# Affine scheme/Cohomological criterion/Introduction/Section

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.

- 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.