On the de Rham cohomology of cyclic covers

TL;DR

The paper provides explicit $k$-bases for the first de Rham cohomology of cyclic covers of the projective line, covering both Kummer and Artin–Schreier cases over algebraically closed fields of any characteristic. It combines Čech cohomology with detailed divisor, differential, and residue computations to construct concrete bases for $H^1(X,\mathcal{O}_X)$, $H^0(X,\Omega_X)$, and $H^1_{\mathrm{dR}}(X/k)$, with duality verified via residue pairings. The results express the bases in closed form in terms of the defining equation, enabling applications to group actions, $p$-cyclic covers, and deformation theory. The approach clarifies the Hodge–de Rham sequence in this explicit setting and provides tools for studying the Galois module structure of cohomology on cyclic covers. Overall, the work delivers fully explicit, computable cohomology bases that facilitate analysis of symmetries and p-adic phenomena in cyclic covers.

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 $H^{0}(X,Ω_{X})$, and we use these to construct an explicit basis for the first de Rham cohomology group $H^{1}_{\mathrm{dR}}(X/k)$ via Čech cohomology. Our approach relies on detailed computations of divisors of functions and differentials, together with residue calculations and the duality pairing between $H^{0}(X,Ω_{X})$ 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.

Theorems & Definitions (10)

• Lemma 1
• Proposition 2
• proof
• Theorem 3
• proof
• Remark
• Proposition 4
• proof
• Theorem 5
• proof