Deformations of vector bundles and the localization problem in tight closure theory
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 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 ?
Why does localization often holds?
Because an element belongs to the tight closure because of a certain reason, and this reason often localizes/globalizes.
A typical reason for belonging to the tight closure is through 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.
One should think of the left hand side as a geometric condition and of the right hand side as a cohomological conditon (later).
In dimension two tight closure always localizes. However, a positive answer to the tantalizing question in dimenson two in general would imply the localization property in dimension three.
It is difficult in general to get hold of tight closure and plus closure, they are difficult to compute. We will describe now a specific situation where one has a kind of geometric interpretation for and a detailed understanding of tight closure.
- Forcing algebras
Forcing algebras, introduced by Hochster, give a new interpretation of tight closure.
If we fix a point , then the fiber over of the spectrum of the forcing algebra, , is just
-
This is the solution set to an inhomogeneous linear equation over , so it is an affine space (in the sense of linear algebra) (may be empty). If the ideal is primary to , and , then the fiber over has dimension , whereas the dimensions of the other fibers are .
The relation between tight closure and forcing algebras is given in the following theorem.
Let
be a normal excellent local domain with maximal ideal
over a field of positive characteristic. Let
generate an
-primary ideal
and let
be another element in
. Then
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. If the dimension is two, then we have to look whether the first cohomology of the structure sheaf vanishes. This is true if and only if the open subset is an affine scheme (the spectrum of a ring).
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.
- Geometric interpretation in dimension two
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
-
An element
-
defines a cohomology class
-
With this notation we have
-
(homogeneous of some degree )
such that
for all . This cohomology class lives in
-
For the plus closure we have a similar correspondence:
-
if and only if there exists a curve
such that the pull-back of the cohomology class vanishes.
For a vector bundle and a cohomology class one can construct a geometric object: Because of , the class defines an extension
-
and hence projective bundles
-
and the corresponding torsor
-
This geometric object is the torsor or principal fiber bundle or affine-linear bundle given by . The sheaf acts on it by translations.
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 (and if and only if contains a projective curve). For these properties, positivity (ampleness) properties of the syzygy bundle are crucial. We need the concept of semistability.
For strongly semistable syzygy bundles we get the following degree formula; the rational number occuring in it is called the degree bound for tight closure.
Suppose that
is strongly semistable. Then
-
In general, there exists an exact criterion depending on and the strong Harder-Narasimhan filtration of .
If is finite, then the same criterion holds for plus closure.
Therefore we get:
The first chance for a counterexample to the localization property is in dimension three. We will look at a special case of the localization problem. Since we understand now the two-dimensional situation quite well, it is natural to look at a family of two-dimesional rings and corresponding families of projective curves.
- Geometric deformations of two-dimensional tight closure problems
Let (think of ) and let be flat, and in .
For every , a field,
we can consider and in .
In particular for , .
How does the property
-
vary with ?
We call such a situation a geometric or equicharacteristic deformation of tight closure problems.
We will look at the easiest possibilty of such a deformation:
-
where has degree and have degree one and is homogeneous. Then (for )
-
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.
Note that in the second case tight closure is plus closure. 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.
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.
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! 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 .
This is an elementary but tedious computation.
Tight closure does not commute with localization.
One knows in our situation that
is a so-called test element. Hence the previous Lemma shows that
.
Tight closure is not plus closure in graded dimension two for fields with transcendental elements.
Consider
-
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.
There is an example of a smooth projective
(relatively over the affine line)
variety and an effective divisor
and a morphism
-
such that
is not an affine variety over the generic point
, but for every algebraic point
the fiber
is an affine variety.
Take
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.