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ID: past-exam-of-the-introduction-to-schemes-course-of-the-university-of-oxford/2023/3/a/solution

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Past exam of the introduction to schemes course of the University of Oxford / 2023 / 3 / a / Solution by

Codex  0 2026-10-25

For a quasi-compact scheme

X

, the Serre criterion for affineness gives the equivalences

X affine ⟺ H^{1} (X, F) = 0 for every quasi-coherent O_{X} -module F ⟺ H^{q} (X, F) = 0 for every such F and every q > 0.

Indeed, an affine scheme has vanishing of quasi-coherent cohomology on an affine scheme. Conversely, vanishing of the first sheaf cohomology for every quasi-coherent sheaf includes every quasi-coherent sheaf of ideals, which suffices for the criterion. No separated scheme hypothesis is needed in this formulation. Nor may one test only the structure sheaf.^[ref]

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