# Kurs:Vector bundles, forcing algebras and local cohomology (Tehran 2012)/Lecture 2

*Geometric vector bundles*

We have seen that the forcing algebra has locally the form and its spectrum has locally the form . This description holds on the union . Moreover, in the homogeneous case () the transition mappings are linear. Hence is a geometric vector bundle according to the following definition.

Let denote a scheme. A scheme equipped with a morphism

is called a
*geometric vector bundle*
of rank over if there exists an open covering
and -isomorphisms

such that for every open affine subset the transition mappings

are linear automorphisms, i.e. they are induced by an automorphism of the polynomial ring given by .

Here we can restrict always to affine open coverings. If is separated then the intersection of two affine open subschemes is again affine and then it is enough to check the condition on the intersections. The trivial bundle of rank is the -dimensional affine space over , and locally every vector bundle looks like this. Many properties of an affine space are enjoyed by general vector bundles. For example, in the affine space we have the natural addition

and this carries over to a vector bundle, that is, we have an addition

The reason for this is that the isomorphisms occurring in the definition of a geometric vector bundle are linear, hence the addition on coming from an isomorphism with some affine space over is independent of the choosen isomorphism. For the same reason there is a unique closed subscheme of called the *zero-section* which is locally defined to be
.
Also, multiplication by a scalar, i.e. the mapping

carries over to a scalar multiplication

In particular, for every point the fiber is an affine space over .

For a geometric vector bundle and an open subset one sets

so this is the set of sections in over . This gives in fact for every scheme over a set-valued sheaf. Because of the observations just mentioned, these sections can also be added and multiplied by elements in the structure sheaf, and so we get for every vector bundle a locally free sheaf, which is free on the open subsets where the vector bundle is trivial.

A coherent -module on a scheme is called
*locally free*
of rank , if there exists an open covering
and -module-isomorphisms
for every
.

Vector bundles and locally free sheaves are essentially the same objects.

Let denote a scheme.

** Then the category of locally free sheaves on and the category of geometric vector bundles on are equivalent. **

A geometric vector bundle corresponds to the sheaf of its sections, and a locally free sheaf corresponds to the (relative) spectrum of the symmetric algebra of the dual module .

The free sheaf of rank corresponds to the affine space over .

*Torsors of vector bundles*

We have seen that

acts on the spectrum of a forcing algebra by addition. The restriction of to is a vector bundle, and restricted to becomes a -torsor.

Let denote a geometric vector bundle over a scheme . A scheme together with an action

is called a geometric
(Zariski)-*torsor*
for
(or a
-*principal fiber bundle*
or a
*principal homogeneous space*)
if there exists an open covering
and isomorphisms

such that the diagrams (we set and )

commute, where is the addition on the vector bundle.

The torsors of vector bundles can be classified in the following way.

Let denote a noetherian separated scheme and let

denote a geometric vector bundle on with sheaf of sections .

** Then there exists a correspondence between first cohomology classes
and geometric -torsors. **

We describe only the correspondence. Let denote a -torsor. Then there exists by definition an open covering such that there exist isomorphisms

which are compatible with the action of on itself. The isomorphisms induce automorphisms

These automorphisms are compatible with the action of on itself, and this means that they are of the form

with suitable sections
.
This family defines a Čech cocycle for the covering and gives therefore a cohomology class in .

For the reverse direction, suppose that the cohomology class
is represented by a Čech cocycle
for an open covering
.
Set
.
We take the morphisms

given by to glue the together to a scheme over . This is possible since the cocycle condition guarantees the glueing condition for schemes.

The action of
on itself glues also together to give an action on .

It follows immediately that for an affine scheme
(i.e. a scheme of type )
there is no non-trivial torsor for any vector bundle. There will however be in general many non-trivial torsors on the punctured spectrum
(and on a projective variety).

*Forcing algebras and induced torsors*

As is a -torsor, and as every -torsor is represented by a unique cohomology class, there should be a natural cohomology class coming from the forcing data. To see this, let be a noetherian ring and be an ideal. Then on we have the short exact sequence

An element defines an element and hence a cohomology class . Hence defines in fact a -torsor over . We will see that this torsor is induced by the forcing algebra given by and .

Let denote a noetherian ring, let denote an ideal and let be another element. Let be the corresponding cohomology class and let

denote the forcing algebra for these data.

** Then the scheme together with the natural action of the syzygy bundle on it is isomorphic to the torsor given by . **

We compute the cohomology class and the cohomology class given by the forcing algebra. For the first computation we look at the short exact sequence

On , the element is the image of (the non-zero entry is at the th place). The cohomology class is therefore represented by the family of differences

On the other hand, there are isomorphisms

The composition of two such isomorphisms on is the identity plus the same section as before.

Let denote a two-dimensional normal local noetherian domain and let and be two parameters in . On we have the short exact sequence

and its corresponding long exact sequence of cohomology,

The connecting homomorphism sends an element to . The torsor given by such a cohomology class can be realized by the forcing algebra

Note that different forcing algebras may give the same torsor, because the torsor depends only on the spectrum of the forcing algebra restricted to the punctured spectrum of . For example, the cohomology class defines one torsor, but the two fractions yield the two forcing algebras and , which are quite different. The fiber over the maximal ideal of the first one is empty, whereas the fiber over the maximal ideal of the second one is a plane.

If is regular, say (or the localization of this at or the corresponding power series ring) then the first cohomology classes are -linear combinations of , .

They are realized by the forcing algebrasIn the next lectures we will deal with global properties of torsors and forcing algebras and how these properties are related to closure operations of ideals.

Exercise for Saturday: Show that belongs to the radical of the ideal if and only if the spectrum morphism

is surjective.