# nLab Serre's criterion of affineness

Serre’s criterion of affineness characterizes affine morphisms of schemes in terms of exactness properties of the corresponding functors among the categories of quasicoherent sheaves.

If $f\colon X\to Y$ is a quasicompact morphism of algebraic schemes and $X$ is separated, then $f$ is affine iff it is cohomologically affine, that is, the direct image functor $f_*$ is exact.

Last revised on February 5, 2014 at 07:53:02. See the history of this page for a list of all contributions to it.