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

,^[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 $H 1 (X, I) = 0$ 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 $i \geq 1$ 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, vol. 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. Series 9. 36: 1–16. Zbl 0078.34604.
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. Vol. 197. AMS. ISBN 978-0-8218-1357-7.

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

[1] Stacks 01XF.

[2] Serre (1957).

[4] Stacks 01XF.

[5] Stacks 01XE, Lemma 29.3.2.

[1]

[2]

[a]

[3]

[4]