Locally free sheaf: Difference between revisions

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In sheaf theory, a field of mathematics, a sheaf of ${\mathcal {O}}_{X}$ -modules ${\mathcal {F}}$ on a ringed space $X$ is called locally free if for each point $p\in X$ , there is an open neighborhood $U$ of $p$ such that ${\mathcal {F}}|_{U}$ is free as an ${\mathcal {O}}_{X}|_{U}$ -module. Taking an inductive limit, this implies that ${\mathcal {F}}_{p}$ , the stalk of ${\mathcal {F}}$ at $p$ , is free as a $({\mathcal {O}}_{X})_{p}$ -module for all $p$ . The converse is true if ${\mathcal {F}}$ is moreover coherent. If ${\mathcal {F}}_{p}$ is of finite rank $n$ for every $p\in X$ , then ${\mathcal {F}}$ is said to be of rank $n.$

Example: Let $X=\operatorname {Spec} (R)$ . Then any finitely generated projective module over R can be viewed as a locally free ${\mathcal {O}}_{X}$ -module. (cf. Hartshorne.)

In mathematics, an algebraic vector bundle is a vector bundle for which all the transition maps are algebraic functions. All $SU(2)$ -instantons over the sphere $S^{4}$ are algebraic vector bundles.

References

Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
Sections 0.5.3 and 0.5.4 of Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4. doi:10.1007/bf02684778. MR 0217083.

External links

This article incorporates material from Locally free on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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 Example: 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.
+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.-->
 In [[mathematics]], an '''algebraic vector bundle''' is a [[vector bundle]] for which all the [[transition map]]s are [[algebraic function]]s. All <math>SU(2)</math>-[[instanton]]s over the [[sphere]] <math>S^4</math> are algebraic vector bundles.

Revision as of 02:42, 14 June 2014

See also

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