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  "body": "\n## Definition\n\nLet $(X, \\mathcal{O}_X)$ be a [[ringed space|KNDXDE7K8M]] and let $\\mathrm{Ab}(X)$ be the category of [[sheaves|TB6TM6R4MD]] of [[abelian groups|082GHNPJ53]] on $X$.\nThe global sections functor $\\Gamma(X, -) \\colon \\mathrm{Ab}(X) \\to \\mathrm{Ab}$ is left exact, and $\\mathrm{Ab}(X)$ has enough injective objects.\nFor a sheaf $\\mathcal{F}$ in $\\mathrm{Ab}(X)$, the *$i$-th cohomology group* $H^i(X, \\mathcal{F})$ is the value at $\\mathcal{F}$ of the $i$-th right [[derived functor|NPMC3K6J2F]] of $\\Gamma(X, -)$.\nConcretely, for an [[injective resolution|ZW3W6FX4PW]] $0 \\to \\mathcal{F} \\to \\mathcal{I}^0 \\to \\mathcal{I}^1 \\to \\cdots$ in $\\mathrm{Ab}(X)$,\n$$H^i(X, \\mathcal{F}) = H^i\\big(\\Gamma(X, \\mathcal{I}^0) \\to \\Gamma(X, \\mathcal{I}^1) \\to \\cdots\\big).$$\nFor a [[sheaf of $\\mathcal{O}_X$-modules|S33B2H3MYD]] $\\mathcal{F}$, the groups $H^i(X, \\mathcal{F})$ are taken of its underlying sheaf of abelian groups; they are modules over $\\Gamma(X, \\mathcal{O}_X)$.\n\n## Remarks\n\nThe groups $H^i(X, \\mathcal{F})$ for an $\\mathcal{O}_X$-module $\\mathcal{F}$ agree with the derived functors of $\\Gamma$ computed in the category of $\\mathcal{O}_X$-modules, since injective $\\mathcal{O}_X$-modules are flasque and flasque sheaves are acyclic.\nThis is the same construction as [[sheaf cohomology in topology|KB76923A1M]], applied to the topological space of a scheme.\n\nExample: always $H^0(X, \\mathcal{F}) = \\mathcal{F}(X)$.\nFor $X = \\mathbb{P}^n_k$ and $\\mathcal{F} = \\mathcal{O}(d)$, $H^0$ is the space of homogeneous polynomials of degree $d$, $H^n \\cong k$ when $d = -n-1$, and $H^i = 0$ for $0 \u003c i \u003c n$.\n\nNon-example: a resolution of $\\mathcal{F}$ by sheaves that are not acyclic, for example by arbitrary locally free sheaves, does not in general compute $H^i(X, \\mathcal{F})$.\nIn practice one uses flasque resolutions or, on separated schemes with quasi-coherent sheaves, the Čech complex of an affine cover.\n",
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