Bimodule herds

Details Bimodule herds Bimodule herds

2008-05-16

arxiv.org/abs/0805.2510v2

Mathematics

Rings and Algebras

34 pages; v.2: main theorems reformulated, references added

Scientific paper

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.

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