### The Coates-Sinnott conjecture and eigenspaces of K-groups, by Pietro Cornacchia and Paul Arne Østvær

Let E signify a totally real Abelian number field with a prime power conductor and ring of p--integers R_E for a prime p. Let G denote the Galois group of E over the rationals, and let \chi be a p--adic character of G of order prime to p. The odd--primary results in this paper depend on the Bloch--Kato conjecture, while the two--primary results are non--conjectural. Theorem A calculates, under a minor restriction on \chi, the Fitting ideals of K_n(R_E;Z_p)(\chi) over Z_p[G](\chi). Here we require that n = 2 mod(4). These Fitting ideals are principal, and generated by a Stickelberger element. This gives a partial verification and also a strong indication of the Coates--Sinnott conjecture. We also discuss (co)--descent for higher K--groups, and prove in Theorem B a Hilbert Theorem 90 type of result for the transfer map in higher K--theory of number fields.

Pietro Cornacchia <cornac@dm.unipi.it>
Paul Arne Østvær <paularne@math.uio.no>