Continuous cohomology and homology of profinite groups

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 (see [Wi], [R–Z] and [S-W]). 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 (see Chapter VIII of [Br]). 1 Complete totally disconnected R-modules In this and the following four sections, we are going to carry over the algebraic preliminaries needed in order to define in a natural way cohomology and homology of profinite groups with continuous coefficients. As already mentioned, the categories of coefficients considered are not abelian, so one has to state and prove again all the basic results which naturally hold in abelian categories. Often, this is done just rephrasing definitions, results and proofs given in that context. Let R be a topological compact unitary ring, by Corollary 4.2.24 in [A-G-M], R has a basis {Ii} of open neighborhoods of zero consisting of two-sided ideals and R ∼= lim ←− R/Ii, where R/Ii is a finite discrete ring, i.e. R is a profinite ring.