# Kurs:Vector bundles and ideal closure operations (MSRI 2012)/Lecture 3

In the last lecture we will focus on 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.

*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 or its induced torsor.

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 defines a torsor over the projective curve.

*Semistability of vector bundles*

In the situation of a forcing algebra of 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 (Mumford-) 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.

An important property of a semistable bundle of negative degree is that it can not have any global section . Note that a semistable vector bundle need not be strongly semistable, the following is probably the simplest example.

Let be the smooth Fermat quartic given by and consider on it the syzygy bundle (which is also the restricted cotangent bundle from the projective plane). This bundle is semistable. Suppose that the characteristic is . Then its Frobenius pull-back is . The curve equation gives a global non-trivial section of this bundle of total degree . But the degree of is negative, hence it can not be semistable anymore.

The following example is related to Beispiel *****.

Let , where is a field of positive characteristic , , and

The equation yields the short exact sequence

This shows that is strongly 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 result rests on the ampleness of occuring in the dual exact sequence given by
(this rests on work of Gieseker and Hartshorne).
It implies for a strongly semistable syzygy bundle the following *degree formula* for tight closure.

If we take on the right hand side , the *Frobenius closure* of the ideal, instead of , then this statement is true for all characteristics. As stated, it is true in a relative setting for large enough.

We indicate the proof of the inclusion result. The degree condition implies that is such that has non-negative degree. Then also all Frobenius pull-backs have non-negative degree. Let be a twist of the tautological line bundle on such that its degree is larger than the degree of , the dual of the canonical sheaf. Let be a non-zero element. Then , and by Serre duality we have

On the right hand side we have a semistable sheaf of negative degree, which can not have a non-trivial section. Hence

and therefore belongs to the tight closure.

*Harder-Narasimhan filtration*

In general, there exists an exact criterion depending on and the *strong Harder-Narasimhan filtration* of . For this we give the definition of the Harder-Narasimhan filtration.

Let be a vector bundle on a smooth projective curve over an algebraically closed field . Then the (uniquely determined) filtration

of subbundles such that all quotient bundles are semistable with decreasing slopes
,
is called the *Harder-Narasimhan filtration* of .

The Harder-Narasimhan filtration exists uniquely (by a Theorem of Harder and Narasimhan). 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 .

*Plus closure in dimension two*

Let be a field and let be a normal two-dimensional standard-graded domain over with corresponding smooth projective curve . A homogeneous -primary ideal with homogeneous ideal generators and another homogeneous element of degree yield a cohomology class

Let be the corresponding torsor. We have seen that the affineness of this torsor over is equivalent to the affineness of the corresponding torsor over
.
Now we want to understand what the property
means for and for . Instead of the plus closure we will work with the *graded plus closure* , where
holds if and only if there exists a finite graded extension
such that
.
The existence of such an translates into the existence of a finite morphism

such that . Here we may assume that is also smooth. Therefore we discuss the more general question when a cohomology class , where is a locally free sheaf on , can be annihilated by a finite morphism

of smooth projective curves. The advantage of this more general approach is that we may work with short exact sequences (in particular, the sequences coming from the Harder-Narasimhan filtration) in order to reduce the problem to semistable bundles which do not necessarily come from an ideal situation.

Let denote a smooth projective curve over an algebraically closed field , let be a locally free sheaf on and let be a cohomology class with corresponding torsor . Then the following conditions are equivalent.

- There exists a finite morphism
from a smooth projective curve such that .

- There exists a projective curve .

If (1) holds, then the pull-back is trivial (as a torsor), as it equals the torsor given by . Hence is isomorphic to a vector bundle and contains in particular a copy of . The image of this copy is a projective curve inside .

If (2) holds, then let be the normalization of . Since dominates , the resulting morphism

.

We want to show that the cohomological criterion for
(non)-affineness of a torsor along the Harder-Narasimhan filtration of the vector bundle also holds for the existence of projective curves inside the torsor, under the condition that the projective curve is defined over a finite field. This implies that tight closure is
(graded)
plus closure for graded -primary ideals in a two-dimensional graded domain over a finite field.

*Annihilation of cohomology classes of strongly semistable sheaves*

We deal first with the situation of a strongly semistable sheaf of degree . The following two results are due to Lange and Stuhler. We say that a locally free sheaf is *étale trivializable* if there exists a finite étale morphism
such that
.
Such bundles are directly related to linear representations of the étale fundamental group.

Let denote a finite field (or the algebraic closure of a finite field) and let be a smooth projective curve over . Let be a locally free sheaf over .

** Then is étale trivializable if and only if there exists some such that
. **

Let denote a finite field (or the algebraic closure of a finite field) and let be a smooth projective curve over . Let be a strongly semistable locally free sheaf over of degree .

** Then there exists a finite morphism
**

We consider the family of locally free sheaves , . Because these are all semistable of degree , and defined over the same finite field, we must have (by the existence of the moduli space for vector bundles) a repetition, i.e.

for some . By Lemma 3.9 the bundle admits an étale trivialization . Hence the finite map trivializes the bundle.

Let denote a finite field (or the algebraic closure of a finite field) and let be a smooth projective curve over . Let be a strongly semistable locally free sheaf over of nonnegative degree and let denote a cohomology class.

** Then there exists a finite morphism
**

**such that is trivial. **

If the degree of is positive, then a Frobenius pull-back has arbitrary large degree and is still semistable. By Serre duality we get that . So in this case we can annihilate the class by an iteration of the Frobenius alone.

So suppose that the degree is . Then there exists by Theorem 3.10 a finite morphism which trivializes the bundle. So we may assume that . Then the cohomology class has several components and it is enough to annihilate them separately by finite morphisms. But this is possible by the parameter theorem of K. Smith (or directly using Frobenius and Artin-Schreier extensions).

*The general case*

We look now at an arbitrary locally free sheaf on , a smooth projective curve over a finite field. We want to show that the same numerical criterion (formulated in terms of the Harder-Narasimhan filtration) for non-affineness of a torsor holds also for the finite annihilation of the corresponding cohomomology class (or the existence of a projective curve inside the torsor).

Let denote a finite field (or the algebraic closure of a finite field) and let be a smooth projective curve over . Let be a locally free sheaf over and let denote a cohomology class. Let be a strong Harder-Narasimhan filtration of . We choose such that has degree and that has degree . We set .

** Then the following are equivalent.
**

- The class can be annihilated by a finite morphism.
- Some Frobenius power of the image of inside is .

Suppose that (1) holds. Then the torsor is not affine and hence by Theorem 3.7 also (2) holds.

So suppose that (2) is true. By applying a certain power of the Frobenius we may assume that the image of the cohomology class in is . Hence the class stems from a cohomology class . We look at the short exact sequence

where the sheaf of the right hand side has a nonnegative degree. Therefore the image of in can be annihilated by a finite morphism due to Fakt *****. Hence after applying a finite morphism we may assume that stems from a cohomology class . Going on inductively we see that can be annihilated by a finite morphism.

Let denote a smooth projective curve over the algebraic closure of a finite field , let be a locally free sheaf on and let be a cohomology class with corresponding torsor .

** Then is affine if and only if it does not contain any projective curve. **

Due to Theorem 3.7 and Theorem 3.12, for both properties the same numerical criterion does hold.

These results imply the following theorem in the setting of a two-dimensional graded ring.

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

** Then
**

This is also true for non-primary graded ideals and also for submodules in finitely generated graded submodules. Moreover, G. Dietz has shown that one can get rid also of the graded assumption (of the ideal or module, but not of the ring).

*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.