Self-duality and torsion Galois modules in number fields

Abstract

Let K/k be a Galois extension of number fields with Galois group r, and let D, o be the rings of integers of K, k, respectively. In this paper we investigate the structure of 0 as an or-module by comparing 0 with its dual, Horn@, 0). M. J. Taylor [ 11, Theorem, p. 1731 proved in 1978 that these two modules are stably isomorphic over Zr, assuming that all the primes of o dividing the order of r are unramified in K. This result had been conjectured by Frohlich [5, p. 4231 in the case k = Q. Its generalization to all tame extensions is an easy consequence of Taylor’s subsequent proof of Frohlich’s conjecture describing the class of 0 in the class group of locally free ZTmodules in terms of the Artin root number [ 12, Theorem 1, p. 4 1; 5, Proposition 3, p. 4391. To study the Galois module structure of 0 in the wild case Queyrut [7] introduced the S-Grothendieck groups Gi(oT) and K”(oT), where S is a set of primes of o. Namely, G&(oT) is the Grothendieck group corresponding to the category of finitely generated o-torsion free or-modules, with relations arising from short exact sequences splitting outside S; Ki(oT) is the Grothendieck group of all finitely generated o-torsion free or-modules which are locally projective outside S, with relations arising from short exact sequences. With Cassou-Nogues [3, Corollaire 6.3, p. 231, Queyrut generalized Taylor’s duality result to the wild case by showing that if S is a set of rational primes containing those with a divisor in o wildly ramified in K, then the class of 0 is equal to the class of its dual in K:(U). As in the tame case, this result could be obtained as a consequence of Queyrut’s conjecture concerning the class of 0 in K~(ZI’) [3, p. 81, which was proved in several cases [3]. The proofs of the above results used the description of certain Grothendieck groups as quotients of groups of character functions, as well as results on modules introduced by Swan in [lo].