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 locally compact Hausdorff 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 the sheaf f_!(F) defined as a subsheaf of the direct image sheaf f_∗(F) by the formula

f_!(F)(U) := {s ∈ F(f ⁻¹(U)) | f|_supp(s): supp(s) → U is proper},

where U is an open subset of Y. Here, the notion of a proper map of spaces is unambiguous since the spaces in question are locally compact Hausdorff.^[1] The functoriality of this construction now follows from basic properties of the support and the definition of sheaves.

Olaf Schnürer and Wolfgang Soergel have introduced the notion of a "locally proper" map of spaces and shown that the functor of direct image with compact support remains well-behaved when defined in the generality of separated and locally proper continuous maps between arbitrary spaces.^[2]

Properties

• If f is proper, then f_! equals f_∗.
• If f is an open embedding, then f_! identifies with the extension by zero functor.^[3]

References

1. "Section 5.17 (005M): Characterizing proper maps—The Stacks project". stacks.math.columbia.edu. Retrieved 2022-09-25.
2. Schnürer, Olaf M.; Soergel, Wolfgang (2016-05-19). "Proper base change for separated locally proper maps". Rendiconti del Seminario Matematico della Università di Padova. 135: 223–250. doi:10.4171/rsmup/135-13. ISSN 0041-8994.
3. "general topology - Proper direct image and extension by zero". Mathematics Stack Exchange. Retrieved 2022-09-25.

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