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