Paper detail

Calculabilité de la cohomologie étale modulo l

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 étale 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.