K₀ and the dimension filtration for p-torsion Iwasawa modules

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 whose dimension is smaller than the dimension of the centralizer of any p-regular element of G, then the Euler characteristic of M is trivial. Writing F_i 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 F_i to that of F_d, 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.