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 |
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direct image f∗ |
inverse image f∗ |
direct image with compact support f! |
exceptional inverse image Rf! |
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Let f: X → Y be a continuous mapping of topological spaces, and Sh(–) 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: Springer-Verlag, ISBN 978-3-540-16389-3, MR 842190, esp. section VII.1
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