Contents

Idea

The notion of coring is a generalization of a $k$-coalgebra. While for a coalgebra $k$ must be a commutative ring (often a field), a coring is defined over a general noncommutative ring $k$ or even an associative algebra $A$.

Whereas a coalgebra structure is defined on a $k$-module (if $k$ is a field, it is a vector space) – which may be regarded as a central $k$-bimodule – a coring structure is defined on a general 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.

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

and the counit

Morphisms of corings over varying bases

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

(i) $\alpha : A\to B$ 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 $\alpha$. (ii) $\gamma : {}_A C_A\to {}_A D_A$ is a map of $A$-$A$-bimodules, that is commutes where the vertical arrows are the combined bimodule actions

(iii) $\gamma$ commutes with counit $\alpha \circ \epsilon_C = \epsilon_D\circ \gamma$

(iv) $p\circ (\gamma\otimes_A\gamma)\circ \Delta_C = \Delta_D\circ \gamma$, or diagramatically, Map $b\otimes c\otimes b'\mapsto b\gamma(c)b'$, $B\otimes_A C\otimes_A B\to D$ from the base extension of $C$ to $D$ is by construction a map of $B$-bimodules (externally we just use the $B$-actions on $B$ and on $D$ which are compatible by action axioms) and the conditions (iii),(iv) express that this map of $B$-bimodules is a morphism of $B$-corings.

Morphism $(\alpha,\gamma)$ factorizes into a morphism of $A$-corings ${}_A C_A\to {}_A B\otimes_A C\otimes_A B_A$, $c\mapsto 1\otimes c\otimes 1$ into the base extension coring (determined by $\alpha$, $A$, $B$ and $C$), followed by a morphism of $B$-corings ${}_B B\otimes_A C\otimes_A B_B\to {}_B D_B$.

Every morphism $(\alpha,\gamma)$ as above induces an induction functor ${}^C\mathcal{M}\to{}^D\mathcal{M}$, among the categories of (say, left) comodules, $M\mapsto B\otimes_A M$, $f\mapsto id_B\otimes f$ with, which for $C$-coaction $\rho^M: M\to C\otimes_A M, m\mapsto \sum m_{(-1)}\otimes m_{(0)}$ gives $D$-comodule structure

Similarly, one defines the induction functor for right comodules, and in particular coaction $M'\otimes_A B\to M'\otimes_A B\otimes_B D\cong M'\otimes_A D$, $m'\otimes b\mapsto m_{(0)}\otimes\gamma(m_{(1)})b$. In particular, one can start with $\Delta_C$ and induce $C\otimes_A B\to C\otimes_A D$ via $c\otimes b\mapsto c_{(1)}\otimes \gamma(c_{(2)})b$.

Under the assumption that the morphism $(\alpha,\gamma)$ is a pure morphism of corings (left hand version), which is for example satisfied automatically when $C$ is flat as a right $A$-module, the induction functor has a right adjoint, the coinduction functor. It is given by a cotensor product which is a comodule under the purity condition. Thus if $N$ is a left $D$-comodule, consider the cotensor product written in two ways as the equalizer where $p:D\otimes_A N\to D\otimes_B N$ is the canonical projection.

Comodules over a coring

Given an $A$-coring $C = ({}_A C_A, \Delta,\epsilon)$, a left $C$-comodule $M$ is a left $A$-module with a map of left $A$-modules $\nu: M\to M\otimes_A {}_A C_A$ (left $C$-coaction), such that the composition $M\stackrel{\rho}\to M\otimes_A C \stackrel{\rho\otimes C}\to (M\otimes_A C)\otimes_A C$ after rebracketing becomes identical to the composition $M\stackrel{\rho}\to M\otimes_A C \stackrel{M\otimes \Delta_C}\to M\otimes_A (C \otimes_A C)$ and the counitality axiom holds.

A morphism of left $C$-comodules $f:(M,\rho^M)\to(N,\rho^N)$ is a morphism of underlying $A$-modules $f:M\to N$ such that $(M\otimes_A f)\circ\rho^M = \rho^N\circ f$. Thus there is a category of left $C$-comodules equipped with a forgetful functor to the category of left $A$-modules.

The functor $F: M\mapsto M\otimes_A C$ is canonically a comonad on ${}_A Mod$ with comultiplication $\Delta^F = Id\otimes_A\Delta_C : F\to F F$ and the Eilenberg-Moore category of the comonad $F$ is isomorphic to the category of $C$-comodules.

Analogously, one considers right $C$-comodules as right $A$-comodules with right $C$-coactions.

Examples

Canonical (Sweedler) coring

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

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

Coring of a projective module

Suppose we are given ring $R,S$ and an $S$-$R$-bimodule, finitely generated as right projective $R$-module $M = {}_S M_R$. Clearly, $M^* = Hom_R(M,R)$ is an $R$-$S$-bimodule. Let $M$ be given as a direct summand of a free module $F = \oplus_i R_i$; the decomposition $F = M\oplus L$ induces projection $p:F\to M$, $\sum_i r_i\mapsto \sum_i r_i x_i$, a section $s: M\hookrightarrow F$, $s(m) = \sum_i x_i^*(m)\in\oplus_i R_i$, and dual basis $x_1,\ldots,x_n\in M$, $x_1^*,\ldots,x_n^*\in Hom_R(M,R)$ characterized by $m = \sum_i x_i^*(m) x_i$ for all $m\in M$. Then define an $R$-coring $C$ over $R$ as $R$-bimodule $M^*\otimes_S M$ with comultiplication

and $\epsilon:M^*\otimes_S M, f\otimes m\mapsto f(m)\in R$.

Matrix coring

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

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 (1996) [ISBN 0-8218-0495-2, ams:surv-45, MR 97k:16001, errata and updates]

  (discussion internal to the category of associative algebras)

• 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.

• T. Brzeziński, L. El Kaoutit, J. Gómez-Torrecillas, The bicategories of corings, J. Pure & Appl. Alg. 205:3 (2006) 510-541 doi:10.1016/j.jpaa.2005.07.013 math.RA/0408042

There is a generalization of corings:

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