Direct image functor

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.

Definition

Image functors for sheaves
direct image f∗
inverse image f∗
direct image with compact support f!
exceptional inverse image Rf!
$f^* \leftrightarrows f_*$
$(R)f_! \leftrightarrows (R)f^!$
v · continuous mapping of topological spaces, and Sh(–) the category of sheaves of abelian groups on a topological space. The direct image functor $f_*: Sh(X) \to Sh(Y)$ sends a sheaf F on X to its direct image presheaf $f_*F : U \mapsto F(f^{-1}(U)),$ 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∗(φ): f∗(F) → f∗(G) on Y. Example If Y is a point, then the direct image equals the global sections functor. Let f: X → Y be a continuous map of topological spaces or a morphism of schemes. Then the exceptional inverse image is a functor Rf!: D(Y) → D(X). Variants A similar definition applies to sheaves on topoi, such as etale sheaves. Instead of the above preimage f−1(U) the fiber product of U and X over Y is used. Higher direct images The direct image functor is left exact, but usually not right exact. Hence one can consider the right derived functors of the direct image. They are called higher direct images and denoted Rq f∗. One can show that there is a similar expression as above for higher direct images: for a sheaf F on X, Rq f∗(F) is the sheaf associated to the presheaves $U \mapsto H^q(f^{-1}(U), F)$ Properties The direct image functor is right adjoint to the inverse image functor, which means that for any continuous $f: X \to Y$ and sheaves $\mathcal F, \mathcal G$ respectively on X, Y, there is a natural isomorphism: $\mathrm{Hom}_{\mathbf {Sh}(X)}(f^{-1} \mathcal G, \mathcal F ) = \mathrm{Hom}_{\mathbf {Sh}(Y)}(\mathcal G, f_*\mathcal F)$. If f is the inclusion of a closed subspace X ⊂ Y then f∗ is exact. Actually, in this case f∗ is an equivalence between sheaves on X and sheaves on Y supported on X. References Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR842190 , esp. section II.4 This article incorporates material from Direct image (functor) on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. Wikimedia Foundation. 2010. Игры ⚽ Нужно решить контрольную? Regan Hagar Macrauchenia Look at other dictionaries: Inverse image functor — In mathematics, the inverse image functor is a contravariant construction of sheaves. The direct image functor is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle… …   Wikipedia Direct image with compact support — In mathematics, in the theory of sheaves the direct image with compact (or proper) support is an image functor for sheaves. 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