WebMar 10, 2024 · Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under … WebThe exact condition for locally free sheaves on a ringed space ( X, O X) to be coherent is exactly that O X be coherent. a) The condition is clearly necessary since O X is locally free.
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WebAug 17, 2024 · The direct sum of a family of sheaves is the sheafification of the direct sum of the underlying presheaves. This construction is justified by a general fact from category theory: left adjoints commute with colimits. WebIn 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 … server.tomcat.threads.max
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Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free … See more 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 definition of … See more On an arbitrary ringed space quasi-coherent sheaves do not necessarily form an abelian category. On the other hand, the quasi-coherent … See more Let $${\displaystyle f:X\to Y}$$ be a morphism of ringed spaces (for example, a morphism of schemes). If $${\displaystyle {\mathcal {F}}}$$ is a quasi-coherent sheaf on $${\displaystyle Y}$$, then the inverse image $${\displaystyle {\mathcal {O}}_{X}}$$-module … See more For a morphism of schemes $${\displaystyle X\to Y}$$, let $${\displaystyle \Delta :X\to X\times _{Y}X}$$ be … See more A quasi-coherent sheaf on a ringed space $${\displaystyle (X,{\mathcal {O}}_{X})}$$ is a sheaf $${\displaystyle {\mathcal {F}}}$$ of $${\displaystyle {\mathcal {O}}_{X}}$$-modules which … See more • An $${\displaystyle {\mathcal {O}}_{X}}$$-module $${\displaystyle {\mathcal {F}}}$$ on a ringed space $${\displaystyle X}$$ is called locally free of finite rank, or a vector bundle, if every point in $${\displaystyle X}$$ has an open neighborhood $${\displaystyle U}$$ such … See more An important feature of coherent sheaves $${\displaystyle {\mathcal {F}}}$$ is that the properties of $${\displaystyle {\mathcal {F}}}$$ at a point $${\displaystyle x}$$ control the behavior of $${\displaystyle {\mathcal {F}}}$$ in a neighborhood of $${\displaystyle x}$$, … See more WebMay 1, 2024 · The correct definition is that a sheaf F of O X -Modules is coherent if : 1) F is locally finitely generated : X can be covered by open subsets U on which there exist surjections O U N → F ∣ U . and 2) For any open subset V ⊂ X and any morphism f: O V s → F ∣ V the sheaf K e r ( f) on V is locally finitely generated . the telling movie