In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support.

Let ${\displaystyle X}$ be a topological space. Then

By definition, this is the cohomology of the sub–chain complex ${\displaystyle C_{c}^{\ast }(X;R)}$ consisting of all singular cochains ${\displaystyle \phi :C_{i}(X;R)\to R}$ that have compact support in the sense that there exists some compact ${\displaystyle K\subseteq X}$ such that ${\displaystyle \phi }$ vanishes on all chains in ${\displaystyle X\setminus K}$.

Let ${\displaystyle X}$ be a topological space and ${\displaystyle p:X\to \star }$ the map to the point. Using the direct image and direct image with compact support functors ${\displaystyle p_{*},p_{!}:{\text{Sh}}(X)\to {\text{Sh}}(\star )={\text{Ab}}}$, one can define cohomology and cohomology with compact support of a sheaf of abelian groups ${\displaystyle {\mathcal {F}}}$ on ${\displaystyle X}$ as

Taking for ${\displaystyle {\mathcal {F}}}$ the constant sheaf with coefficients in a ring ${\displaystyle R}$ recovers the previous definition.

Given a manifold X, let ${\displaystyle \Omega _{\mathrm {c} }^{k}(X)}$ be the real vector space of k-forms on X with compact support, and d be the standard exterior derivative. Then the de Rham cohomology groups with compact support ${\displaystyle H_{\mathrm {c} }^{q}(X)}$ are the homology of the chain complex ${\displaystyle (\Omega _{\mathrm {c} }^{\bullet }(X),d)}$:

i.e., ${\displaystyle H_{\mathrm {c} }^{q}(X)}$ is the vector space of closed q-forms modulo that of exact q-forms.

Despite their definition as the homology of an ascending complex, the de Rham groups with compact support demonstrate covariant behavior; for example, given the inclusion mapping j for an open set U of X, extension of forms on U to X (by defining them to be 0 on X–U) is a map ${\displaystyle j_{*}:\Omega _{\mathrm {c} }^{\bullet }(U)\to \Omega _{\mathrm {c} }^{\bullet }(X)}$ inducing a map