Direct image with compact support: Difference between revisions

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

==Definition==

{{Images of sheaves}}

Let ''<math>f'': ''X''\to → ''Y''</math> be a [[continuous mapping]] of [[locally compact]] [[Hausdorff space|Hausdorff]] [[topological space]]s, and let <math>\mathrm{Sh}(–-)</math> denote the [[category (mathematics)|category]] of sheaves of [[abelian group]]s on a topological space. The '''direct image with compact (or proper) [[support (mathematics)|support]]''' is the [[functor]]

:''f''<submath>f_{!</sub>}: \mathrm{Sh}(''X'')\to → \mathrm{Sh}(''Y'')</math>

that sends a sheaf ''<math>\mathcal{F''}</math> on ''<math>X''</math> to the sheaf ''f''<submath>f_{!</sub>}(''\mathcal{F''})</math> given by the formula

:''f''<submath>f_{!</sub>}(''\mathcal{F''})(''U'') := \{''s'' ∈ ''\in\mathcal{F''}(''f''^{&nbsp;&minus;^{-1}}(''U'')) |\mid {f|_{\vert}_{\operatorname{supp}(''s'')}}: [[support (mathematics)|\operatorname{supp]]}(''s'')&nbsp;→&nbsp;''\to U'' \text{ is [[proper map|proper]]}\}</math>

for every open subset ''<math>U''</math> of ''<math>Y</math>.'' 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''<submath>f_{!</sub>}(''\mathcal{F''})</math> as a subsheaf of the [[Direct image functor|direct image]] sheaf ''f''<sub>∗</submath>f_*(''\mathcal{F''}),</math> 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>

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==Properties==

* If ''<math>f''</math> is proper, then ''f''<submath>f_!</submath> equals ''f''<submath>∗f_*</submath>.

* If ''<math>f''</math> is an open [[embedding]], then ''f''<submath>f_!</submath> 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/q/2768645 |access-date=2022-09-25 |website=Mathematics Stack Exchange |language=en}}</ref>

==References==

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[[Category:Sheaf theory]]

[[Category:Theory of continuous functions]]

[[Category:Functors]]