Compactly supported cohomology with local coefficients

Let $X$ be a locally compact Hausdorff space and let $\mathcal{L}$ be a left $R$-module local system on $X$. Its compactly supported cohomology with local coefficients is For compact sets $K\subseteq K^{\mathrm{\prime }}$, the identity is a map of pairs $(X,X\setminus K^{\mathrm{\prime }})\to (X,X\setminus K)$; contravariance gives the displayed system's transition $H^k(X,X\setminus K;\mathcal{L})\to H^k(X,X\setminus K^{\mathrm{\prime }};\mathcal{L})$. Thus the coefficient system is always the restriction of one ambient system, and no extension from $X\setminus K$ is being assumed.

Equivalently, an element is represented by $(K,a)$ with $K$ compact and $a\in H^k(X,X\setminus K;\mathcal{L})$. Two representatives $(K,a)$ and $(K^{\mathrm{\prime }},a^{\mathrm{\prime }})$ are equal precisely when their images agree for some compact $N\supseteq K\cup K^{\mathrm{\prime }}$. Compact union makes the support poset filtered, so this relation is transitive and addition is performed after transition to $K\cup K^{\mathrm{\prime }}$. This is the same explicit filtered-colimit construction as Compactly supported singular cohomology, now applied to the relative local-cochain groups.

If $f:X\to Y$ is proper, $\mathcal{K}$ is a local system on $Y$, and $\theta :f^{\ast }\mathcal{K}\to \mathcal{L}$ is a coefficient morphism, then each compact $K\subseteq Y$ has compact inverse image and Functoriality with coefficient morphisms gives $H^k(Y,Y\setminus K;\mathcal{K})⟶H^k(X,X\setminus f^{-1}K;\mathcal{L})\mathrm{.}$ These maps commute with enlargement of supports and hence induce the proper pullback $f^{\ast }:H_c^k(Y;\mathcal{K})\to H_c^k(X;\mathcal{L})$. Identity and composite proper maps give identity and composite pullbacks because both assertions already hold before taking the colimit.

When $X$ is compact, $K=X$ is terminal and yields $H_c^k(X;\mathcal{L})\cong H^k(X;\mathcal{L})$. For $X=\mathrm{\varnothing }$, for the zero system, and in negative degrees the group is zero. Local compactness permits the cofinal use of closures of relatively compact open sets exactly as in the published constant-coefficient definition. No simultaneous selection of supports or neighborhoods is made, so the construction is choice-free.