Category: Sheaf theory

Torsion sheaf
In mathematics, a torsion sheaf is a sheaf of abelian groups on a site for which, for every object U, the space of sections is a torsion abelian group. Similarly, for a prime number p, we say a sheaf
Presheaf with transfers
In algebraic geometry, a presheaf with transfers is, roughly, a presheaf that, like cohomology theory, comes with pushforwards, “transfer” maps. Precisely, it is, by definition, a contravariant additi
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 com
Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The de
Stalk (sheaf)
The stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point.
Cousin problems
In mathematics, the Cousin problems are two questions in several complex variables, concerning the existence of meromorphic functions that are specified in terms of local data. They were introduced in
Sheaf cohomology
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructio
Flat topology
In mathematics, the flat topology is a Grothendieck topology used in algebraic geometry. It is used to define the theory of flat cohomology; it also plays a fundamental role in the theory of descent (
Presheaf (category theory)
In category theory, a branch of mathematics, a presheaf on a category is a functor . If is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion
Constant sheaf
In mathematics, the constant sheaf on a topological space associated to a set is a sheaf of sets on whose stalks are all equal to . It is denoted by or . The constant presheaf with value is the preshe
Abstract differential geometry
The adjective abstract has often been applied to differential geometry before, but the abstract differential geometry (ADG) of this article is a form of differential geometry without the calculus noti
De Rham–Weil theorem
In algebraic topology, the De Rham–Weil theorem allows computation of sheaf cohomology using an acyclic resolution of the sheaf in question. Let be a sheaf on a topological space and a resolution of b
Injective sheaf
In mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext). There is a further group of
Germ (mathematics)
In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind that captures their shared local properties. In particula
Coherent sheaf cohomology
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questio
Inverse image functor
In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map , the inve
Base change theorems
In mathematics, the base change theorems relate the direct image and the inverse image of sheaves. More precisely, they are about the base change map, given by the following natural transformation of
Leray's theorem
In algebraic topology and algebraic geometry, Leray's theorem (so named after Jean Leray) relates abstract sheaf cohomology with Čech cohomology. Let be a sheaf on a topological space and an open cove
Kripke semantics
Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 195
Sheaf of modules
In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and the restriction maps F(U) → F(V) ar
In mathematics, a topos (UK: /ˈtɒpɒs/, US: /ˈtoʊpoʊs, ˈtoʊpɒs/; plural topoi /ˈtoʊpɔɪ/ or /ˈtɒpɔɪ/, or toposes) is a category that behaves like the category of sheaves of sets on a topological space (
Algebraic analysis
Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of func
Direct image with compact support
In mathematics, the direct image with compact (or proper) support is an image functor for sheaves that extends the compactly supported global sections functor to the relative setting. It is one of Gro
Sheaf on an algebraic stack
In algebraic geometry, a quasi-coherent sheaf on an algebraic stack is a generalization of a quasi-coherent sheaf on a scheme. The most concrete description is that it is the data consists of, for eac
Six operations
In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra, also known as the six-functor formalism. It originally sprang from the relatio
Spencer cohomology
In mathematics, Spencer cohomology is cohomology of a manifold with coefficients in the sheaf of solutions of a linear partial differential operator. It was introduced by Donald C. Spencer in 1969.
In topology, a branch of mathematics, a cosheaf with values in an ∞-category C that admits colimits is a functor F from the category of open subsets of a topological space X (more precisely its nerve)
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 that belong together in various senses. Gi
In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order
Invertible sheaf
In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry
Ringed space
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is
Limit and colimit of presheaves
In category theory, a branch of mathematics, a limit or a colimit of presheaves on a category C is a limit or colimit in the functor category . The category admits small limits and small colimits. Exp
Ringed topos
In mathematics, a ringed topos is a generalization of a ringed space; that is, the notion is obtained by replacing a "topological space" by a "topos". The notion of a ringed topos has applications to
Gabriel–Rosenberg reconstruction theorem
In algebraic geometry, the Gabriel–Rosenberg reconstruction theorem, introduced in , states that a quasi-separated scheme can be recovered from the category of quasi-coherent sheaves on it. The theore
Leray spectral sequence
In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectra
In mathematics, an ∞-topos is, roughly, an ∞-category such that its objects behave like sheaves of spaces with some choice of Grothendieck topology; in other words, it gives an intrinsic notion of she
In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations. Since aro
Exponential sheaf sequence
In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry. Let M be a complex manifold, and write OM for the sheaf of holomorphic functio
Locally constant function
In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on w
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For e
Sheaf of algebras
In algebraic geometry, a sheaf of algebras on a ringed space X is a sheaf of commutative rings on X that is also a sheaf of -modules. It is quasi-coherent if it is so as a module. When X is a scheme,
Sheaf of logarithmic differential forms
In algebraic geometry, the sheaf of logarithmic differential p-forms on a smooth projective variety X along a smooth divisor is defined and fits into the exact sequence of locally free sheaves: where
Local system
In mathematics, a local system (or a system of local coefficients) on a topological space X is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian
Deligne cohomology
In mathematics, Deligne cohomology is the hypercohomology of the Deligne complex of a complex manifold. It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for
Local cohomology
In algebraic geometry, local cohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by , and in 1961-2 at IHES writt
Reflexive sheaf
In algebraic geometry, a reflexive sheaf is a coherent sheaf that is isomorphic to its second dual (as a sheaf of modules) via the canonical map. The second dual of a coherent sheaf is called the refl
Direct image functor
In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic
Constructible sheaf
In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space X, such that X is the union of a finite number of locally closed subsets on each of which the sheaf is a
Borel–Moore homology
In topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Armand Borel and John Moore in 1960. For reasonable compact spaces, Bor
In mathematics, a gerbe (/dʒɜːrb/; French: [ʒɛʁb]) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non
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 fo
Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C that makes the objects of C act like the open sets of a topological space. A category together with
Ideal sheaf
In algebraic geometry and other areas of mathematics, an ideal sheaf (or sheaf of ideals) is the global analogue of an ideal in a ring. The ideal sheaves on a geometric object are closely connected to
Restriction (mathematics)
In mathematics, the restriction of a function is a new function, denoted or obtained by choosing a smaller domain for the original function The function is then said to extend
Godement resolution
The Godement resolution of a sheaf is a construction in homological algebra that allows one to view global, cohomological information about the sheaf in terms of local information coming from its stal
Verdier duality
In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by Jean-Louis Verdier as an ana
Étale topos
In mathematics, the étale topos of a scheme X is the category of all étale sheaves on X. An étale sheaf is a sheaf on the étale site of X.