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.{{harvtxt|Stacks 01XF}}. The theorem was first published by Serre in 1957.{{harvtxt|Serre |1957}}.

Statement

Let {{math|X}} be a scheme with structure sheaf {{math|O_X.}} If:

:(1) {{math|X}} is quasi-compact, and

:(2) for every quasi-coherent ideal sheaf {{math|I}} of {{math|O_X}}-modules, {{math|H¹(X, I) {{=}} 0}},{{efn|Some texts, such as {{harvtxt |Ueno |2001 |pp=128–133 }}, require that {{math|Hⁱ(X,I) {{=}} 0}} for all {{math|i ≥ 1}} as a condition for the theorem. In fact, this is equivalent to condition (2) above.}}

then {{math|X}} is affine.{{harvtxt|Stacks 01XF}}.

Related results

A special case of this theorem arises when {{mvar|X}} is an algebraic variety, in which case the conditions of the theorem imply that {{mvar|X}} is an affine variety.
A similar result has stricter conditions on {{mvar|X}} but looser conditions on the cohomology: if {{mvar|X}} is a quasi-separated, quasi-compact scheme, and if {{math|H¹(X, I) {{=}} 0}} for any quasi-coherent sheaf of ideals {{mvar|I}} of finite type, then {{mvar|X}} is affine.{{harvtxt|Stacks 01XE}}, Lemma 29.3.2.

Notes

References

Bibliography

{{Hartshorne AG}}
{{cite journal |last=Serre |first=Jean-Pierre |authorlink=Jean-Pierre Serre |title=Sur la cohomologie des variétés algébriques |journal=J. Math. Pures Appl. |series=Series 9 |year=1957 |volume=36 |pages=1–16 |zbl=0078.34604}}
{{cite web |url=http://stacks.math.columbia.edu/tag/01XE |author=The Stacks Project authors |title=Section 29.3 (01XE):Vanishing of cohomology—The Stacks Project |ref=CITEREFStacks 01XE}}
{{cite web |url=http://stacks.math.columbia.edu/tag/01XF |author=The Stacks Project authors |title=Lemma 29.3.1 (01XF)—The Stacks Project |ref=CITEREFStacks 01XF}}
{{cite book |last=Ueno |first=Kenji |title=Algebraic Geomety II: Sheaves and Cohomology |publisher=AMS |year=2001 |isbn=978-0-8218-1357-7 |series=Translations of Mathematical Monographs |volume=197 }}

Category:Theorems in algebraic geometry