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  "body": "\n## Statement\n\nLet $X$ be an [[affine scheme|B3V3XJ2W8G]] and let $\\mathcal{F}$ be a [[quasi-coherent sheaf|KPKND0K7W6]] on $X$.\nThen the [[sheaf cohomology|PQ6Q143ARJ]] groups vanish in positive degrees:\n$$H^i(X, \\mathcal{F}) = 0 \\quad \\text{for all } i \u003e 0.$$\n\n## Remarks\n\nThe idea of the proof for $X = \\operatorname{Spec} A$ with $A$ Noetherian: if $I$ is an injective $A$-module, the associated sheaf $\\tilde{I}$ is flasque.\nResolve $\\mathcal{F} = \\tilde{M}$ by $\\tilde{I}^\\bullet$ for an injective resolution $M \\to I^\\bullet$; this is a flasque resolution, and taking global sections gives back the exact complex $I^\\bullet$.\nIn general one uses that the Čech complex of a quasi-coherent sheaf for a cover by standard open sets $D(f_i)$ is exact in positive degrees.\n\nExample: on a separated scheme $X$ with an affine open cover, the theorem shows that the cover is acyclic for quasi-coherent sheaves, so Čech cohomology of this cover computes $H^i(X, \\mathcal{F})$.\n\nThe affine hypothesis is necessary: on $X = \\mathbb{P}^1_k$, which is not affine, $H^1(X, \\mathcal{O}(-2)) \\cong k \\ne 0$.\nConversely, Serre's criterion says that a Noetherian scheme $X$ is affine if $H^1(X, \\mathcal{F}) = 0$ for every quasi-coherent sheaf $\\mathcal{F}$, and it suffices to ask this for coherent ideal sheaves.\n\n## Explanations\n\n### L1 human\n\nOn an affine scheme, every quasi-coherent sheaf comes from a module, and global sections of a short exact sequence of such sheaves stay exact.\nSo there is never an obstruction to passing from local to global data, and all higher cohomology vanishes.\nAffine schemes play for quasi-coherent sheaves the role that contractible open sets play in topology.\n\n### L2 human\n\nThe theorem is the base of every computation of coherent cohomology: one covers a separated scheme by affine open sets and computes the Čech complex.\nIt gives the long exact sequence argument that a surjection of quasi-coherent sheaves on an affine scheme is surjective on global sections.\nWith Serre's converse criterion, it characterizes affine schemes cohomologically, in the same way as Cartan's Theorem B for Stein manifolds.\n",
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