K 0 and the dimension filtration for p -torsion Iwasawa modules

Abstract

Let G be a compact p-adic analytic group. We study K-theoretic questions related to the representation theory of the completed group algebra kG of G with coefficients in a finite field k of characteristic p. We show that if M is a finitely generated kG-module with canonical dimension smaller than the dimension of the centralizer, as a p-adic analytic group, of any p-regular element of G, then the Euler characteristic of M is trivial. Writing ℱi for the abelian category consisting of all finitely generated kG-modules of dimension at most i, we provide an upper bound for the rank of the natural map from the Grothendieck group of ℱi to that of ℱd, where d denotes the dimension of G. We show that this upper bound is attained in some special cases, but is not attained in general.