Direct image with compact support

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
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 with compact (or proper) 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)) : f_\|supp(s):supp(s)→U is 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. Properties If f is proper, then f_! equals f_∗. In general, f_!(F) is only a subsheaf of f_∗(F) References Iversen, Birger (1986), Cohomology of sheaves, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16389-3, MR 842190 , esp. section VII.1 Categories: Sheaf theory Continuous mappings Wikimedia Foundation. 2010. Игры ⚽ Нужно решить контрольную? Bernardo I of Kongo 1995–96 St. Louis Blues season Look at other dictionaries: 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 1 Definition 1.1 Example 1.2 Variants … Wikipedia Image functors for sheaves — In mathematics, especially in sheaf theory, a domain applied in areas such as topology, logic and algebraic geometry, there are four image functors for sheaves which belong together in various senses.Given a continuous mapping f : X rarr; Y of… … 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 Compact fluorescent lamp — Low energy light bulb redirects here. For other low energy bulbs, see LED lamp. The tubular type compact fluorescent lamp is one of the most popular types in Europe … Wikipedia Verdier duality — In mathematics, Verdier duality is a generalization of the Poincaré duality of manifolds to spaces with singularities. The theory was introduced by Jean Louis Verdier (1965), and there is a similar duality theory for schemes due to Grothendieck.… … Wikipedia Coherent duality — In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the local… … 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 Étale cohomology — In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil… … 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 Vector space — This article is about linear (vector) spaces. For the structure in incidence geometry, see Linear space (geometry). Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is… … Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”