Direct image with compact support
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In mathematics, in the theory of sheaves the direct image with compact support is an image functor for sheaves.
[edit] Definition
| Image functors for sheaves |
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direct image f∗
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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 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.
[edit] Properties
If f is proper, then f! equals f∗. In general, f!(F) is only a subsheaf of f∗(F)
[edit] Reference
- Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, MR842190, ISBN 978-3-540-16389-3, esp. section VII.1
