Duality and Rational Modules in Hopf Algebras over Commutative Rings

Abstract

Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and $A^{\circ }$ is pure in $R^A$, then the categories of rational left $A$-modules and right $A^{\circ }$-comodules are isomorphic. In the Hopf algebra case, we can also strengthen the Blattner-Montgomery duality theorem. Finally, we give sufficient conditions to get the purity of $A^{\circ }$ in $R^A$.

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