Homology group for trivial group action

Definition

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 $(g-1)m, g\in G, m \in M$.