[Paper Review] Bimodule herds

Tomasz Brzeziński, Joost Vercruysse|arXiv (Cornell University)|May 16, 2008

Algebraic structures and combinatorial models14 references4 citations

TL;DR

This paper introduces the concept of a bimodule herd—a structure comprising a bimodule, its formal dual (a pen), and a shepherd map satisfying unitality and coassociativity. It shows that every bimodule herd induces a pair of corings and coactions, and when tame (faithfully flat and a progenerator), it becomes a Galois comodule over entwining structures, with dual bicomodule coherds yielding non-unital rings that may be isomorphic to the original algebras.

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.

Motivation & Objective

• To formalize the notion of a bimodule herd as a categorical structure unifying bimodules, their duals (pens), and shepherd maps.
• To establish a correspondence between bimodule herds and pairs of corings with coactions, particularly under tameness conditions.
• To introduce the dual concept of a bicomodule coherd and explore its connection to non-unital rings.
• To investigate conditions under which the derived rings from a bicomodule coherd are isomorphic to the original algebras A and B.
• To demonstrate the composition of bimodule herds via tensor products and illustrate the framework with Galois co-objects over commutative, faithfully flat Hopf algebras.

Proposed method

• Define a bimodule herd as a triple (M, P, s), where M is a B-A bimodule, P is its formal dual (a pen), and s is a shepherd map satisfying unitality and coassociativity.
• Construct two corings from a bimodule herd using the shepherd map and the bimodule structure, showing they admit coactions.
• Apply the notion of tameness—faithful flatness and progenerator property—to ensure the corings arise from entwining structures.
• Dualize the bimodule herd construction to define a bicomodule coherd, which induces a pair of non-unital rings.
• Use tensor products to compose bimodule herds, preserving structural compatibility.
• Illustrate the framework using Galois co-objects of a commutative, faithfully flat Hopf algebra as a concrete example.

Experimental results

Research questions

• RQ1 How can a bimodule, its formal dual, and a shepherd map be combined to form a coherent algebraic structure with coassociative and unital properties?
• RQ2 What coring and coaction structures emerge from a bimodule herd, and how do they relate to entwining structures when tameness is assumed?
• RQ3 In what conditions does the dual construction of a bicomodule coherd yield rings isomorphic to the original algebras A and B?
• RQ4 How can bimodule herds be composed via tensor products, and what structure is preserved under such composition?
• RQ5 What role do Galois co-objects over commutative, faithfully flat Hopf algebras play in realizing the bimodule herd framework?

Key findings

• Every bimodule herd gives rise to a pair of corings and associated coactions, establishing a categorical link to coring theory.
• When a bimodule herd is tame (faithfully flat and a progenerator), the induced corings are associated to entwining structures.
• Under tameness, the bimodule herd becomes a Galois comodule for the constructed corings, generalizing classical Galois theory to this setting.
• The formal dual of a bimodule herd yields a bicomodule coherd, which defines a pair of non-unital rings.
• Sufficient conditions are provided under which the derived rings from the bicomodule coherd are isomorphic to the original algebras A and B.
• The composition of bimodule herds via the tensor product is well-defined and preserves the structural properties of the components.