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, stated in terms of sheaf cohomology.^[1] The theorem was first published by Serre in 1957.^[2]

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, H¹(X, I) = 0,^[a]

then X is affine.^[3]

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 H¹(X, I) = 0 for any quasi-coherent sheaf of ideals I of finite type, then X is affine.^[4]

Notes

1. Some texts, such as Ueno (2001, pp. 128–133), require that Hⁱ(X,I) = 0 for all i ≥ 1 as a condition for the theorem. In fact, this is equivalent to condition (2) above.