Calculabilit\'e de la cohomologie \'etale modulo l

TL;DR

This paper demonstrates that the Betti numbers modulo e5 of algebraic schemes over algebraically closed fields are algorithmically computable, using a novel approach involving colimits of toposes and computational algebraic geometry.

Contribution

It introduces a method to compute e5-modulo Betti numbers of schemes via colimits of toposes on finite e5-groups, linking algebraic geometry with computability theory.

Findings

Betti numbers modulo e5 are algorithmically computable.

Expressed cohomology as colimits of toposes on finite e5-groups.

Developed a universal formalism for computation in fields.

Abstract

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.