  Let $X$ be an algebraic scheme over an algebraically closed field and $\ell$
a prime number invertible on $X$. According to classical results (due
essentially to A. Grothendieck, M. Artin and P. Deligne), the \'etale
cohomology groups $\mathrm{H}^i(X,\mathbb{Z}/\ell\mathbb{Z})$ are
finite-dimensional. Using an $\ell$-adic variant of M. Artin's good
neighborhoods and elementary results on the cohomology of pro-$\ell$ groups, we
express the cohomology of $X$ as a well controlled colimit of that of toposes
constructed on $BG$ where the $G$ are computable finite $\ell$-groups. From
this, we deduce that the Betti numbers modulo $\ell$ of $X$ are algorithmically
computable (in the sense of Church-Turing). The proof of this fact, along with
certain related results, occupies the first part of this paper. This relies on
the tools collected in the second part, which deals with computational
algebraic geometry. Finally, in the third part, we present a "universal"
formalism for computation on the elements of a field.
