Benutzer:Bocardodarapti/Talk in Fez June 2008
Geometric deformations of strong semistability 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
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). The two main results on this question are the following:
Theorem ((K. Smith))
Theorem ((Brenner))
Let be a standard-graded, two-dimensional normal domain over(the algebraic closure of) a finite field. Let be an -primary graded ideal.
Then
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.
The result just mentioned rests on a detailed study of tight closure in dimension two with the help of vector bundles on the corresponding curve.
The first chance for a counterexample is thus in dimension three, no parameter ideal. We look at a special case of the localization problem:
GEOMETRIC DEFORMATIONS OF 2-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 .
Proposition
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 .
Beweis
,,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
We will look for 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.
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 thatfor all . This cohomology class lives in
For the plus closure we have a similar correspondence:
- if and only if there exists a curvesuch that the pull-back of the cohomology class vanishes.
Definition ((semistable and strongly semistable))
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.
Theorem ((Brenner))
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.
A COUNTEREXAMPLE TO THE LOCALIZATION PROPERTY
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.
Example ((Monsky))
Let
Consider
Then Monsky proved the following results on the Hilbert-Kunz multiplicityof 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 , , 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
- in
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 .
Theorem ((Brenner-Monsky))
Beweis
Corollary
Beweis
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.