# Kurs:Vector bundles, forcing algebras and local cohomology (Medellin 2012)/Lecture 6

If is a normal local domain of dimension and an -primary ideal, then (or inside the solid closure) if and only if is not an affine scheme, where denotes the forcing algebra. Here we will discuss in general, with this application in mind, when a scheme is affine.

*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.

*Codimension condition*

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 .

*Ring of global sections of affine schemes*

Let be a noetherian ring and an open subset.

** Then the following hold.
**

- is an affine scheme if and only if .
- If this holds, and with and , then . In particular, the ring of global sections over is finitely generated over .

We only give a sketch. (1). There always exists a natural scheme morphism

and is affine if and only if this morphism is an isomorphism. It is always an open embedding (because it is an isomorphism on the , ), and the image is . This is everything if and only if the extended ideal is the unit ideal.

(2). We write and consider the natural morphism

corresponding to the ring inclusion . This morphism is again an open embedding and its image is everything.

An application of this is the following computation.

We consider the Fermat cubic , the ideal and the element . We claim that for characteristic the element does not belong to the solid closure of . Equivalently, the open subset

is affine. For this we show that the extended ideal inside the ring of global sections is the unit ideal. First of all we get the equation

or, equivalently,

We write this as

which yields on the rational function

This shows that belongs to the extended ideal. Similarly, one can show that also the other coefficients belong to the extended ideal. Therefore in characteristic different from , the extended ideal is the unit ideal.

We will see later also examples where the ring of global sections is not finitely generated.

We consider the Fermat cubic , the ideal and the element . We claim that in positive characteristic the element does belong to the tight closure of . Equivalently, the open subset

is not affine. The element defines the cohomology class

and its Frobenius pull-backs are

This cohomology module has a -graded structure (the degree is given by the difference of the degree of the numerator and the degree of the denominator) and, moreover, it is in positive degree (this is related to the fact that the corresponding projective curve is elliptic). Therefore for any homogeneous element of positive degree we have and so belongs to the tight closure.

From this it follows also that in characteristic the element belongs to the solid closure, because affineness is an open property in an arithmetic family.