Direct image with compact support

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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 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

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)