On the de Rham cohomology of cyclic covers

TL;DR

This paper explicitly computes the de Rham cohomology bases for cyclic covers of the projective line over algebraically closed fields, including Kummer and Artin-Schreier extensions, facilitating applications in group actions and p-cyclic covers.

Contribution

It provides explicit, closed-form bases for the de Rham cohomology groups of cyclic covers, advancing computational methods in algebraic geometry in characteristic p.

Findings

Explicit bases for H^1(O_X) and H^0(Ω_X) for cyclic covers

Closed-form expressions for de Rham cohomology in terms of defining equations

Applications to group actions and p-cyclic cover analysis

Abstract

We compute explicit bases for the de Rham cohomology of cyclic covers of the projective line defined over an algebraically closed field of characteristic $p\geq 0$. For both Kummer and Artin-Schreier extensions, we describe precise $k$-bases for the cohomology groups $H^{1}(X,\mathcal{O}_{X})$ and $H^{0}(X,\Omega_{X})$, and we use these to construct an explicit basis for the first de Rham cohomology group $H^{1}_{\mathrm{dR}}(X/k)$ via \v{C}ech cohomology. Our approach relies on detailed computations of divisors of functions and differentials, together with residue calculations and the duality pairing between $H^{0}(X,\Omega_{X})$ and $H^{1}(X,\mathcal{O}_{X})$. The resulting expressions are given in closed form in terms of the defining equation of the cover, making the cohomology fully explicit and readily applicable to questions involving group actions, and the study of $p$-cyclic covers.