Bimodule herds

JOURNAL OF ALGEBRA

卷:321

Article

Brzezinski, Tomasz¹  Vercruysse, Joost² 

[1] Swansea Univ, Dept Math, Swansea SA2 8PP, W Glam, Wales

[2] Vrije Univ Brussel, Fac Engn, B-1050 Brussels, Belgium

关键词: Bimodule herd;    Torsor;    Coring;    Galois comodule;    Entwining structure;   

DOI  :  10.1016/j.jalgebra.2009.01.020

来源: Elsevier

【 摘 要 】

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 unitality 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, or if it is a progenerator and the underlying ring extensions are split, 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. (C) 2009 Elsevier Inc. All rights reserved.

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