Sur la K-théorie du foncteur norme

Abstract

The K-theory of a functor may be viewed as a relative version of the K-theory of a ring. After its description, we prove a Mayer–Vietoris exact sequence in this framework. In the case of a Galois extension of a number field F / L with rings of integers A, B respectively, this K-theory of the “norm functor” is an extension of a subgroup of the ideal class group Cl ( A ) of F by the Tate cohomology group H ˆ 0 ( G , A ∗ ) . The Mayer–Vietoris exact sequence enables us to describe in a quite explicit way a quotient of the subgroup Cl N ( A ) : = ker N : Cl ( A ) → Cl ( B ) of the ideal class group Cl ( A ) , where N is the norm. We also prove a short exact sequence 1 → B ∗ / B ∗ ∩ N ( F ∗ ) → H ˆ 0 ( G , F ∗ ) ∩ H ˆ 0 ( G , U F ) → Cl N ( A ) / I G Cl ( A ) → 1 where U F is the group of semi-local units of F. Finally, we conclude this paper by applications of our methods to Number Theory.