The relative Brauer group and generalized cross products for a cyclic covering of affine space

Abstract

We study algebras and divisors on a normal affine hypersurface defined by an equation of the form zn=f(x1,…,xm). The coordinate ring is T=k[x1,…,xm,z]/(zn−f), and if R=k[x1,…,xm][f−1] and S=R[z]/(zn−f), then S is a cyclic Galois extension of R. We show that if the Galois group is G, the natural map H1(G,Cl(T))→H1(G,Pic(S)) factors through the relative Brauer group B(S/R) and that all of the maps are onto. Sufficient conditions are given for H1(G,Cl(T)) to be isomorphic to B(S/R). As an example, all of the groups, maps, divisors and algebras are computed for an affine surface defined by an equation of the form zn=(y−a1x)⋯(y−anx)(x−1).