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| In [[mathematics]], in the field of [[sheaf theory]] and especially in [[algebraic geometry]], the '''direct image functor''' generalizes the notion of a [[section of a sheaf]] to the relative case.
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| ==Definition==
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| {{Images of sheaves}}
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| 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 [[functor]]'''
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| :<math>f_*: Sh(X) \to Sh(Y)</math>
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| sends a sheaf ''F'' on ''X'' to its direct image presheaf
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| :<math>f_*F : U \mapsto F(f^{-1}(U)),</math>
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| which turns out be a sheaf on ''Y''. This assignment is functorial, i.e. a [[morphism of sheaves]] φ: ''F'' → ''G'' on ''X'' gives rise to a morphism of sheaves ''f''<sub>∗</sub>(φ): ''f''<sub>∗</sub>(''F'') → ''f''<sub>∗</sub>(''G'') on ''Y''.
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| === Example ===
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| If ''Y'' is a point, then the direct image equals the [[global sections functor]].
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| Let f: X → Y be a continuous map of topological spaces or a morphism of schemes. Then the exceptional inverse image is a functor
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| f<sup>!</sup>: D(Y) → D(X).
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| === Variants ===
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| A similar definition applies to sheaves on [[topos|topoi]], such as [[etale|etale sheaves]]. Instead of the above preimage ''f''<sup>−1</sup>(''U'') the [[fiber product]] of ''U'' and ''X'' over ''Y'' is used.
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| == Higher direct images ==
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| The direct image functor is left exact, but usually not right exact. Hence one can consider the right [[derived functor]]s of the direct image. They are called '''higher direct images''' and denoted ''R<sup>q</sup> f''<sub>∗</sub>.
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| One can show that there is a similar expression as above for higher direct images: for a sheaf ''F'' on ''X'', ''R<sup>q</sup> f''<sub>∗</sub>(''F'') is the sheaf associated to the presheaf
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| :<math>U \mapsto H^q(f^{-1}(U), F)</math>
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| == Properties ==
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| * The direct image functor is [[adjoint functor|right adjoint]] to the [[inverse image functor]], which means that for any continuous <math>f: X \to Y</math> and sheaves <math>\mathcal F, \mathcal G</math> respectively on ''X'', ''Y'', there is a natural isomorphism:
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| :<math>\mathrm{Hom}_{\mathbf {Sh}(X)}(f^{-1} \mathcal G, \mathcal F ) = \mathrm{Hom}_{\mathbf {Sh}(Y)}(\mathcal G, f_*\mathcal F)</math>. | |
| * If ''f'' is the inclusion of a closed subspace ''X'' ⊂ ''Y'' then ''f''<sub>∗</sub> is exact. Actually, in this case ''f''<sub>∗</sub> is an [[equivalence of categories|equivalence]] between sheaves on ''X'' and sheaves on ''Y'' supported on ''X''. It follows from the fact that the stalk of <math>(f_* \mathcal F)_y</math> is <math>\mathcal F_y</math> if <math>y \in X</math> and zero otherwise (here the closeness of ''X'' in ''Y'' is used).
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| == See also ==
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| *[[Proper base change theorem]]
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| ==References==
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| * {{Citation | last1=Iversen | first1=Birger | title=Cohomology of sheaves | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Universitext | isbn=978-3-540-16389-3 | mr=842190 | year=1986}}, esp. section II.4
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| {{PlanetMath attribution|id=1101|title=Direct image (functor)}}
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| {{DEFAULTSORT:Direct Image Functor}}
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| [[Category:Sheaf theory]]
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| [[Category:Continuous mappings]]
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