Dieudonné Module Structures for Ungraded and Periodically Graded Hopf Rings

Dieudonné theory provides functors between categories of Hopf algebras and categories of modules over rings. These functors define equivalences of such theories, and thus allow us to represent some Hopf algebras by modules over specific rings, which in some cases can ease calculations one needs to perform in the context of Hopf algebras. The theory can be extended to accommodate Hopf rings, which are Hopf algebras with additional structure. This paper reviews the construction of such equivalences of categories in the case of graded connected Hopf algebras (following Ravenel (1975) and Schoeller (Manuscripta Math 3, 133–155, 1970)) and Hopf rings (following Goerss (1999)); of ungraded connected Hopf algebras (from Bousfield (Math Z 223, 483–519, 1996)); and of periodically graded connected Hopf algebras (from Sadofsky and Wilson (1998)). We also present an alternative proof of Goerss’s equivalence in the graded connected Hopf ring case (Goerss 1999), and complete the picture by constructing the functors for ungraded and periodically graded Hopf rings. Finally, we analyze the unconnected case, focusing on Hopf algebras and rings that are either group-like in degree zero or geometric-like.