Affines Schema/Kohomologisches Kriterium/Einführung/Textabschnitt
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.