Singular cohomology with coefficients

For a space $X$ and an abelian group $G$, a degree-$n$ singular cocycle is a cochain in $Z^n(X;G)=\ker {\delta }^n$ and a degree-$n$ singular coboundary belongs to $B^n(X;G)=\mathrm{im}{\delta }^{n-1}$, for the positive coboundary convention of Singular cochain complex with coefficients. The square-zero lemma The singular coboundary squares to zero implies $B^n\subseteq Z^n$: if $\phi ={\delta }^{n-1}\psi$, then ${\delta }^n\phi =0$.

The singular cohomology group with coefficients in $G$ is $H^n(X;G)=Z^n(X;G)\mathrm{/}{B}^n(X;G)(n\ge 0),H^n(X;G)=0(n<0)\mathrm{.}$ This is the abelian quotient with addition $[\phi ]+[\psi ]=[\phi +\psi ]$: changing either representative by a coboundary changes the sum by a coboundary. Thus two cocycles represent the same class exactly when their difference is ${\delta }^{n-1}\eta$ for some degree-$(n-1)$ cochain $\eta$. In degree zero, $B^0=0$ because $C^{-1}=0$; hence $H^0=\ker {\delta }^0$. The zero convention in negative degrees agrees with the same kernel/image construction on the zero cochain groups.

If coefficients are a module over a commutative ring, the same groups and maps are modules and linear maps, and this is their module quotient. Empty $X$ or zero coefficients give zero cohomology in every degree. The notation does not define cohomology as the dual of homology; such identifications require their own theorems. No choice of cocycle representatives and no AC is part of this definition.