g];#G) = 0 and so Corollary 1.3 implies that [[pi].sub.G][[omega].sup.m](N/Q) = 0. In this case however the class group Cl([M.sub.g]) is a rather weak measure of structure. Indeed, writing [K.sub.0].sup.'](R) for the Grothendieck group attached to the category of finitely generated modules over a commutative ring R, then for any abelian p-group G the regularity of [M.sub.g] implies that the composite map

[Mathematical Expression Omitted]

is injective (cf. [Sw], Theorems 1.1 and 4.17). In this case, therefore, one can give a direct proof of Corollary 1.3 by using the exact sequences used in ([C1], [subsection]VIII, IX) together with the fact that any finite G-module of p-power order has trivial class in [K.sub.0].sup.'](Z[G]). Our main aim in this paper, however, is to show that for certain special classes of prime power degree abelian extensions one can obtain much finer information by working modulo Swan subgroups [T.sub.h](G) for suitable subgroups H, and then incorporating the fine structure results of [F4].

In explicitly choosing a set of places S which is multiplicatively tame for N/K the requirement that [Cl.sub.s](N) be of finite projective dimension as a G-module is in general difficult to satisfy. For example, specialising to the case that K = Q and #G is a power of an odd prime l one knows that the G-module Cl(N) is of finite projective dimension if and only if l #Cl(N). In turn, the requirement that l #Cl(N) implies that N is a genus field (that is, that G is isomorphic to the direct product over all its inertia subgroups). In [F1] Frohlich used central class field theoretic techniques to completely characterize the l-power degree genus fields N for which l #Cl(N).

Definition 1.5 Let l be an odd prime. An l-power degree abelian genus field extension N/Q will be said to be of type I if there is exactly one rational prime which ramifies in N/Q, of type II if there are exactly two rational primes which ramify in N/Q, and each ramified prime has decomposition subgroup G; of type III if there are exactly two rational primes which ramify in N/Q, precisely one of which has decomposition group G, and for the other ramified prime the decomposition and inertia subgroups coincide. Any field N of type I, II or III will be said to be a multiplicatively tame genusfield (of l-power degree). Extensions of type …