Sheaf cohomology
Sheaf cohomology In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian ...
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Cohomology
Cohomology In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence ... defined from a cochain complex. That is, cohomology is defined as the abstract study of cochains, cocycles and coboundaries. Cohomology can be viewed as a method of ... more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of ...
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Sheaf (mathematics)
Sheaf (mathematics) In mathematics, a sheaf F on a topological space X is ... one. A presheaf is similar to a sheaf, but it may not be possible to ... to F(V) then form a sheaf of sets on X. Furthermore, each F ... maps are ring homomorphisms, making F a sheaf of rings on X. For a ...
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Étale cohomology
Étale cohomology In mathematics, the étale cohomology theory of algebraic geometry is a refined ... remaining conjectures awaited the development of étale cohomology. This theory is an example of a Weil cohomology theory in algebraic geometry, and as such ... been found. The formal definition of étale cohomology is as the derived functor of ...
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Local cohomology
Local cohomology In mathematics, local cohomology is a chapter of homological algebra and sheaf theory introduced into algebraic geometry by Alexander ... sections Γ Y are considered of a sheaf F of abelian groups, on a topological ... derived functors of Γ Y form local cohomology groups H Y i (X,F) ...
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Coherent sheaf
Coherent sheaf In mathematics, especially in algebraic geometry and ... the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism ... n . Here O X is the structure sheaf of local rings, given by definition on ... taken to be finitely generated.) For a sheaf of rings R, a sheaf F ...
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Dolbeault cohomology
Dolbeault cohomology In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology is a special generalization of de Rham cohomology to complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups H p,q (M,C) depend ... of degree (p,q). Construction of the cohomology groups Let Ω p,q be ...
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Injective sheaf
Injective sheaf In mathematics, injective sheaves of abelian groups ... to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext.). There is a further group of ... for that application. Injective sheaves An injective sheaf F is just a sheaf that ...
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Intersection cohomology
Intersection cohomology In mathematics, intersection cohomology is a theory from algebraic topology, initially ... to apply to spaces with singularities. The cohomology groups of a topological manifold have an ... does not look like R n . Intersection cohomology is a modified definition of cohomology which recovers the property of Poincaré ...
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De Rham cohomology
De Rham cohomology In mathematics, de Rham cohomology is a tool belonging both to algebraic ... to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential ... to singular homology, and to Alexander-Spanier cohomology. Definition The set of smooth, differentiable ...
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