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K 0 and the dimension filtration for p ‐torsion Iwasawa modules
Author(s) -
Ardakov Konstantin,
Wadsley Simon
Publication year - 2008
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdm053
Subject(s) - mathematics , abelian group , dimension (graph theory) , finitely generated abelian group , upper and lower bounds , centralizer and normalizer , euler characteristic , torsion (gastropod) , combinatorics , rank (graph theory) , pure mathematics , discrete mathematics , mathematical analysis , medicine , surgery
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 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.