Content deleted Content added
m Open access bot: arxiv added to citation with #oabot. |
|||
Line 12:
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.<ref>{{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}}</ref> This defines ''f''<sub>!</sub>(''F'') as a subsheaf of the [[Direct image functor|direct image]] sheaf ''f''<sub>∗</sub>(''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.<ref>{{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 }}</ref>
==Properties==
| |||