On Fitting ideals of certain etale K groups Matthieu Le Floc'h

On Fitting ideals of certain etale K-groups Matthieu Le Floc'h March 4, 2004 Abstract Let F be an abelian number field and S the set of primes of F that are either ramified or over p, with p an odd prime. In this paper we compute the (first) Fitting ideal of K et2i?2(OSF )(?) for i ≥ 2, where OSF is the ring of S-integers of F and ? is a character of Gal(F/Q) of order prime to p different from the ith power of the Teichmuller character. This Fitting ideal proves to be principal and generated by a Stickelberger element. keywords: etale K-groups, Fitting ideals, Iwasawa modules, Stickelberger elements. Introduction Let F be an abelian number field. The classical Stickelberger's Theorem states that the first Stickelberger ideal annihilates the ideal class group of F (see [W]). Inspired by Stickelberger, Coates and Sinnott made similar guesses about annihilator ideals for higher even K-groups. Namely, they conjectured in [CS] that the ith “twisted” Stickelberger element annihilates K2i(OF ). Adopting a p-adic approach, one can try to annihilate K2i(OF ) ? Zp, with p a fixed odd prime. The Quillen-Lichtenbaum conjecture (which is true for i = 2) affirms that K2i(OF ) ? Zp is canonically isomorphic to the higher etale K-theory group K et2i(OF [1/p]).

• char- acter ?


• zp


• trivial finite


• galois group over


• zp-torsion free


• commutative ring


• iwasawa theory


• ring zp


• over ?