Locally free sheaf: Difference between revisions
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#redirect [[Coherent sheaf]] |
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{{mergeto|coherent sheaf|date=June 2014}} |
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In [[sheaf theory]], a field of mathematics, a sheaf of <math>\mathcal{O} _X</math>-modules <math>\mathcal{F}</math> on a [[ringed space]] <math>X</math> is called ''locally free'' if for each point <math>p\in X</math>, there is an [[topological space|open]] [[neighborhood (mathematics)| neighborhood]] <math>U</math> of <math>p</math> such that <math>\mathcal{F}| _U</math> is [[free module|free]] as an <math>\mathcal{O} _X| _U</math>-module. This implies that <math>\mathcal{F}_p</math>, the [[Stalk of a sheaf|stalk]] of <math>\mathcal{F}</math> at <math>p</math>, is free as a <math>(\mathcal{O} _X)_p</math>-module for all <math>p</math>. The converse is true if <math>\mathcal{F}</math> is moreover [[coherent sheaf|coherent]]. If <math>\mathcal{F}_p</math> is of finite rank <math>n</math> for every <math>p\in X</math>, then <math>\mathcal{F}</math> is said to be of rank <math>n.</math> |
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==Examples== |
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Let <math>X = \operatorname{Spec}(R)</math>. Then any [[finitely generated projective module]] over ''R'' can be viewed as a locally free <math>\mathcal{O}_X</math>-module. (cf. Hartshorne.) |
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Example: any topological vector bundle on a [[compact space]] ''X'' can be thought of as a locally free <math>\mathcal{O}_X</math>-module where <math>\mathcal{O}_X</math> is the sheaf of rings of continuous functions on ''X'', by [[Swan's theorem]]. Indeed, take ''R'' to be the ring of continuous functions on ''X''. Then Swan's theorem says the functor <math>E \mapsto \Gamma(X, E)</math> from the category of vector bundles on ''X'' to the category of finitely generated projective module over ''R'' is an equivalence of categories.--> |
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==See also== |
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* [[Coherent sheaf]], in particular [[Picard group]] |
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* [[Swan's theorem]] |
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* [[Algebraic vector bundle]] |
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==References== |
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*{{Hartshorne AG}} |
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*Sections 0.5.3 and 0.5.4 of {{EGA|book=I}} |
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==External links== |
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*{{PlanetMath attribution|id=4618|title=Locally free}} |
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[[Category:Algebraic geometry]] |
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[[Category:Sheaf theory]] |
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[[Category:Vector bundles]] |
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Latest revision as of 13:35, 16 July 2023
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