direct image with compact support

{{Images of sheaves}}

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

:$f\_\{!\}:\backslash mathrm\{Sh\}(X)\backslash to\; \backslash mathrm\{Sh\}(Y)$

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

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

for every open subset $U$ of $Y$. Here, the notion of a proper map of spaces is unambiguous since the spaces in question are locally compact Hausdorff.{{Cite web |title=Section 5.17 (005M): Characterizing proper maps—The Stacks project |url=https://stacks.math.columbia.edu/tag/005M |access-date=2022-09-25 |website=stacks.math.columbia.edu}} This defines $f\_\{!\}(\backslash mathcal\{F\})$ as a subsheaf of the direct image sheaf $f\_*(\backslash mathcal\{F\})$ 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.{{Cite journal |last=Schnürer |first=Olaf M. |last2=Soergel |first2=Wolfgang |date=2016-05-19 |title=Proper base change for separated locally proper maps |url=https://ems.press/journals/rsmup/articles/13889 |journal=Rendiconti del Seminario Matematico della Università di Padova |language=en |volume=135 |pages=223–250 |doi=10.4171/rsmup/135-13 |issn=0041-8994|arxiv=1404.7630 }}

• If $f$ is proper, then $f\_!$ equals $f\_*$.
• If $f$ is an open embedding, then $f\_!$ identifies with the extension by zero functor.{{Cite web |title=general topology - Proper direct image and extension by zero |url=https://math.stackexchange.com/q/2768645 |access-date=2022-09-25 |website=Mathematics Stack Exchange |language=en}}

{{reflist}}

• {{Citation | last1=Iversen | first1=Birger | title=Cohomology of sheaves | publisher=Springer-Verlag | location=Berlin, New York | series=Universitext | isbn=978-3-540-16389-3 | mr=842190 | year=1986}}, esp. section VII.1

Category:Sheaf theory

Category:Theory of continuous functions

Category:Functors