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.


Let f: XY 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

f!(F)(U) := {sF(f 1(U)) | f|supp(s): supp(s)  U is proper},

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.


If f is proper, then f! equals f. In general, f!(F) is only a subsheaf of f(F)


  • Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR 0842190, esp. section VII.1
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