Computing the cohomology of constructible étale sheaves on curves

Levrat, Christophe

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 1085-1122

We present an explicit expression for 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 this complex and study its complexity. We illustrate this algorithm with several examples.

Nous présentons une expression explicite du complexe de cohomologie d’un faisceau constructible de groupes abéliens sur le site étale d’une courbe algébrique irréductible sur un corps algébriquement clos, dans le cas où la torsion du faisceau est inversible dans le corps. Cette expression fait intervenir uniquement des groupes finis, et est fonctorielle en la courbe et le faisceau. En particulier, nous montrons comment calculer l’action galoisienne sur ce complexe. Nous présentons également un algorithme qui calcule cette expression, et étudions sa complexité. Nous illustrons cet algorithme par plusieurs exemples.

Classification : 14F20, 11G20, 11Y16
Keywords: étale cohomology, constructible sheaf, algebraic curve, algorithm, complexity

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