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, stated in terms of sheaf cohomology.. The theorem was first published by Serre in 1957..
Statement Let be a scheme with structure sheaf If: (1) is quasi-compact, and (2) for every quasi-coherent ideal sheaf of -modules, , then is affine..
Related results A special case of this theorem arises when is an algebraic variety, in which case the conditions of the theorem imply that is an affine variety. A similar result has stricter conditions on but looser conditions on the cohomology: if is a quasi-separated, quasi-compact scheme, and if for any quasi-coherent sheaf of ideals of finite type, then is affine., Lemma 29.3.2.
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