Direct image with compact support: Difference between revisions

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

==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''' is the [[functor]]

:''f''_!: Sh(''X'') → Sh(''Y'')

that sends a sheaf ''F'' on ''X'' to the sheaf ''f''_!(''F'') defined as a subsheaf of the [[Direct image functor|direct image]] sheaf ''f''_∗(''F'')given by the formula

:''f''_!(''F'')(''U'') := {''s'' ∈ ''F''(''f''^{ −1}(''U'')) | f|_supp(''s''): [[support (mathematics)|supp]](''s'') → ''U'' is [[proper map|proper]]},

wherefor every open subset ''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> TheThis defines ''f''_!(''F'') as a subsheaf of the [[Direct image functor|direct image]] sheaf ''f''_∗(''F''), and the functoriality of this construction nowthen 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 in the generality offor 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==