ℓ-adic cohomology

Summary

Let $X$ be a variety over a separably closed field, and let $\ell$ be a prime different from the characteristic. The $\ell$-adic cohomology of $X$ is the inverse limit of the étale cohomology groups of $X$ with coefficients $\Z/\ell^n\Z$. It gives modules over the $\ell$-adic integers, and after tensoring, vector spaces over $\Q_\ell$, a field of characteristic $0$. These groups behave like the singular cohomology of a complex variety, with Poincaré duality, a Künneth formula and a Lefschetz trace formula. They are the cohomology used in the proof of the Weil conjectures, and the Galois action on them gives Galois representations.