Generalizations of the Sweedler dual

TL;DR

This paper generalizes Sweedler's finite dual construction from fields to noetherian commutative rings, revealing new dual adjunctions for Hopf algebras over such rings.

Contribution

It identifies conditions under which the dual algebra functor's left adjoint extends to noetherian rings and establishes new dual adjunctions involving Hopf algebras.

Findings

Generalization of Sweedler's dual to noetherian rings

Existence of dual adjunctions for R-bialgebras

Dual adjunctions restricted to Hopf R-algebras

Abstract

As left adjoint to the dual algebra functor, Sweedler's finite dual construction is an important tool in the theory of Hopf algebras over a field. We show in this note that the left adjoint to the dual algebra functor, which exists over arbitrary rings, shares a number of properties with the finite dual. Nonetheless the requirement that it should map Hopf algebras to Hopf algebras needs the extra assumption that this left adjoint should map an algebra into its linear dual. We identify a condition guaranteeing that Sweedler's construction works when generalized to noetherian commutative rings. We establish the following two apparently previously unnoticed dual adjunctions: For every commutative ring $R$ the left adjoint of the dual algebra functor on the category of $R$-bialgebras has a right adjoint. This dual adjunction can be restricted to a dual adjunction on the category of Hopf $R$-algebras, provided that $R$ is noetherian and absolutely flat.