Kurs:Computation of tight closure (Ann Arbor 2012)/Lecture 1

Aus Wikiversity
Zur Navigation springen Zur Suche springen

In these lectures we want to demonstrate how tight closure can be understood and computed with the help of geometric and cohomological methods. We recall briefly the definition of tight closure.



Tight closure

Let be a noetherian domain of positive characteristic, let

be the Frobenius homomorphism, and

its th iteration. Let be an ideal and set

Then define the tight closure of to be the ideal

If is not a domain, then one requires that does not belong to any minimal prime ideal of . This definition is not well suited for computations. The problem is that it one has to check infinitely many conditions. The tight closure of an ideal in a regular ring is just the ideal itself. The following observations translate the containments and into statements on certain cohomology classes.

Let be a noetherian local ring of dimension (we treat the case of a standard-graded ring at the same time) and let be an -primary ideal. Then we have a free (not necessarily minimal) resolution (in fact it is enough that the complex is exact on the punctured spectrum)

An element belongs to if and only if it is mapped to in . We split up the long exact sequence into several short exact sequences of -modules, namely into

etc. These syzygy modules do not have especially nice properties. This changes if we consider the restriction of these sequences to the open subset

This scheme is called the punctured spectrum of , and restriction means that we consider the restrictions of the coherent sheaves . Because the support of is just , the restriction of to becomes , hence we get the short exact sequence

(we do not distinguish in the symbols between the modules and the sheaves, with the exception of the structure sheaf). That this last mapping is surjective is also clear since the corresponding module-mapping is surjective when localized at and since (and since sheaf surjectivity is a local property). Now for a surjective sheaf homomorphism

between locally free sheaves the kernel is itself locally free. So in particular is locally free

(on ). By induction it follows that all are locally free.

If has dimension at least and is normal (or is at least ), then

Hence if and only if , and this property can be checked over . Because is itself not an affine scheme, this property is not a local property, but a global property. Locally belongs to the ideal sheaf given by on . The difference between local and global properties are usually controlled by sheaf cohomology (or by local cohomology). The short exact sequence

from above gives rise to a long exact cohomology sequence

The element is mapped to some element

Now

because comes from the left if and only if it is mapped to on the right.

The short exact sheaf sequence

yields again a long exact cohomology sequence, and we write down the part

In particular we get a cohomology class

in . Suppose that . Then

is injective and so if and only if . From the other short exact sheaf sequences

we obtain

and hence the inductively defined cohomology classes

Now suppose that is Cohen-Macaulay of dimension . Then

for and therefore

for between and and

is injective

(). Thus if and only if for any . We will in particular work with

and call this the top-dimensional cohomology class inside the top-dimensional syzygy sheaf.


Example  

Suppose that the (primary) ideal ( is local and Cohen-Macaulay) has finite projective dimension. Then we have a finite free resolution

and the length of this resolution is due to the Auslander-Buchsbaum formula, since the depth of is . Then the top-dimensional syzygy module is free, just because

The cohomology class is then described as

and it is if and only if all components are . These components lie in the (often) well understood cohomology module

If is even regular, then every ideal has a finite free resolution and the ideal containment problem reduces to the computation of (the components of) and deciding whether they are or not.


Example  

Let be a Cohen-Macaulay local ring of dimension and let be a (full) parameter ideal. We consider the Koszul-resolution of these parameters. This is a finite resolution and the top-dimensional syzygy sheaf is just the structure sheaf. A direct computation using Čech-cohomology shows that for an element we get



Cohomological criteria for tight closure

The following theorem says that tight closure (for a Cohen-Macaulay ring of dimension at least ) is a “cohomological closure operation”, i.e. it depends only on the induced cohomological class over the punctured spectrum . This is the base for understanding tight closure on and (in the graded case) on .


Theorem  

Suppose that is a Cohen-Macaulay ring of positive characteristic and of dimension . Let be an -primary ideal. Let be a free (not necessarily minimal) resolution of , let be the corresponding syzygy sheaves, let be an element and let be the corresponding cohomology classes.

Then for each , , we have the equivalence that if and only if is tightly in the sense that there exists not in any minimal prime ideal such that

in for all .[1]

Proof  

We consider the short exact sheaf sequences

on coming from the resolution for ( is just the structure sheaf). Because all these sheaves are locally free, taking the absolute Frobenius (and all its iterations) is exact, therefore we get short exact sequences[2]

and cohomology pull-backs . Note also that for and we get

so the image of this map inside is exactly . By the universal property of the absolute Frobenius and of the connecting homomorphisms in cohomology we have

and also

Because of the injectivity of in the given range we have that belongs to the ideal if and only if if and only if .


In general it is difficult to control the sequence , , of locally free sheaves. It is one of the goals of these lectures to discuss situations where it can be controlled. The easiest case is when is free (which is only possible for ). In this case we can deduce two well-known theorems in tight closure theory. The presented proofs are different from the classical proofs and give a hint how we will argue in the next lectures.

The standard proof of the following theorem uses the fact that the Frobenius is flat for regular rings. We use instead that every ideal in a regular ring has a finite free resolution or, equivalently, that the top-dimensional syzygy sheaf is free.


Theorem  

Suppose that is a regular local ring of positive characteristic and of dimension .

Then for every ideal we have .

Proof  

We assume , lower dimensions may be treated directly. Because of we can also reduce to the case of a primary ideal . Suppose that , and let be the corresponding non-zero class arising from a finite free resolution. At least one component, say is then also non-zero, and we can write it in terms of Čech-cohomology as

where is a regular system of parameters of and . We have to show that there is no such that for all . Multiplying the class with some element of we may assume that is a unit.[3]

We have (with )

and its annihilator is . But then


The standard proof of the following fact is based on the Briançon-Skoda theorem. It is also true without the Cohen-Macaulay condition.


Theorem  

Suppose that is a graded Cohen-Macaulay ring of positive characteristic and of dimension . Let be a homogeneous system of parameters.

Then .

Proof  

We assume , lower dimensions may be treated directly. We consider the Koszul-resolution of the parameter ideal . A homogeneous element of degree gives rise to the graded cohomology class

Under the condition that this cohomology class has nonnegative degree. It is known that is for sufficiently large degree, i.e. there is a number[4] such that for all . Now choose a homogeneous element which does not belong to any minimal prime ideal. Then the degree of

is at least , so this class must be and therefore belongs to the tight closure of the parameter ideal.


A theorem of Hara states that the “converse” of this theorem is also true for prime numbers , i.e. that an element of degree smaller than the sum of the degrees of the parameters can belong to the tight closure only if it belongs already to the ideal itself.

A classical example of this inclusion criterion is that

in the Fermat ring in characteristic . The same holds for any equation under the condition that this

(hyper)-surface is a normal domain and and are parameters.

In these lectures we are in particular interested in determining degree bounds for the tight closure of primary ideals which are not parameter ideals. An easy looking question for a non-parameter ideal was raised by M. McDermott, namely whether

This was answered positively by A. Singh by a long “equational” argument.


Example  

Let , where is a field of positive characteristic , and . We consider the short exact sequence

and the cohomology class

We want to show that for all (here the test element equals the element in the ring). It is helpful to work with the graded structure on this syzygy sheaf (or to work on the corresponding elliptic curve directly). Now the equation can be considered as a syzygy (of total degree ) for , yielding an inclusion

Since this syzygy does not vanish anywhere on the quotient sheaf is invertible and in fact isomorphic to the structure sheaf. Hence we have

and the cohomology sequence

where denotes the degree-th piece. Our cohomology class lives in , so its Frobenius pull-backs live in , and we can have a look at the cohomology of the pull-backs of the sequence, i.e.

The class lives in . It is mapped on the right to , which is (because we are working over an elliptic curve), hence it comes from the left, which is . So and .



Footnotes
  1. We work with the Frobenius pull-back of the sheaves and of the class.
  2. Note that these sequences come also from the Frobenius pull-backs of the resolution complex by restriction to . The Frobenius pull-backs of the resolution complex are however not exact anymore. Hence it is better to work only on . So it is also allowed that the “resolution” we start with is only exact on .
  3. First we write the class as a sum of fractions where the numerators are units and the denominators are several monomials. Then we can multiply with a monomial so that only one summand remains.
  4. This number is called the -invariant of .


Pdf-version