View source for Homology group for trivial group action

==Definition==

Let <math>G</math> be a [[group]] and <math>A</math> be an [[abelian group]].

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

===Definition in terms of classifying space===

<math>\! H_n(G,A)</math> can be defined as the homology group <math>H_n(BG,A)</math>, where <math>BG</math> is the [[classifying space]] of <math>G</math> 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''' <math>H_n(G,A)</math> are defined as the [[defining ingredient::homology group]]s <math>H_{n,\varphi}(G,A)</math> where <math>\varphi:G \to \operatorname{Aut}(A)</math> is the trivial map. In other words, we treat <math>A</math> as a <math>G</math>-module with ''trivial'' action of <math>G</math> on <math>A</math> (i.e., every element of <math>G</math> fixes every element of <math>A</math>. We thus also treat <math>A</math> as a trivial <math>\mathbb{Z}G</math>-module, where <math>\mathbb{Z}G</math> is a [[group ring]] of <math>G</math> over the [[ring of integers]] <math>\mathbb{Z}</math>.

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