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

Abstract

The Stickelberger splitting map in the case of abelian extensions $F / \Q$ 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 \cite{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 $K_2 ({\mathcal O}_{F_{l^k}})$ for the Iwasawa tower $F_{l^k} := F(\mu_{l^k})$, for $k \geq 1.$ 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 $K_2 ({\mathcal O}_{F_{l^k}})_l$ for all $k \geq 1$, while this is proved in full generality in [GP], under the assumption that the Iwasawa $\mu$--invariant $\mu_{F,l}$ vanishes. As a consequence, our Stickelberger splitting map leads to annihilation results as predicted by the original Coates-Sinnott conjecture for the subgroups $div(K_{2n}(F)_l)$ of $K_{2n}(O_F)_l$ consisting of all the $l$--divisible elements in the even Quillen $K$-groups of $F$, for all odd primes $l$ and all $n$. } In \S6, we construct a Stickelberger splitting map for \'etale $K$--theory. Finally, we construct both the Quillen and \'etale 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 $K_{2m} ({\mathcal O}_{F_k})_l$ (respectively $K^{et}_{2m} ({\mathcal O}_{F_k})_l$), for all $k$