Serre's theorem on affineness

In the mathematical discipline of algebraic geometry, Serre's theorem on affineness (also called Serre's cohomological characterization of affineness or Serre's criterion on affineness) is a theorem due to Jean-Pierre Serre which gives sufficient conditions for a scheme to be affine. The theorem was first published by Serre in 1957.

Statement
Let $X$ be a scheme with structure sheaf $O_{X}.$ If:
 * (1) $X$ is quasi-compact, and
 * (2) for every quasi-coherent ideal sheaf $I$ of $O_{X}$-modules, $H^{1}(X, I) = 0$,

then $H^{i}(X,I) = 0$ is affine.

Related results

 * A special case of this theorem arises when $X$ is an algebraic variety, in which case the conditions of the theorem imply that $X$ is an affine variety.
 * A similar result has stricter conditions on $X$ but looser conditions on the cohomology: if $X$ is a quasi-separated, quasi-compact scheme, and if $i ≥ 1$ for any quasi-coherent sheaf of ideals $I$ of finite type, then $X$ is affine.