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Benutzer:Bocardodarapti/Talk in Fez June 2008

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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:



Let be local and excellent. If is a parameter ideal (generated by a (sub-)system of parameters), then
and the tight closure of commutes with localization.



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 .



    Let

    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 .

    By localization, there exists

    , , 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

    for almost all .


    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 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.



    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.



    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.



    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.



    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 , , 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 .



    Let

    , , . Set . Then

    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.