TIGHT CLOSURE IS DEAD - LONG LIVE TIGHT CLOSURE
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.
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 .
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.
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.
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 .
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.