Direct image with compact support
In mathematics, in the theory of sheaves the direct image with compact (or proper) support is an image functor for sheaves.
Definition
Image functors for sheaves 

direct image f_{∗} 
inverse image f^{∗} 
direct image with compact support f_{!} 
exceptional inverse image Rf^{!} 

Base change theorems 
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 (or proper) support
 f_{!}: Sh(X) → Sh(Y)
sends a sheaf F on X to f_{!}(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_{!}(F) is only a subsheaf of f_{∗}(F)
References
 Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: SpringerVerlag, ISBN 9783540163893, MR 0842190, esp. section VII.1
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