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Homology group for trivial group action


Let G be a group and A be an abelian group.

The homology groups for trivial group action \! H_n(G,A), also denoted \! H_n(G;A) (n = 0,1,2,3,\dots) are abelian groups defined in the following equivalent ways.

Definition in terms of classifying space

\! H_n(G,A) can be defined as the homology group H_n(BG,A), where BG is the classifying space of G and theohomology group is understood to be in the topological sense (singular homology or cellular homology, or any of the equivalent homology theories satisfying the axioms).

Definition as homology group for an action taken as the trivial action

The homology groups for trivial group action H_n(G,A) are defined as the homology groups H_{n,\varphi}(G,A) where \varphi:G \to \operatorname{Aut}(A) is the trivial map. In other words, we treat A as a G-module with trivial action of G on A (i.e., every element of G fixes every element of A. We thus also treat A as a trivial \mathbb{Z}G-module, where \mathbb{Z}G is a group ring of G over the ring of integers \mathbb{Z}.

No. Shorthand Detailed description of H_{n,\varphi}(G,A), the n^{th} homology group
1 Explicit, using the bar resolution H_n(G,A), is defined as the quotient Z_n(G,A)/B_n(G,A) where Z_n(G,A) is the group of cycles for the action and B_n(G,A) is the group of boundaries.
2 Complex based on arbitrary projective resolution Let \mathcal{F} be a projective resolution for \mathbb{Z} as a \mathbb{Z}G-module with the trivial action. Let \mathcal{C} be the complex \mathcal{F} \otimes_{\mathbb{Z}G} A. The homology group H_n(G,A) is defined as the n^{th} homology group for this complex.
3 As a \operatorname{Tor} functor \operatorname{Tor}_n^{\mathbb{Z}G}(\mathbb{Z},A) where \mathbb{Z} is a trivial \mathbb{Z}G-module and A has the module structure specified by \varphi.
4 As a left derived functor H_n(G,A) = L^n(-_G)(A), i.e., it is the n^{th} left derived functor of the coinvariants functor for G (denoted -_G) evaluated at A. The coinvariants functor sends a \mathbb{Z}G-module M to M/N where N is generated by all elements of the form (g-1)m, g\in G, m \in M.