Affine scheme/Cohomological criterion/Introduction/Section

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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.
  1. is an affine scheme.
  2. For every quasicoherent sheaf on and all we have .
  3. 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.