Locally free sheaf

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.)

Example: any topological vector bundle on a topological space X can be thought of as a locally free ${\mathcal {O}}_{X}$-module where ${\mathcal {O}}_{X}$ 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 $E\to \Gamma (X,E)$ 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 maps are algebraic functions. All $SU(2)$-instantons over the sphere $S^{4}$ are algebraic vector bundles.

See also

• Coherent sheaf, in particular Picard group
• Swan's theorem

References

• Template:Hartshorne-AG
• 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.