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In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group.

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Resolution of coefficient system in group homology

This is an example of the hyperhomology spectral sequences in the case of chain complexes of $G$-modules. See chapter 5.7 (especially proposition 5.7.6) of Weibel's book "An introduction to homologica …
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