# Kurs:Vector bundles and ideal closure operations (MSRI 2012)/Lecture 2

*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 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).

We mention an alternative description of the torsor corresponding to a first cohomology class in a locally free sheaf which is better suited for the projective situation.

Let denote a locally free sheaf on a scheme . For a cohomology class one can construct a geometric object: Because of , the class defines an extension

This extension is such that under the connecting homomorphism of cohomology, is sent to . The extension yields a projective subbundle

If is the corresponding geometric vector bundle of , one may think of as which consists for every base point of all the lines in the fiber passing through the origin. The projective subbundle has codimension one inside , for every point it is a projective space lying (linearly) inside a projective space of one dimension higher. The complement is then over every point an affine space. One can show that the global complement

is another model for the torsor given by the cohomology class. The advantage of this viewpoint is that we may work, in particular when is projective, in an entirely projective setting.

*Forcing algebras and induced torsors*

Let be the spectrum of a forcing algebra. 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 algebrasThe closure operations we have considered in the first lecture and in the tutorial can be characterized by some property of the forcing algebra. However, they can not be characterized by a property of the corresponding torsor alone. For example, for , we may write

so the torsors given by the forcing algebras

are all the same (the restriction over ), but their global properties are quite different. We have a non-surjection, a surjective non submersion, a submersion which does not admit (for ) a continuous section and a map which admits a continuous section.

*Tight closure and solid closure*

We deal now with closure operations which depend only on the torsor which the forcing algebra defines, so they only depend on the cohomology class of the forcing data inside the syzygy bundle. Our main example is tight closure, a theory developed by Hochster and Huneke, and related closure operations like solid closure and plus closure.

Let be a noetherian domain of positive characteristic, let

be the *Frobenius homomorphism* and

(mit ) its th iteration. Let be an ideal and set

Then define the *tight closure* of to be the ideal

The element defines the cohomology class . Suppose that is normal and that has height at least
(think of a local normal domain of dimension at least and an -primary ideal ).
Then the th Frobenius pull-back of the cohomology class is

() and this is the cohomology class corresponding to . By the height assumption, if and only if , and if this holds for all then by definition. This shows already that tight closure under the given conditions does only depend on the cohomology class.

This is also a consequence of the following theorem of Hochster which gives a characterization of tight closure in terms of forcing algebra and local cohomology.

if and only if

where

denotes the forcing algebra of these elements.If the dimension is at least two, then

This means that we have to look at the cohomological properties of the complement of the exceptional fiber over the closed point, i.e. the torsor given by these data. If the dimension is two, then we have to look whether the first cohomology of the structure sheaf vanishes. This is true
(by Serre's
cohomological criterion
for affineness)
if and only if the open subset is an *affine scheme*
(the spectrum of a ring).

The right hand side of this equivalence
- the non-vanishing of the top-dimensional local cohomology -
is independent of any characteristic assumption, and can be taken as the basis for the definition of another closure operation, called *solid closure*. So the theorem above says that in positive characteristic tight closure and solid closure coincide. There is also a definition of tight closure for algebras over a field of characteristic by reduction to positive characteristic.

In the following we will deal with some important questions from tight closure theory, namely the relation to plus closure and whether tight closure commutes with localization. The geometric interpretation with bundles will guide us through these questions.

*Plus closure*

For an ideal
in a domain define its *plus closure* by

Equivalent: Let be the *absolute integral closure* of . This is the integral closure of in an algebraic closure of the quotient field
(first considered by Artin).
Then

The plus closure commutes with localization.

We also have the inclusion . Here the question arises:

Question: Is ?

This question is known as the *tantalizing question* in tight closure theory.

In terms of forcing algebras and their torsors, the containment inside the plus closure means that there exists a -dimensional closed subscheme inside the spectrum of the forcing algebra which meets the exceptional fiber (the fiber over the maximal ideal) in only finitely many points. This also means that the superheight of the extended ideal is . In this case the local cohomological dimension of the torsor must be as well, since it contains a closed subscheme with this cohomological dimension. So also the plus closure depends only on the torsor. One should think of tight closure as a cohomological property and of plus closure as a geometric property of torsors.

One of the main results about tight closure and plus closure is the following theorem due to K. Smith.