Computing the cohomology of constructible étale sheaves on curves

TL;DR

The paper delivers an explicit, functorial expression for the cohomology complex $\mathop{\mathrm{R\Gamma}}(X,\mathcal{F})$ of a constructible étale sheaf on a (possibly singular) curve $X$ over a field, under the hypothesis that the torsion $\mathbb{Z}/n\mathbb{Z}$ is invertible in the base field. It builds a finite, explicit model using a minimal Galois cover $X^{\langle n\rangle}$ that trivialises $\mathbb{Z}/n\mathbb{Z}$-torsors, together with a detailed, cone-based representation of $\mathop{\mathrm{R\Gamma}}$ in terms of local inertia data, ramification groups, and a chosen trivialising cover $V\to U$. The authors provide an algorithm to compute this complex, with complexity analyses and discuss improvements over existing methods, especially in the finite-field setting, as well as potential applications to point counting on higher-dimensional varieties. They illustrate the approach with explicit examples on subschemes of $\mathbb{P}^1$ and of an elliptic curve, demonstrating concrete computations of the cohomology groups and the corresponding Galois actions.

Abstract

We present an explicit expression of the cohomology complex of a constructible sheaf of abelian groups on the small étale site of an irreducible curve over an algebraically closed field, when the torsion of the sheaf is invertible in the field. This expression only involves finite groups, and is functorial in both the curve and the sheaf. In particular, we explain how to compute the Galois action on this complex. We also present an algorithm which computes it and study its complexity.

Figures (1)

• Figure 1: Ramification at infinity of the cover ${U^{\langle n\rangle}}\to U$

Theorems & Definitions (49)

• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Proposition 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5
• proof