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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]].
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that sends a sheaf <math>\mathcal{F}</math> on <math>X</math> to the sheaf <math>f_{!}(\mathcal{F})</math> given by the formula
:<math>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}\}</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 <math>f_{!}(\mathcal{F})</math> as a subsheaf of the [[Direct image functor|direct image]] sheaf <math>f_*(\mathcal{F})</math> and the functoriality of this construction then follows from basic properties of the support and the definition of sheaves.
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[[Category:Sheaf theory]]
[[Category:Theory of continuous functions]]
[[Category:Functors]]
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