Tameness and local normal bases for objects of finite Hopf algebras

Abstract:

Let $R$ be a commutative ring, $S$ an $R$-algebra, $H$ a Hopf $R$algebra, both finitely generated and projective as $R$-modules, and suppose $S$ is an $H$-object, so that ${H^{\ast }} = {\operatorname {Hom} _R}(H,R)$ acts on $S$ via a measuring. Let $I$ be the space of left integrals of ${H^{\ast }}$. We say $S$ has normal basis if $S \cong H$ as ${H^{\ast }}$modules, and $S$ has local normal bases if ${S_p} \cong {H_p}$ as $H_p^{\ast }$-modules for all prime ideals $p$ of $R$. When $R$ is a perfect field, $H$ is commutative and cocommutative, and certain obvious necessary conditions on $S$ hold, then $S$ has normal basis if and only if $IS = R = {S^{{H^{\ast }}}}$. If $R$ is a domain with quotient field $K$, $H$ is cocommutative, and $L = S \otimes {}_RK$ has normal basis as $({H^{\ast }} \otimes K)$-module, then $S$ has local normal bases if and only if $IS = R = {S^{{H^{\ast }}}}$.