Étale cohomology

Remarks

Example: let $X = \operatorname{Spec} k$ for a field $k$ with separable closure $k^{s}$ and Galois group $G = \Gal(k^{s}/k)$. Étale $X$-schemes are disjoint unions of spectra of finite separable extensions of $k$. An étale sheaf $F$ is then the same as the discrete $G$-module $M = \varinjlim F(\operatorname{Spec} k')$ over finite subextensions $k'$. The global sections are $M^G$, so $H^i_{\text{ét}}(X, F) = H^i(G, M)$ is Galois cohomology. Non-example: if the étale morphisms in clause 1 are replaced by open immersions, one gets the Zariski site. For 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 $i > 0$. Étale 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, $H^1_{\text{ét}}(X, \Z/n\Z) \cong (\Z/n\Z)^{2g}$.