# Kurs:Vector bundles and their torsors (Kolkata 2011)/Lecture 2

We consider a linear homogeneous equation

and also a linear inhomogeneous equation

where are elements in a field . The solution set to the homogeneous equation is a vector space over of dimension or (if all are ). For the solution set of the inhomogeneous equation there exists an action

(supposing that one solution exists), then there exists a bijection

Suppose now that is a geometric object (a topological space, a manifold, a variety, the spectrum of a ring) and that

(maybe of non-constant dimension) and where again acts on . Locally, there are bijections . Let

is fiberwise an affine space over the base and locally an affine space over , so locally it is an easy object. We are interested in global properties of and of .

*Group schemes and their actions*

We have seen in the first lecture that a vector bundle is in particular a group scheme, i.e. there is a scheme morphism (the addition)

a morphism

(the zero section) and a negative morphism

fulfilling several natural conditions. In general, as a group may act on a set, a group scheme may act on another scheme. We give the precise definition.

Let denote a group scheme over a scheme and let

denote a scheme over . A morphism

is called a
*group scheme action*
of on , if the diagram

commutes and if the composition

is the identity on .

The multiplication of a group scheme may be considered as an operation of the group scheme on itself. These are in some sense the easiest group operations. The next easiest case is the situation of an operation which looks locally like the group acting on itself. This leads to the following natural definition.

Let denote a group scheme over a scheme . A scheme together with a group scheme 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.

*Torsors of vector bundles*

We look now at the torsors of vector bundles. They 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 there are 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).
This is already true if we take the affine line over
(corresponding to the structure sheaf)
as vector bundle and consider the -torsors. These are in particular an interesting class of schemes for a two-dimensional ring.

Let denote a two-dimensional local noetherian domain and let and be two parameters in , i.e. elements which generate the maximal ideal up to radical. Then the punctured spectrum is

with some (). If is normal then the cohomology class is if and only if . To see this, we work with , which does not make a difference as powers of powers are again parameters. We note that under the normality assumption the syzygy module is free of rank one ( is a generator). Then we look at the short exact sequence on ,

and its corresponding long exact sequence of cohomology,

Here, the connecting homomorphisms for an element works in the following way. On both open subsets and we get the local representatives and and their difference, considered in , defines the cohomology class. This difference is just . By the exactness of the long cohomology sequence, if and only if comes from the left, which is true if and only if belongs to the ideal generated by .

If we want to realize the geometric torsor corresponding to such a cohomology class, we start with two affine lines over and , which we write as and . According to Proposition 2.3 these have to be glued with the identification .

*Forcing algebras and induced torsors*

(and in general a syzygy group scheme). Now we make the transition from homogeneous linear equations to inhomogeneous linear equations with the following definition.

Let be a commutative ring and let and be elements in . Then the -algebra

is called the *forcing algebra* of these elements
(or these data).

The forcing algebra forces to lie inside the extended ideal . For every -algebra such that there exists a (non unique) ring homomorphism by sending to the coefficient in an expression .

The forcing algebra induces the spectrum morphism

Over a point , the fiber of this morphism is given by

If we localize the forcing algebra at we get

(as in the first lecture) two identifications with affine space, but the transition morphisms are now not linear anymore, only affine-linear (because of the translation with ).

Let denote a commutative ring, an ideal with the syzygy group scheme given by . Let be another element and let be the spectrum of the corresponding forcing algebra.

** Then there is a natural action of on . The restriction of this action to the open subset makes to a torsor for the vector bundle . **

The action is induced by the co-operation which sends . In terms of points this is just the mapping which sends a syzygy and a solution of the forcing equation to the new solution of the forcing equation.

For the second statement let denote the syzygy bundle over . We may consider the situation on , where we have the isomorphism

With these isomorphisms the natural diagram

commutes, so locally the natural action of the vector bundle on the restricted spectrum of the forcing algebra is isomorphic to the addition of the vector bundle on itself.

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.

We continue with Beispiel *****, so let denote a two-dimensional normal local noetherian domain and let and be two parameters in . The torsor given by 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 quotients 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 algebras . Since the fiber over the maximal ideal is empty, the spectrum of the forcing algebra equals the torsor. Or, the other way round, the torsor is itself an affine scheme.

It is a difficult question when a torsor is an affine scheme. In the next two 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 Sunday: Show that belongs to the radical of the ideal if and only if the spectrum morphism

is surjective.