On the de Rham cohomology of cyclic covers

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.