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. It is one of Grothendieck's six operations.

Definition

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.

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 0842190, esp. section VII.1