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) |
[2] | R. Hartshorne, "Algebraic geometry" , Springer (1977) |
Locally free sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Locally_free_sheaf&oldid=23891