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

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