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
, and restriction means that we consider the restrictions of the coherent sheaves
. Because the support of
, the restriction of
, 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
By induction it follows that all are locally free.
If has dimension at least and is normal
(or is at least ),
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
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
and hence the inductively defined cohomology classes
Now suppose that is Cohen-Macaulay of dimension . Then
for and therefore
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.
- 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 .
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 .
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
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
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.
Suppose that is a regular local ring of positive characteristic and of dimension .
Then for every ideal we have .
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
is a regular system of parameters of
. We have to show that there is no
. Multiplying the class with some element of
we may assume that
is a unit.
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.
Suppose that is a graded Cohen-Macaulay ring of positive characteristic and of dimension . Let be a homogeneous system of parameters.
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 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
. The same holds for any equation under the condition that this
(hyper)-surface is a normal domain and
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.
- ↑ We work with the Frobenius pull-back of the sheaves and of the class.
- ↑ 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 .
- ↑ 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.
- ↑ This number is called the -invariant of .