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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $\\mathrm{Sh}(X)$ be the category of sheaves of [[abelian groups|082GHNPJ53]] on $X$.\nThe global sections functor $\\Gamma(X, -) \\colon \\mathrm{Sh}(X) \\to \\mathrm{Ab}$, $F \\mapsto F(X)$, is left exact, and $\\mathrm{Sh}(X)$ has enough injective objects.\nFor a sheaf $F$, choose an injective resolution $0 \\to F \\to I^0 \\to I^1 \\to I^2 \\to \\cdots$ in $\\mathrm{Sh}(X)$.\nThe *$q$-th cohomology group of $X$ with coefficients in $F$* is\n$$H^q(X, F) = H^q\\big(\\Gamma(X, I^0) \\to \\Gamma(X, I^1) \\to \\Gamma(X, I^2) \\to \\cdots\\big).$$\nSo $H^q(X, -)$ is the $q$-th right derived functor of $\\Gamma(X, -)$; it does not depend on the choice of resolution up to canonical isomorphism.\n\n## Remarks\n\nAlways $H^0(X, F) = F(X)$, by left exactness of $\\Gamma$.\nA short exact sequence of sheaves gives a long exact sequence of cohomology groups.\n\nExample: if $X$ is a single point, a sheaf is just an abelian group $A$, and $\\Gamma$ is the identity functor, which is exact.\nSo $H^0(X, A) = A$ and $H^q(X, A) = 0$ for $q \u003e 0$.\n\nNon-example: the rule $U \\mapsto$ (constant functions $U \\to \\Z$) is a presheaf but in general not a sheaf.\nTwo different constants on two disjoint open sets do not glue to a constant function on their union.\nThe definition needs a sheaf, here its sheafification, the sheaf of locally constant functions.\n\nIn practice one uses acyclic resolutions instead of injective ones: flasque, soft or fine resolutions compute the same groups.\nIf $X$ is paracompact, Hausdorff and locally contractible, then $H^q(X, \\underline{A})$ for the constant sheaf $\\underline{A}$ is isomorphic to singular cohomology $H^q(X; A)$.\n",
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