  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.
