abstract sheaf

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• 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

• Abstract variety — In mathematics, in the field of algebraic geometry, the idea of abstract variety is to define a concept of algebraic variety in an intrinsic way. This followed the trend in the definition of manifold independent of any ambient space (Hassler… …   Wikipedia

• Sheaf cohomology — In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in… …   Wikipedia

• 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 related concepts applied to sheaves: flabby …   Wikipedia

• 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 O X modules. That is, we have : S otimes; T isomorphic to O X , which acts as identity element for… …   Wikipedia

• Leray's theorem — In algebraic geometry, Leray s theorem relates abstract sheaf cohomology with Cech cohomology.Let mathcal F be a sheaf on a topological space X and mathcal U={U i} a countable cover of X. If mathcal F is acyclic on every finite intersection of… …   Wikipedia

• Category theory — In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects and morphisms . Categories now appear in most branches of mathematics and in… …   Wikipedia

• Differentiable manifold — A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the middle chart the Tropic of Cancer is a smooth curve, whereas in the first it has a sharp… …   Wikipedia

• List of important publications in mathematics — One of the oldest surviving fragments of Euclid s Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.[1] This is a list of important publications in mathematics, organized by field. Some… …   Wikipedia

• Gluing axiom — In mathematics, the gluing axiom is introduced to define what a sheaf F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor : F : O ( X ) rarr; C to a category C which initially one… …   Wikipedia

• Tensor contraction — In multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components… …   Wikipedia