Direct image with compact support
In mathematics, in the theory of sheaves the direct image with compact support is an image functor for sheaves.Definition
Let f: X → Y be a continuous mapping of topological spaces, and let Sh denote the category of sheaves of abelian groups on a topological space. The direct image with compact support
sends a sheaf F on X to f! defined by
where U is an open subset of Y. The functoriality of this construction follows from the very basic properties of the support and the definition of sheaves.Properties
If f is proper, then f! equals f∗. In general, f! is only a subsheaf of f∗
