Continuous cohomology and homology of profinite groups

Details Continuous cohomology and homology of profinite groups Continuous cohomology and homology of profinite groups

2003-06-26

arxiv.org/abs/math/0306381v1

Mathematics

Group Theory

33 pages

Scientific paper

Let $G$ be a profinite group with a countable basis of neighborhoods of the identity. A cohomology and homology theory for the group $G$ with non-discrete topological coefficients is developed, improving previous expositions of the subject. Though the category of topological $G$-modules considered is additive but not abelian, there is a theory of derived functors. All standard properties of group cohomology and homology are then obtained rephrasing the standard proofs given in the abelian categories' setting. In this way, one gets the universal coefficients Theorem, Lyndon/Hochschild-Serre spectral sequence and Shapiro's Lemma. Another interesting feature of this theory is that it allows to rephrase and easily prove for profinite groups, all definitions and results about cohomological dimension and duality which hold for discrete groups. No claim is made on the originality of the results here exposed but rather on the presentation of the subject.

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