Direct image with compact support

In mathematics, the direct image with compact (or proper) support is an image functor for sheaves that extends the compactly supported global sections functor to the relative setting. It is one of Grothendieck's six operations.

Definition

Let ${\displaystyle f:X\to Y}$ be a continuous mapping of locally compact Hausdorff topological spaces, and let ${\displaystyle \mathrm {Sh} (-)}$ denote the category of sheaves of abelian groups on a topological space. The direct image with compact (or proper) support is the functor

${\displaystyle f_{!}:\mathrm {Sh} (X)\to \mathrm {Sh} (Y)}$

that sends a sheaf ${\displaystyle {\mathcal {F}}}$ on ${\displaystyle X}$ to the sheaf ${\displaystyle f_{!}({\mathcal {F}})}$ given by the formula

${\displaystyle f_{!}({\mathcal {F}})(U):=\{s\in {\mathcal {F}}(f^{-1}(U))\mid {f\vert }_{\operatorname {supp} (s)}:\operatorname {supp} (s)\to U{\text{ is proper}}\}}$

for every open subset ${\displaystyle U}$ of ${\displaystyle Y}$. Here, the notion of a proper map of spaces is unambiguous since the spaces in question are locally compact Hausdorff.^[1] This defines ${\displaystyle f_{!}({\mathcal {F}})}$ as a subsheaf of the direct image sheaf ${\displaystyle f_{*}({\mathcal {F}})}$ and the functoriality of this construction then follows from basic properties of the support and the definition of sheaves.

The assumption that the spaces be locally compact Hausdorff is imposed in most sources (e.g., Iversen or Kashiwara–Schapira). In slightly greater generality, 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 for separated and locally proper continuous maps between arbitrary spaces.^[2]

Properties

• If ${\displaystyle f}$ is proper, then ${\displaystyle f_{!}}$ equals ${\displaystyle f_{*}}$.
• If ${\displaystyle f}$ is an open embedding, then ${\displaystyle f_{!}}$ 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. arXiv:1404.7630. 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