nLab
coring

Idea

The notion of coring is a generalization of that of coalgebra.

Whereas a coalgebra structure is defined on a vector space – which may be regarded as a bimodule over the ground field – a coring structure is defined on a bimodule over a general ring.

Definition

An $A$-coring is a comonoid in the monoidal category of bimodules over a fixed (typically noncommutative) unital ring $A$.

This generalizes the notion of $A$-coalgebras which are defined only if $A$ is commutative and where the bimodules in question are central?.

Examples

Sweedler corings

The classical example of a coring is the Sweedler coring corresponding to an extension $R\hookrightarrow S$ of unital rings. The category of descent data for this extension is equivalent to the category of comodules over the Sweedler coring.

Corings are in general useful for the treatment of descent in noncommutative algebraic geometry.

Matrix corings

Another major class of examples are the so-called matrix coring?s.

References

The notion of an $A$-coring is introduced by M. Sweedler and recently lived through a renaissance in works of T. Brzeziński, R. Wisbauer, G. Böhm, L. Kaoutit, Gómez-Torrecillas, S. Caenepeel, J. Y. Abuhlail, J. Vercruysse and others, including the creation of Galois theory for corings. Some prefer to speak about $A$-cocategories.

There is already a monograph:

• T. Brzeziński, R. Wisbauer, Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge 2003.