Étale cohomology ($H_{\mathrm{et}}^i(X,\mathcal{F})$)

Étale cohomology is sheaf cohomology on the site whose objects are étale morphisms $Y\to X$ and whose coverings are jointly surjective families of such morphisms. It provides cohomological invariants for algebraic varieties over fields of positive characteristic. In Deligne-Lusztig theory, compactly supported groups with $\ell$-adic coefficients, for $\ell$ different from the field characteristic, give finite-dimensional representations of the finite groups acting on the varieties.