Fitting ideals of isotypic parts of even K-groups

Abstract

For a real abelian number field F with Galois group $$G=\mathrm {Gal}(F/\mathbf {Q})$$ , an odd prime p and an odd integer $$m\ge 3$$ , we study the Fitting ideal of the dual of the $$\chi $$ -part of $$\frac{K_{2m-1}(O_{F})\otimes \mathbf {Z}_{p}}{D_{p,m}(F)}$$ . Here, $$\chi $$ is a semi-simple p-adic character of G, $$K_{2m-1}(O_{F})$$ is the K-theory group of the ring of integers of F, and $$D_{p,m}(F)$$ is the submodule generated by the elements of Deligne–Soule. This Fitting ideal is then compared to the Fitting ideal of the $$\chi $$ -part of $$K_{2m-2}O_{F}$$ . Finally, an example is given, where we eliminate the dependency of the previous result on the character $$\chi $$ .