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