# Kurs:Vector bundles, forcing algebras and local cohomology (Tehran 2012)/Lecture 5

*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 a 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.

*Affineness and superheight*

One can show 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. The following example is a standardexample for this phenomenon and is in fact given by a forcing algebra.

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.

If is a two-dimensional local ring with parameters and if is the forcing algebra for some -primary ideal, then the ring of global sections of the torsor is just

In the following two examples we use results from tight closure theory to establish (non)-affineness properties of certain torsors.

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, the superheight is one because plus closure is trivial for normal domains in characteristic , and therefore by Theorem 5.4 the algebra is not finitely generated. For and one can also show that is, considered as a complex space, a Stein space.

Let be a field of positive characteristic and consider the ring

together with the ideal and . Since has a rational singularity, it is -regular, i.e. all ideals are tightly closed. Therefore and so the torsor

is an affine scheme. In characteristic zero this can be proved by either using that is a quotient singularity or by using the natural grading () where the corresponding cohomology class gets degree and then applying the geometric criteria on the corresponding projective curve (rather the corresponding curve of the standard-homogenization ).