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  "body": "\n## Summary\n\nSheaf cohomology assigns to a sheaf $\\mathcal{F}$ on a space $X$ a sequence of groups $H^i(X, \\mathcal{F})$, the right derived functors of the global sections functor.\nIn algebraic geometry it is applied to sheaves of modules on schemes, above all to [[quasi-coherent|KPKND0K7W6]] and [[coherent sheaves|56RXEJ8Z6S]].\nFor these the groups are modules, and in good cases finite-dimensional vector spaces.\nIt measures the failure of local sections to glue and of surjections of sheaves to be surjective on global sections.\nThe topological side of the theory, with constant and locally constant coefficients, is described in [[sheaf cohomology in topology|KB76923A1M]].\n\n## Explanations\n\n### L1 human\n\nGlobal sections of a sheaf are the objects you can define on the whole space at once, such as functions or sections of a bundle.\nWhen a short exact sequence of sheaves is given, the global sections need not form an exact sequence: something can exist locally but not globally.\nSheaf cohomology measures exactly this defect, in degree $1$, $2$, and so on.\nFor example, on the projective line over a field, $H^0$ of the structure sheaf is the field itself, the constant functions.\nThe bundle $\\mathcal{O}(-2)$ has no sections, but its first cohomology group is one-dimensional.\n",
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