= Cohomological criterion for affineness
{title2=$X\text{ affine}\iff H^1(X,\mathcal I)=0\text{ for all coherent ideals }\mathcal I$}
= Serre criterion for affineness
{c}
{synonym}
A <Noetherian scheme> is affine exactly when every <coherent ideal sheaf> has zero first <sheaf cohomology>. One direction is <vanishing of quasi-coherent cohomology on an affine scheme>. For the converse, ideal-sheaf vanishing produces <affine principal neighbourhoods from ideal-sheaf vanishing>. A finite such cover yields a <unit-ideal certificate from a principal affine cover>, and those <affine charts> glue to the <spectrum of a commutative ring> of <global sections>. The criterion, including the finite-type ideal version for quasi-compact quasi-separated schemes, is recorded in https://stacks.math.columbia.edu/tag/01XE[Stacks Project, Section 30.3].
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