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.
EGA II 5.2.2, EGA IV 1.7.17
Akhil Mathew, Serre’s criterion for affineness as Morita theory, a blog post
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