Benutzer:Holger Brenner/Talk in Mumbai/Local cohomology and ideal closure operations I
In these two talks I want to discuss three topics related to local cohomology: the affineness of
(quasi-affine) schemes, the relation of local cohomology to closure operations, in particular tight closure, and the behavior of local cohomology
(and cohomological dimension) in an
(arithmetic or geometric)
deformation.
- Affine schemes
A scheme is called affine if it is isomorphic to the spectrum of some commutative ring . If the scheme is of finite type over a field (or ring) (if we have a variety), then this is equivalent to saying that there exist global functions
such that the mapping
is a closed embedding. The relation to cohomology is given by the following well-known theorem of Serre.
Let denote a noetherian scheme. Then the following properties are equivalent.
- is an affine scheme.
- For every quasicoherent sheaf on and all we have .
- For every coherent ideal sheaf on we have .
It is in general a difficult question whether a given scheme is affine. For example, suppose that is an affine scheme and
is an open subset (such schemes are called quasiaffine) defined by an ideal . When is itself affine? The cohomological criterion above simplifies to the condition that for .
Of course, if is a principal ideal (or up to radical a principal ideal), then is affine. On the other hand, if is a local ring of dimension , then
is not affine, since
by the relation between sheaf cohomology and local cohomology and a theorem of Grothendieck. A variant of this observation shows that for an open affine subset the closed complement must be of pure codimension one ( must be the complement of the support of an effective divisor). In a regular or (locally )- factorial domain the complement of every divisor is affine, since the divisor can be described (at least locally geometrically) by one equation. But it is easy to give examples to show that this is not true for normal threedimensional domains.
Let be a field and consider the ring
The ideal is a prime ideal in of height one. Hence the open subset is the complement of an irreducible hypersurface. However, is not affine. For this we consider the closed subscheme
and . If were affine, then also the closed subscheme would be affine, but this is not true, since the complement of the punctured plane has codimension .
The argument employed in this example rests on the following definition and the next theorem.
Let be a noetherian commutative ring and let be an ideal. The (noetherian) superheight is the supremum
Let be a noetherian commutative ring and let be an ideal and . Then the following are equivalent.
- is an affine scheme.
- has superheight and is a finitely generated -algebra.
It is not true at all that the ring of global sections of an open subset of the spectrum of a noetherian ring is of finite type over this ring. This is not even true if is an affine variety. This problem is directly related to Hilbert's fourteenth problem, which has a negative answer. We will later present examples where has superheight one, yet is not affine, hence its ring of global sections is not finitely generated.
- Forcing algebras and their torsors
We want to deal now with a very special class of open subsets and ask whether they are affine or not and what their cohomological dimension is. Though it is in some sense a very special class it exhibits already a very rich behaviour. These open subsets are given by so-called forcing equations and forcing algebras.
Let be a commutative ring and let and be elements in . Then the -algebra
is called the forcing algebra of these elements (or these data).
This algebra was introduced by Hochster. The forcing algebra forces that belongs to the extended ideal . It yields a scheme morphism
We are interested in the relationship:
How is related to ?
Does belong to certain closure operations of ? Properties of .
Examples
- Tight closure
We want to deal with tight closure, a closure operation introduced by Hochster and Huneke.
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
The relation between tight closure and forcing algebras is given in the following theorem.
if and only if
where
denotes the forcing algebra of these elements.If the dimension is at least two, then
This means that we have to look at the cohomological properties of the complement of the exceptional fiber over the closed point. Such an open subset is called a torsor. If the dimension is two, then we have to look whether the first cohomology of the structure sheaf vanishes. This is true if and only if the open subset is an affine scheme (the spectrum of a ring).
The right hand side of this equivalence - the non-vanishing of the top-dimensional local cohomology - is independent of any characteristic assumption, and the basis for solid closure.
It is a fact that tight closure is difficult to compute. Since tight closure can be formulated with local cohomology, it follows that it must be quite difficult to give a general criterion for vanishing of local cohomology.
An important property of tight closure is that it is trivial for regular rings, i.e. for every ideal . This implies the following cohomological property.
Let denote a regular local ring of dimension and of positive characteristic, let be an -primary ideal and be an element with . Let be the corresponding forcing algebra.
Then the extended ideal satisfies
This follows from Fakt and .
In dimension two this is true in every
(even mixed)
characteristic.
Let denote a two-dimensional regular local ring, let be an -primary ideal and an element with . Let
be the corresponding forcing algebra.
Then for the extended ideal we have
In particular, the open subset is an affine scheme if and only if .
The example above, the equation can be considered as the forcing algebra for the ideal and the element . The non-affineness of corresponds to this containment.
We will continue in the next lecture with a detailed study of the situation of a two-dimensional graded base ring.
In higher dimension in characteristic zero it is not true that a regular ring is solidly closed, as was shown by the following example of Paul Roberts.
Let be a field of characteristic and let
Then the ideal has the property that . This means that in , the element belongs to the solid closure of the ideal , and hence the three-dimensional polynomial ring is not solidly closed.
This example uses a forcing equation of a special type: For parameters in a -dimensional local ring and some one considers the forcing algebra given by
The monomial conjecture states that this equation does not have a solution in . It is open only in mixed characteristic. The equation expresses that the Čech cohomology class is mapped to in the forcing algebra. Robert's computation shows that this does not imply that the complete local cohomology module vanishes. Therefore solid closure is not a characteristic-free replacement for tight closure. There is a variant, called parasolid closure, which is characteristic free and has all the properties of tight closure (over a field). A detailed understanding of the top-dimensional local cohomology of the torsors given by the forcing algebras for these special equations could solve the monomial conjecture.
- Plus closure
The above mentioned (finite) superheight condition is also related to another closure operation, the 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.
In terms of forcing algebras and their torsors, the containment inside the plus closure means that there exists a -dimensional closed subscheme inside the torsor which meet the exceptional fiber (the fiber over the maximal ideal) in one point, and this means that the superheight of the extended ideal is . In this case the local cohomological dimension of the torsor must be as well, since it contains a closed subscheme with this cohomological dimension.
In characteristic zero, the plus closure behaves very differently compared with positive characteristic. If is a normal domain of characteristic , then the trace map shows that the plus closure is trivial, for every ideal . This implies also that if is a twodimensional normal local ring of characteristic and an -primary ideal and an element with , then the extendend ideal inside the forcing algebra has superheight . If moreover belongs to the solid closure of , then is not affine and so by Fakt its ring of global sections is not finitely generated.
Let be a field and consider the Fermat ring
together with the ideal and . For we have . This element is however in the tight closure of the ideal in positive characteristic (assume that the characteristic does not divide ) and is therefore also in characteristic inside the tight closure and inside the solid closure. Hence the open subset
is not an affine scheme. In positive characteristic, is also contained in the plus closure and therefore this open subset contains punctured surfaces (the spectrum of the forcing algebra contains two-dimensional closed subschemes which meet the exceptional fiber in only one point; the ideal has superheight two in the forcing algebra). In characteristic zero however, due to Fakt the superheight is one and therefore by Fakt the algebra is not finitely generated. For and one can also show that is, considered as a complex space, a Stein space.