Homology group

Definition

Let $G$ be a group acting on an abelian group $A$, via an action $\varphi:G \to \operatorname{Aut}(A)$. Equivalently, $A$ is a module over the (possibly non-commutative) unital group ring $\mathbb{Z}G$ of $G$ over the ring of integers.

The homology groups $H_{n,\varphi}(G,A)$ ($n = 0,1,2,3,\dots$) are abelian groups defined in the following equivalent ways.

When $\varphi$ is understood from context, the subscript ${}_\varphi$ may be omitted in the notation for the homology group, as well as the notation for the groups of $n$-cycles and $n$-boundaries.

No.	Shorthand	Detailed description of $H_{n,\varphi}(G,A)$, the $n^{th}$ homology group
1	Explicit, using the bar resolution	$H_{n,\varphi}(G,A)$, is defined as the quotient $Z_{n,\varphi}(G,A)/B_{n,\varphi}(G,A)$ where $Z_{n,\varphi}(G,A)$ is the group of cycles for the action and $B_{n,\varphi}(G,A)$ 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 $H_{n,\varphi}(G,A)$ 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	$H_{n,\varphi}(G,A) = L^n(-_G)(A)$, 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 $(g-1)m, g\in G, m \in M$.

Equivalence of definitions

Further information: Equivalence of definitions of homology group

The equivalence of (1) and (2) follows from the fact that (1) is the special case of (2) that arises if we choose our projective resolution as the bar resolution. The equivalence between (2) and (3) is by the definition of $\operatorname{Tor}$. The equivalence between (3) and (4) follows from the fact that the coinvariants functor (defined in (4)) sends $A$ to $\mathbb{Z} \otimes_{\mathbb{Z}G} A$ where $\mathbb{Z}$ is given the structure of a trivial $\mathbb{Z}G$-module.