On the Stickelberger splitting map in the K--theory of number fields

الأصل · EN

The Stickelberger splitting map in the case of abelian extensions F / was defined in [Ba1, Chap. IV]. The construction used Stickelebrger's theorem. For abelian extensions F / K with an arbitrary totally real base field K the construction of Ba1 cannot be generalized since Brumer's conjecture (the analogue of Stickelberger's theorem) is not proved yet at that level of generality. In this paper, we construct a general Stickelberger splitting map under the assumption that the first Stickelberger elements annihilate the Quillen K--groups groups K₂ (OFₗₖ) for the Iwasawa tower Fₗₖ:= F(μₗₖ), for k ≥ 1. The results of [Po] give examples of CM abelian extensions F/K of general totally real base-fields K for which the first Stickelberger elements annihilate K₂ (OFₗₖ)ₗ for all k ≥ 1, while this is proved in full generality in [GP], under the assumption that the Iwasawa μ--invariant μF,ₗ vanishes. As a consequence, our Stickelberger splitting map leads to annihilation results as predicted by the original Coates-Sinnott conjecture for the subgroups div(K₂ₙ(F)ₗ) of K₂ₙ(OF)ₗ consisting of all the l--divisible elements in the even Quillen K-groups of F, for all odd primes l and all n. In 6, we construct a Stickelberger splitting map for étale K--theory. Finally, we construct both the Quillen and étale Stickelberger splitting maps under the more general assumption that for some arbitrary but fixed natural number m>0, the corresponding m--th Stickelberger elements annihilate K₂ₘ (OFₖ)ₗ (respectively Ket₂ₘ (OFₖ)ₗ), for all k