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 be a ringed space. A sheaf of modules over is said to be locally free if for every point there is an open neighbourhood , , such that the restriction of to is a free sheaf of modules over , that is, it is isomorphic to the direct sum of a set of copies of the structure sheaf . If is connected and is finite, for example consisting of elements, then does not depend on the point and is called the rank of the locally free sheaf . Let be a vector bundle of rank on and let be the sheaf of germs of its sections. Then is a locally free sheaf of rank . Conversely, for every locally free sheaf of rank there is a vector bundle of rank on such that 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 and the isomorphy classes of vector bundles of rank on .

Example. Let be a smooth connected algebraic variety of dimension . Then the sheaf of regular differential forms is a locally free sheaf of rank .

Let , a connected affine scheme, be the spectrum of the commutative ring (cf. Spectrum of a ring), let be a locally free sheaf of rank and let be the -module of its global sections. Then the -module is projective and the mapping establishes a one-to-one correspondence between the set of classes (up to isomorphisms) of locally free sheaves of rank and the set of classes (up to isomorphisms) of projective -modules of rank (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