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Line 3: ==Definition== {{Images of sheaves}} Let ''f'': ''X'' → ''Y'' be a [[continuous mapping]] of [[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 ''f''<sub>!</sub>(''F'') defined by :''f''<sub>!</sub>(''F'')(''U'') := {''s'' ∈ ''F''(''f''<sup> −1</sup>(''U'')) :\| f\|<sub>\|supp(''s'')</sub>: [[support (mathematics)\|supp]](''s'')→U → ''U'' is [[proper map\|proper]]}, where ''U'' is an open subset of ''Y''. The functoriality of this construction follows from the very basic properties of the support and the definition of sheaves.

Direct image with compact support: Difference between revisions