Building Data for Stacky Covers and the Étale Cohomology Ring of an Arithmetic Curve

Abstract: This thesis consists of two papers with somewhat different flavours. In Paper I we compute the étale cohomology ring H^'(X,Z/nZ) for X the ring of integers of a number field K. As an application, we give a non-vanishing formula for an invariant defined by Minhyong Kim. We also give examples of two distinct number fields whose rings of integers have isomorphic cohomology groups but distinct cohomology ring structures.In Paper II we define stacky building data for stacky covers in the spirit of Pardini and give an equivalence of (2, 1)-categories between the category of stacky covers and the category of stacky building data. We show that every stacky cover is a flat root stack in the sense of Olsson and Borne–Vistoli and give an intrinsic description of it as a root stack using stacky building data. When the base scheme S is defined over a field, we give a criterion for when a stacky building datum comes from a ramified cover for a finite abelian group scheme over k, generalizing a result of Biswas–Borne.