Generalizations of the Sweedler dual
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 the left adjoint of the dual algebra functor on the category of -bialgebras has a right adjoint. This dual adjunction can be restricted to a dual adjunction on the category of Hopf -algebras, provided that is noetherian and absolutely flat.
Keywords
Cite
@article{arxiv.1510.01797,
title = {Generalizations of the Sweedler dual},
author = {Hans-E. Porst and Ross Street},
journal= {arXiv preprint arXiv:1510.01797},
year = {2018}
}
Comments
27 pages. We are grateful to the referee who noticed that Lemma 18 in the old version was false. This prompted a full reorganisation of the paper. We are grateful also to Exequiel Rivas for pointing out that Definition 16 of the published version was too weak to prove Proposition 17