PICARD INVARIANTS OF GALOIS ALGEBRAS OVER DUAL LARSON HOPF ORDERS

Abstract

Let $A$ be a commutative, cocommutative Hopf algebra, finitely generated and projective over its base ring $R$. Waterhouse asked whether the image of the class-invariant map, taking each $A$-Galois algebra to the class in ${\rm Pic}(A)$ of its $R$-linear dual, is the group of primitive classes in ${\rm Pic}(A)$. We discuss functorial aspects of this problem, and relate it to Fr\"ohlich's Hom-description of ${\rm Pic}(A)$ in the case that $R$ is a Dedekind domain with field of fractions $K$, and $A$ is an $R$-Hopf order in a separable $K$-Hopf algebra. We then apply this machinery to a certain class of Hopf orders $\mathfrak A$ in the Hopf algebra ${\rm Map}(G,K)$. More precisely, we give a positive answer to Waterhouse's question for $\mathfrak A$ when the dual $\mathfrak B$ of $\mathfrak A$ is one of the Hopf orders in $KG$ constructed by Larson, and a compatibility condition holds between the filtrations of $G$ determined by the various completions of $\mathfrak B$.

1991 Mathematics Subject Classification: 11R33, 16W30