 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.
Contents
Definition
Image functors for sheaves direct image f_{∗} inverse image f^{∗} direct image with compact support f_{!} exceptional inverse image Rf^{!} v · continuous mapping of topological spaces, and Sh(–) the category of sheaves of abelian groups on a topological space. The direct image functor sends a sheaf F on X to its direct image presheaf
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 R^{q} f_{∗}.
One can show that there is a similar expression as above for higher direct images: for a sheaf F on X, R^{q} f_{∗}(F) is the sheaf associated to the presheaves
Properties
 The direct image functor is right adjoint to the inverse image functor, which means that for any continuous and sheaves respectively on X, Y, there is a natural isomorphism:
 .
 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: SpringerVerlag, ISBN 9783540163893, MR842190 , esp. section II.4
This article incorporates material from Direct image (functor) on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
Categories: Sheaf theory
 Continuous mappings
Wikimedia Foundation. 2010.
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. Definition Image functors for sheaves … Wikipedia
Exceptional inverse image functor — In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier… … Wikipedia
Functor — For functors as a synonym of function objects in computer programming to pass function pointers along with its state, see function object. For the use of the functor morphism presented here in functional programming see also the fmap function of… … Wikipedia
Fibred category — Fibred categories are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which inverse images (or pull backs) of objects such as vector bundles … Wikipedia
Sheaf (mathematics) — This article is about sheaves on topological spaces. For sheaves on a site see Grothendieck topology and Topos. In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.… … Wikipedia
Adjoint functors — Adjunction redirects here. For the construction in field theory, see Adjunction (field theory). For the construction in topology, see Adjunction space. In mathematics, adjoint functors are pairs of functors which stand in a particular… … Wikipedia
List of mathematics articles (D) — NOTOC D D distribution D module D D Agostino s K squared test D Alembert Euler condition D Alembert operator D Alembert s formula D Alembert s paradox D Alembert s principle Dagger category Dagger compact category Dagger symmetric monoidal… … Wikipedia
Derived category — In mathematics, the derived category D(C) of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C. The construction proceeds on the… … Wikipedia
Grothendieck spectral sequence — In mathematics, in the field of homological algebra, the Grothendieck spectral sequence is a technique that allows one to compute the derived functors of the composition of two functors Gcirc F, from knowledge of the derived functors of F and G… … Wikipedia
18+© Academic, 20002024 Contact us: Technical Support, Advertising
Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx.Share the article and excerpts
Direct image functor
 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.
Contents
Definition
Image functors for sheaves direct image f_{∗} inverse image f^{∗} direct image with compact support f_{!} exceptional inverse image Rf^{!}