In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support.
Then By definition, this is the cohomology of the sub–chain complex
consisting of all singular cochains
that have compact support in the sense that there exists some compact
vanishes on all chains in
be a topological space and
the map to the point.
Using the direct image and direct image with compact support functors
, one can define cohomology and cohomology with compact support of a sheaf of abelian groups
the constant sheaf with coefficients in a ring
recovers the previous definition.
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
are the homology of the chain complex
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
inducing a map They also demonstrate contravariant behavior with respect to proper maps - that is, maps such that the inverse image of every compact set is compact.
Let f: Y → X be such a map; then the pullback induces a map If Z is a submanifold of X and U = X–Z is the complementary open set, there is a long exact sequence called the long exact sequence of cohomology with compact support.
It has numerous applications, such as the Jordan curve theorem, which is obtained for X = R² and Z a simple closed curve in X.
De Rham cohomology with compact support satisfies a covariant Mayer–Vietoris sequence: if U and V are open sets covering X, then where all maps are induced by extension by zero is also exact.