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 central 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?.

Base ring extension

More generally, fix a ground commutative ring $R$. Corings will be now over $R$-algebras. So a coring will mean a pair $(A,C)$ where $A$ is an $R$-algebra and $C$ an $A$-coring.

Let $\alpha :A\to B$ be a morphism of rings and $C$ an $A$-coring. Then the $B$-$B$-bimodule $B{\otimes }_{A}C{\otimes }_{A}B$ has an induced structure of a $B$-coring with comultiplication

$$B{\otimes }_{A}C{\otimes }_{A}B\stackrel{B\otimes {\Delta }_C\otimes B}{⟶}B{\otimes }_{A}C{\otimes }_{A}C{\otimes }_{A}B\stackrel{B\otimes C\otimes 1_B\otimes C\otimes B}{⟶}B{\otimes }_{A}C{\otimes }_{A}B{\otimes }_{A}C{\otimes }_{A}B\cong (B{\otimes }_{A}C{\otimes }_{A}B){\otimes }_B(B{\otimes }_{A}C{\otimes }_{A}B)$$B\otimes_A C\otimes_A B \stackrel{B\otimes \Delta_C\otimes B}\longrightarrow B\otimes_A C\otimes_A C\otimes_A B \stackrel{B\otimes C\otimes 1_B\otimes C\otimes B}\longrightarrow B\otimes_A C\otimes_A B\otimes_A C\otimes_A B \cong (B\otimes_A C\otimes_A B)\otimes_B (B \otimes_A C\otimes_A B)

and the counit

$$B{\otimes }_{A}C{\otimes }_{A}B\stackrel{B\otimes {ϵ}_C\otimes B}{⟶}B{\otimes }_{A}A{\otimes }_{A}B\stackrel{B\otimes \varphi \otimes B}{⟶}B{\otimes }_{A}B{\otimes }_{A}B\stackrel{\mathrm{mult}}{⟶}B$$B\otimes_A C\otimes_A B \stackrel{B\otimes\epsilon_C\otimes B}\longrightarrow B\otimes_A A\otimes_A B \stackrel{B\otimes\phi\otimes B}\longrightarrow B\otimes_A B\otimes_A B \stackrel{mult}\longrightarrow B

Morphisms of corings over different bases

A morphism $(A,C)\to (B,D)$ is a pair $(\alpha ,\gamma )$ where

• $\alpha :A\to B$ is an $R$-algebra morphism; by restriction this makes $D$ an $A$-$A$-bimodule by restriction. Denote also by $p:D{\otimes }_{A}D\to D{\otimes }_{B}D$ the canonical projection of bimodules induced by $\alpha$.
• $\gamma :C\to D$ is a map of $A$-$A$-bimodules
• $\gamma$ commutes with counit $\alpha \circ {ϵ}_C={ϵ}_D\circ \gamma$
• $p\circ (\gamma {\otimes }_A\gamma )\circ {\Delta }_C={\Delta }_D\circ \gamma$

The last two conditions can be said that the base ring extension coring $B{\otimes }_{A}C{\otimes }_{A}B$ of $C$ maps to $D$ (via map induced by $\gamma$) as a morphism of $B$-corings.

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.

Special topics:

• T. Brzeziński, Descent cohomology and corings, Comm. Algebra 36:1894-1900, 2008, arxiv:math.RA/0601491

• L. El Kaoutit, J. Gomez-Torrecillas, On the set of grouplikes of a coring, arxiv/0901.4291

• T. Brzeziński, Flat connections and (co)modules, in: New Techniques in Hopf Algebras and Graded Ring Theory, S Caenepeel and F Van Oystaeyen (eds), Universa Press, Wetteren, 2007 pp. 35-52 arxiv:math.QA/0608170

• T. Brzeziński, The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties, Algebras and Representation Theory 5 (2002) 389-410, math.QA/0002105

• Lars Kadison, Depth two and Galois coring, math.RA/0408155

• George M. Bergman, Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, A.M.S. Math. Surveys and Monographs 45, ix+388 pp., 1996; ISBN 0-8218-0495-2. MR 97k:16001 errata and updates.

• T. Brzeziński, L. Kadison, R. Wisbauer, On coseparable and biseparable corings, Hopf algebras in noncommutative geometry and physics, 71–87, Lecture Notes in Pure and Appl. Math., 239, Dekker, New York, 2005.

There is a generalization of corings:

• Jawad Y. Abuhlail, Semicorings and semicomodules, arxiv/1303.3924