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

The Stickelberger splitting map in the case of abelian extensions $F\mathrm{/}\text{Q}$ was defined in [Ba1, Chap. IV]. The construction used Stickelebrger’s theorem. For abelian extensions $F\mathrm{/}K$ 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 $k\ge 1.$ The results of [Po] give examples of CM abelian extensions $F\mathrm{/}K$ of general totally real base-fields $K$ for which the first Stickelberger elements annihilate $K_2({\mathcal{O}}_{F_{l^k}}{)}_l$ for all $k\ge 1$, 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 $K^{et}{2m} ({\mathcal O}{F_k})_l$),forall$k$