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  "body": "\n## Definition\n\nLet $X$ be a [[scheme|5A35TYZD02]].\nThe *small étale site* $X_{\\text{ét}}$ is the [[site|NYE0Z94MME]] defined as follows.\n\n1. Its objects are the [[étale morphisms|H84VHC8PX2]] $U \\to X$.\n2. Its morphisms are the morphisms $U \\to V$ over $X$.\n3. A family $\\set{U_i \\to U}$ is a covering when the union of the images of the $U_i$ is $U$.\n\nAn *étale sheaf* on $X$ is a [[sheaf on the site|RT8A8FJ566]] $X_{\\text{ét}}$.\nThe category of étale sheaves of abelian groups on $X$ has enough injectives.\nFor such a sheaf $F$, the *étale cohomology groups* $H^i_{\\text{ét}}(X, F)$, for $i \\ge 0$, are the right [[derived functors|NPMC3K6J2F]] of the global section functor $F \\mapsto F(X)$.\n\n## Remarks\n\nExample: let $X = \\operatorname{Spec} k$ for a field $k$ with separable closure $k^{s}$ and Galois group $G = \\Gal(k^{s}/k)$.\nÉtale $X$-schemes are disjoint unions of spectra of finite separable extensions of $k$.\nAn étale sheaf $F$ is then the same as the discrete $G$-module $M = \\varinjlim F(\\operatorname{Spec} k')$ over finite subextensions $k'$.\nThe global sections are $M^G$, so $H^i_{\\text{ét}}(X, F) = H^i(G, M)$ is [[Galois cohomology|EESNTXFWHZ]].\nNon-example: if the étale morphisms in clause 1 are replaced by open immersions, one gets the Zariski site.\nFor an irreducible $X$ and a constant sheaf $A$, every constant sheaf is flasque in the Zariski topology, so $H^i(X, A) = 0$ for all $i \u003e 0$.\nÉtale cohomology does not have this defect: for a smooth projective curve of genus $g$ over an algebraically closed field and $n$ prime to the characteristic, $H^1_{\\text{ét}}(X, \\Z/n\\Z) \\cong (\\Z/n\\Z)^{2g}$.\n",
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