Locally free sheaf

A sheaf of modules that is locally isomorphic to the direct sum of several copies of the structure sheaf. More precisely, let $( X , {\mathcal O} _ {X} )$ be a ringed space. A sheaf of modules ${\mathcal F}$ over ${\mathcal O} _ {X}$ is said to be locally free if for every point $x \in X$ there is an open neighbourhood $U \subset X$, $x \in U$, such that the restriction ${\mathcal F} \mid _ {U}$ of ${\mathcal F}$ to $U$ is a free sheaf of modules over ${\mathcal O} _ {X } \mid _ {U }$, that is, it is isomorphic to the direct sum of a set $I ( x)$ of copies of the structure sheaf ${\mathcal O} _ {X} \mid _ {U }$. If $X$ is connected and $I ( x)$ is finite, for example consisting of $n$ elements, then $n$ does not depend on the point $x$ and is called the rank of the locally free sheaf ${\mathcal F}$. Let $V$ be a vector bundle of rank $n$ on $X$ and let ${\mathcal F}$ be the sheaf of germs of its sections. Then ${\mathcal F}$ is a locally free sheaf of rank $n$. Conversely, for every locally free sheaf ${\mathcal F}$ of rank $n$ there is a vector bundle $V$ of rank $n$ on $X$ such that ${\mathcal F}$ is the sheaf of germs of its sections (see [1], [2]); hence there is a natural one-to-one correspondence between the isomorphy classes of locally free sheaves of rank $n$ and the isomorphy classes of vector bundles of rank $n$ on $X$.

Example. Let $X$ be a smooth connected algebraic variety of dimension $n$. Then the sheaf of regular differential forms $\Omega _ {X} ^ {1}$ is a locally free sheaf of rank $n$.

Let $X = \mathop{\rm Spec} A$, a connected affine scheme, be the spectrum of the commutative ring $A$( cf. Spectrum of a ring), let ${\mathcal F}$ be a locally free sheaf of rank $n$ and let $M = \Gamma ( X , {\mathcal F} )$ be the $A$- module of its global sections. Then the $A$- module $M$ is projective and the mapping ${\mathcal F} \mapsto \Gamma ( X , {\mathcal F} )$ establishes a one-to-one correspondence between the set of classes (up to isomorphisms) of locally free sheaves of rank $n$ and the set of classes (up to isomorphisms) of projective $A$- modules of rank $n$( see [2]).

References

 [1] R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958) MR0102797 Zbl 0080.16201 [2] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
How to Cite This Entry:
Locally free sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_free_sheaf&oldid=47697
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article