Calcul effectif de la cohomologie des faisceaux constructibles sur le site \'etale d'une courbe

TL;DR

This thesis develops algorithms for representing and computing the cohomology of constructible sheaves on curves, providing explicit methods and bounds, with applications to Galois actions and surface cohomology.

Contribution

It introduces new algorithms for explicit cohomology computation of constructible sheaves on curves, including bounds and functorial descriptions, advancing computational algebraic geometry.

Findings

Algorithms for cohomology complexes of sheaves on curves

Explicit description of cup-products in cohomology

Bounds on the number of operations in cohomology computation

Abstract

This thesis deals with the algorithmic representation of constructible sheaves of abelian groups on the \'etale site of a variety over an algebraically closed field, as well as the explicit computation of their cohomology. We describe three representations of such sheaves on curves with at worst nodal singularities, as well as algorithms performing various operations (kernels and cokernels of morphisms, pullback and pushforward, internal Hom and tensor product) on these sheaves. We present an algorithm computing the cohomology complex of a locally constant constructible sheaf on a smooth or nodal curve, which in turn allows us to give an explicit description of the functor $\mathrm{R}\Gamma(X,-)\colon \mathrm{D}^b_c(X,\mathbb{Z}/n\mathbb{Z})\to \mathrm{D}^b_c(\mathbb{Z}/n\mathbb{Z})$. This description is functorial in the scheme $X$ and the given complex of constructible sheaves. In particular, if $X$ and the sheaf $\mathcal{F}$ are obtained by base change from a subfield, we describe the Galois action on the complex $\mathrm{R}\Gamma(X,\mathcal{F})$. We give precise bounds on the number of operations performed by the algorithm computing $\mathrm{R}\Gamma(X,\mathcal{F})$. We also give an explicit description of cup-products in the cohomology of locally constant constructible sheaves over smooth projective curves. Finally, we show how to use these algorithms in order to compute the cohomology groups of a constant sheaf on a smooth surface fibered over the projective line.