Bimodule herds

TL;DR

This paper introduces and explores the concept of bimodule herds, establishing their connections to corings, entwining structures, and Galois comodules, with applications to Hopf algebra co-objects.

Contribution

It defines bimodule herds and bicomodule coherds, and demonstrates their relationships to corings, entwining structures, and Galois co-objects, advancing the theoretical framework in module and comodule theory.

Findings

Bimodule herds induce pairs of corings and coactions.

Tame bimodule herds relate to entwining structures and Galois comodules.

Bicomodule coherds define pairs of non-unital rings.

Abstract

The notion of a bimodule herd is introduced and studied. A bimodule herd consists of a $B$-$A$ bimodule, its formal dual, called a pen, and a map, called a shepherd, which satisfies untiality and coassociativity conditions. It is shown that every bimodule herd gives rise to a pair of corings and coactions. If, in addition, a bimodule herd is tame i.e. it is faithfully flat and a progenerator, then these corings are associated to entwining structures; the bimodule herd is a Galois comodule of these corings. The notion of a bicomodule coherd is introduced as a formal dualisation of the definition of a bimodule herd. Every bicomodule coherd defines a pair of (non-unital) rings. It is shown that a tame $B$-$A$ bimodule herd defines a bicomodule coherd, and sufficient conditions for the derived rings to be isomorphic to $A$ and $B$ are discussed. The composition of bimodule herds via the tensor product is outlined. The notion of a bimodule herd is illustrated by the example of Galois co-objects of a commutative, faithfully flat Hopf algebra.