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Definition (Homology). Let $\operatorname{\mathcal{A}}$ be an abelian category and let $C_{\ast }$ be a chain complex with values in $\operatorname{\mathcal{A}}$. For every integer $n$, we let $\mathrm{H}_{n}( C )$ denote the quotient $\mathrm{Z}_{n}(C) / \mathrm{B}_{n}(C)$. We will refer to $\mathrm{H}_{n}(C)$ as the $n$th homology of the chain complex $C_{\ast }$. We say that the chain complex $C_{\ast }$ is acyclic if the homology objects $\mathrm{H}_{n}(C)$ vanish for every integer $n$.

If $\operatorname{\mathcal{A}}= \operatorname{ Ab }$ is the category of abelian groups and if $x \in \mathrm{Z}_ n(C)$ is an $n$-cycle of $C_{\ast }$, we let $[ x ]$ denote its image in the homology group $\mathrm{H}_{n}(C)$: we refer to $[x]$ as the homology class of $x$. We say that a pair of $n$-cycles $x,x' \in \mathrm{Z}_{n}(C)$ are homologous if $[ x ] = [ x' ]$: that is, if there exists an $(n+1)$-chain $y$ satisfying $x' = x + \partial (y)$.