# 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.[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,*H*^{1}(*X*,*I*) = 0,[lower-alpha 1]

## 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*H*^{1}(*X*,*I*) = 0 for any quasi-coherent sheaf of ideals*I*of finite type, then*X*is affine.[4]

## Notes

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

## References

- Stacks 01XF.
- Serre (1957).
- Stacks 01XF.
- Stacks 01XE, Lemma 29.3.2.

## Bibliography

- Hartshorne, Robin (1977),
*Algebraic Geometry*, Graduate Texts in Mathematics,**52**, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157 - Serre, Jean-Pierre (1957). "Sur la cohomologie des variétés algébriques".
*J. Math. Pures Appl. (9)*.**36**: 1–16. - The Stacks Project authors. "Section 29.3 (01XE):Vanishing of cohomology—The Stacks Project".
- The Stacks Project authors. "Lemma 29.3.1 (01XF)—The Stacks Project".
- Ueno, Kenji (2001).
*Algebraic Geomety II: Sheaves and Cohomology*. Translations of Mathematical Monographs.**197**. AMS. ISBN 978-0-8218-1357-7.

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