Nagata's conjecture on curves: Difference between revisions

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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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==See also==
* [[Swan's theorem]]
 
==References==
*Sections 0.5.3 and 0.5.4 of {{EGA|book=I}}
 
==External links==
*{{PlanetMath attribution|id=4618|title=Locally free}}
 
[[Category:Algebraic geometry]]
[[Category:Sheaf theory]]

Latest revision as of 23:47, 27 April 2014

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