# Kurs:Vector bundles and their torsors (Kolkata 2011)/Lecture 5

In this lecture we will continue with the question when are the torsors given by a forcing algebras over a two-dimensional ring affine? We will look at the graded situation to be able to work on the corresponding projective curve.

In particular we want to address the following questions

- Is there a procedure to decide whether the torsor is affine?
- Is it non-affine if and only if there exists a geometric reason for it not to be affine (because the superheight is too large)?
- How does the affineness vary in an arithmetic family, when we vary the prime characteristic?
- How does the affineness vary in a geometric family, when we vary the base ring?

In terms of tight closure, these questions are directly related to the tantalizing question of tight closure (is it the same as plus closure), the dependence of tight closure on the characteristic and the localization problem of tight closure.

*Geometric interpretation in dimension two*

We will restrict now to the two-dimensional homogeneous case in order to work on the corresponding projective curve. We want to find an object over the curve which corresponds to the forcing algebra.

Let be a two-dimensional standard-graded normal domain over an algebraically closed field . Let be the corresponding smooth projective curve and let

be an -primary homogeneous ideal with generators of degrees . Then we get on the short exact sequence

Here is a vector bundle, called the *syzygy bundle*, of rank and of degree

Thus a homogeneous element of degree defines a cohomology class , so this defiens a torsor over the projective curve. 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.

In the situation of a forcing algebra for homogeneous elements, this torsor can also be obtained as , where is the (not necessarily positively) graded forcing algebra. In particular, it follows that the containment is equivalent to the property that is not an affine variety. For this properties, positivity (ampleness) properties of the syzygy bundle are crucial. We need the concept of semistability.

Let be a vector bundle on a smooth projective curve . It is called *semistable*, if
for all subbundles .

Suppose that the base field has positive characteristic
.
Then is called *strongly semistable*, if all
(absolute)
Frobenius pull-backs are semistable.

For a strongly semistable vector bundle on and a cohomology class with corresponding torsor we obtain the following affineness criterion.

Let denote a smooth projective curve over an algebraically closed field and let be a strongly semistable vector bundle over together with a cohomology class .

** Then the torsor is an affine scheme if and only if
and
(
for all in positive characteristic). **

This implies for a strongly semistable syzygy bundles the following *degree formula* for tight closure.

In general, there exists an exact criterion depending on and the *strong Harder-Narasimhan filtration* of . A Harder-Narasimhan filtration is called strong if all the quotients are strongly semistable. A Harder-Narasimhan filtration is not strong in general, however, by a Theorem of A. Langer, there exists some Frobenius pull-back such that its Harder-Narasimhan filtration is strong.

Let denote a smooth projective curve over an algebraically closed field and let be a vector bundle over together with a cohomology class . Let

be a strong Harder-Narasimhan filtration. We choose such that has degree and that has degree . We set .

** Then the following are equivalent.
**

- The torsor is not an affine scheme.
- Some Frobenius power of the image of inside is .

The same criterion holds for plus closure (the existence of projective curves inside the torsor or the trivializing of the cohomology class along a finite curve map). Hence over (the algebraic closure of) a finite field we have that a torsor is not affine if and only if it contains a projective curve. This implies the following theorem.

Let be a standard-graded, two-dimensional normal domain over (the algebraic closure of) a finite field. Let be an -primary graded ideal.

** Then
**

*Local cohomology under deformations*

We consider a base scheme and a morphism

together with an open subscheme . For every base point we get the open subset

inside the fiber . It is a natural question to ask how properties of vary with . In particular we may ask how the cohomological dimension of varies and how the affineness may vary.

In the algebraic setting we have a -algebra and an ideal (so , and ) which defines for every prime ideal the extended ideal in .

This question is already interesting when is an affine one-dimensional integral scheme, in particular in the following two situations.

- .
Then we speak of an
*arithmetic deformation*and want to know how affineness varies with the characteristic and what the relation is to characteristic zero. - ,
where is a field. Then we speak of a
*geometric deformation*and want to know how affineness varies with the parameter , in particular how the behavior over the special points where the residue class field is algebraic over is related to the behavior over the generic point.

It is fairly easy to show that if the open subset in the generic fiber is affine, then also the open subsets are affine for almost all special points.

We deal with this question where is a torsor over a family of smooth projective curves. The arithmetic as well as the geometric variant of this question are directly related to questions in tight closure theory. Because of the above mentioned degree criteria in the strongly semistable case, a weird behaviour of the affineness property of torsors is only possible if we have a weird behaviour of strong semistability.

We start with the arithmetic situation.

Consider and take the ideal and the element . Consider reductions . Then

and

In particular, the bundle is semistable in the generic fiber, but not strongly semistable for any reduction . The corresponding torsor is an affine scheme for infinitely many prime reductions and not an affine scheme for infinitely many prime reductions.

We will look now at geometric deformations:

where has degree and have degree one and is homogeneous. Then (for every field )

is a two-dimensional standard-graded ring over . For residue class fields of points of we have basically two possibilities.

- , the function field. This is the
*generic*or*transcendental*case.

- , the
*special*or*algebraic*or*finite*case.

How does vary with ? To analyze the behavior of tight closure in such a family we can use what we know in the two-dimensional standard-graded situation.

*A counterexample to the localization problem*

In order to establish an example where tight closure does not behave uniformly under a geometric deformation we first need a situation where strong semistability does not behave uniformly. Such an example was given by Paul Monsky in 1997 in terms of Hilbert-Kunz multiplicity. We consider the ring

where

the ideal

and the element

This is our example ( does not work). First, by strong semistability in the transcendental case we have

by the degree formula. If localization would hold, then would also belong to the tight closure of for almost all algebraic instances , . Contrary to that we can show that for all algebraic instances the element belongs never to the tight closure of .

In this ring ,

but it can not belong to the plus closure. Else there would be a curve morphism which annihilates the cohomology class and this would extend to a morphism of relative curves almost everywhere.

(relatively over the affine line) variety and an effective divisor and a morphism

to be the Monsky quartic and consider the syzygy bundle

together with the cohomology class determined by . This class defines an extension

and hence .

Then is an example with the stated properties by the previous results.

It is an open question whether such an example can exist in characteristic zero.