Direct image with compact support: Difference between revisions

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In [[mathematics]], in the theory of [[sheaf (mathematics)|mathematics]] the '''direct image with compact support''' is an [[image (mathematics)|image]] [[functor]] for sheaves.

#REDIRECT [[Direct image functor]]

==Definition==

{{Template:Images of sheaves}}

Let ''f'': ''X'' → ''Y'' be a [[continuous mapping]] of [[topological space]]s, and ''Sh''(–) the [[category (mathematics)|category]] of sheaves of [[abelian group]]s on a topological space. The '''direct image with compact support'''

:''f''_∗: ''Sh''(''X'') → ''Sh''(''Y'')

sends a sheaf ''F'' on ''X'' to ''f''_!(''F'') defined by

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

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.

==Properties==

If ''f'' is proper, then ''f''_! equals ''f''_∗. In general, ''f''_!(''F'') is only a subsheaf of ''f''_∗(''F'')

==Reference==

* {{Citation | last1=Iversen | first1=Birger | title=Cohomology of sheaves | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | series=Universitext | isbn=978-3-540-16389-3 | id={{MathSciNet | id = 842190}} | year=1986}}, esp. section VII.1

[[Category:Sheaf theory]]

[[Category:Continuous mappings]]