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SedulousApe (talk | contribs) →Definition: added locally compact Hausdorff condition and remark about relaxing this condition. Also added extension by zero = lower shriek for open embedding. |
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In [[mathematics]], in the theory of [[sheaf (mathematics)|sheaves]] the '''direct image with compact (or proper) support''' is an [[image (mathematics)|image]] [[functor]] for sheaves. It is one of [[Alexander Grothendieck |
==Definition==
{{Images of sheaves}}
Let ''f'': ''X'' → ''Y'' be a [[continuous mapping]] of [[locally compact]] [[Hausdorff space|Hausdorff]] [[topological space]]s, and let Sh(–) denote the [[category (mathematics)|category]] of sheaves of [[abelian group]]s on a topological space. The '''direct image with compact (or proper) support'''
:''f''<sub>!</sub>: Sh(''X'') → Sh(''Y'')
sends a sheaf ''F'' on ''X'' to the sheaf ''f''<sub>!</sub>(''F'') defined as a subsheaf of the [[Direct image functor|direct image]] sheaf ''f''<sub>∗</sub>(''F'') by the formula
:''f''<sub>!</sub>(''F'')(''U'') := {''s'' ∈ ''F''(''f''<sup> −1</sup>(''U'')) | f|<sub>supp(''s'')</sub>: [[support (mathematics)|supp]](''s'') → ''U'' is [[proper map|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.<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> The functoriality of this construction now follows from
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.<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}}</ref>
==Properties==
* If ''f'' is proper, then ''f''<sub>!</sub> equals ''f''<sub>∗</sub>.
* If ''f'' is an open [[embedding]], then ''f''<sub>!</sub> identifies with the extension by zero functor.<ref>{{Cite web |title=general topology - Proper direct image and extension by zero |url=https://math.stackexchange.com/questions/2768645/proper-direct-image-and-extension-by-zero |access-date=2022-09-25 |website=Mathematics Stack Exchange |language=en}}</ref>
==References==
{{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
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