Direct image with compact support

Image functors for sheaves
direct image f∗
inverse image f∗
direct image with compact support f!
exceptional inverse image Rf!
${\displaystyle f^{*}\leftrightarrows f_{*}}$
${\displaystyle (R)f_{!}\leftrightarrows (R)f^{!}}$
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

f_!(F)(U) := {s ∈ F(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