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–Soulé. 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 $\chi$.