# Kurs:Vector bundles, forcing algebras and local cohomology (Medellin 2012)/Lecture 10

*Affineness 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 (or a torsor over a punctured twodimensional spectrum). 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 behavior of the affineness property of torsors is only possible if we have a weird behavior of strong semistability.

*Arithmetic deformations*

We start with the arithmetic situation, the following example is due to Brenner and Katzman.

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.

In terms of affineness (or local cohomology) this example has the following properties: the ideal

has cohomological dimension if and has cohomological dimension (equivalently, is an affine scheme) if .

*Geometric deformations - A counterexample to the localization problem*

Let
be a multiplicative system and an ideal in . Then the *localization problem* of tight closure is the question whether the identity

holds.

Here the inclusion is always true and is the problem. The problem means explicitly:

- if , can we find an such that holds in ?

be a one-dimensional domain and of finite type, and an ideal in . Suppose that localization holds and that

( is the multiplicative system). Then holds in for almost all in Spec .

, , such that .

By persistence of tight closure (under a ring homomorphism) we get

The element does not belong to for almost all , so is a unit in and hence

In order to get a counterexample to the localization property we will look now at geometric deformations:

where has degree and have degree 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.

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, in terms of Hilbert-Kunz theory, by Paul Monsky in 1998.

Let

Consider

Then Monsky proved the following results on the *Hilbert-Kunz multiplicity*
of the maximal ideal in , a field:

By the geometric interpretation of Hilbert-Kunz theory this means that the restricted cotangent bundle

is strongly semistable in the transcendental case, but not strongly semistable in the algebraic case. In fact, for , , where , the -th Frobenius pull-back destabilizes (meaning that it is not semistable anymore).

The maximal ideal can not be used directly. However, we look at the second Frobenius pull-back which is (characteristic two) just

By the degree formula we have to look for an element of degree . Let's take . 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 show that for all algebraic instances the element belongs never to the tight closure of .

In terms of affineness
(or local cohomology)
this example has the following properties: the ideal

has cohomological dimension if is transcendental and has cohomological dimension (equivalently, is an affine scheme) if is algebraic.

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.