2010/08/05 by Grzegorz Banaszak, Banaszak, Grzegorz, Cristian D. Popescu +2
Mathematics · #11G30 #19D10 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Number Theory (math.NT) #math.KT #math.NT #msc:11G30 #msc:19D10
paper · pdf · doi:10.48550/arxiv.1008.1000
27 pages
arxiv created 2010/08/05 · openalex publication_date 2010/08/05 · arxiv updated 2010/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeBa1 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 (\mathcal O_Flk) 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 (\mathcal O_Flk)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 \S6, 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 (\mathcal OFk)l (respectively Ket2m (\mathcal OFk)l), for all k