Arithmetic and geometric deformations of tight closure problems
We consider a linear homogeneous equation
-
and also a linear inhomogeneous equation
-
where are elements in a field . The solution set to the homogeneous equation is a vector space over of dimension
or
(if all are ).
For the solution set of the inhomogeneous equation there exists an action
-
and if we fix one solution
(supposing that one solution exists), then there exists a bijection
-
Suppose now that is a geometric object
(a topological space, a manifold, a variety, the spectrum of a ring) and that
-
are functions on . Then we get the space
-
together with the projection to . For a fixed point , the fiber of over is the solution set to the corresponding inhomogeneous equation. For , we get a solution space
-
where all fibers are vector spaces
(maybe of non-constant dimension) and where again acts on . Locally, there are bijections . Let
-
Then is a vector bundle and is a -principal fiber bundle.
is fiberwise an affine space over the base and locally an affine space over , so locally it is an easy object. We are interested in global properties of
and of .
We describe now the algebraic setting.
This algebra was introduced by Hochster. The forcing algebra forces that belongs to the extended ideal . It yields a scheme morphism
-
We are interested in the relationship:
How is related to ?
Does belong to certain closure operations of ? Properties of .
Examples
-
-
-
Tight closure
We want to deal with tight closure, a closure operation introduced by Hochster and Huneke.
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 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).
In this talk we will focus on four problems of tight closure.
- Is there a geometric interpretation for tight closure?
- Are there situations where one can determine the tight closure of an ideal?
- How does tight closure depend on the prime characteristic?
- The localization problem of tight closure.
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 ?
- 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
-
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 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. For this 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 .
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.
- Deformations of two-dimensional tight closure problems
Let flat, and in . For every , a field, we can consider and in . In particular for , .
How does the property
-
vary with ?
There are two cases:
- contains . This is a geometric or equicharacteristic deformation. Example .
- contains . This is an arithmetic or mixed characteristic deformation. Example .
We deal first with the arithmetic situation.
In particular, in this example, the syzygy bundle is semistable on the generic fiber in characteristic zero, but not strongly semistable for infinitely many prime numbers. A question of Miyaoka asks whether in such a situation it is always strongly semistable for at least infinitely many prime reductions (it is not for almost all prime reductions).
We now show that the geometric deformations are a special case of the localization problem.
We will look at the easiest possibilty of such a deformation:
-
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.
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
( 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 .
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
.