Serre's theorem on affineness explained

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.^[1] The theorem was first published by Serre in 1957.^[2]

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.^[3]

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.^[4]

Bibliography

• • Serre . Jean-Pierre . Jean-Pierre Serre . Sur la cohomologie des variétés algébriques . J. Math. Pures Appl. . Series 9 . 1957 . 36 . 1–16 . 0078.34604.
• Web site: Section 29.3 (01XE):Vanishing of cohomology—The Stacks Project . The Stacks Project authors . CITEREFStacks 01XE.
• Web site: Lemma 29.3.1 (01XF)—The Stacks Project . The Stacks Project authors . CITEREFStacks 01XF.
• Book: Ueno, Kenji . Algebraic Geomety II: Sheaves and Cohomology . . 2001 . 978-0-8218-1357-7 . Translations of Mathematical Monographs . 197 .

Notes and References

1. .
2. .
3. .
4. , Lemma 29.3.2.