nLab
coring

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Contents

Idea

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

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

Definition

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

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

Base ring extension

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

Let α:AB\alpha:A\to B be a morphism of rings and CC an AA-coring. Then the BB-BB-bimodule B AC ABB\otimes_A C\otimes_A B has an induced structure of a BB-coring with comultiplication

B AC ABBΔ CBB AC AC ABBC1 BCBB AC AB AC AB(B AC AB) B(B AC AB) 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 AC ABBϵ CBB AA ABBϕBB AB ABmultB 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 varying bases

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

(i) α:AB\alpha : A\to B is an RR-algebra morphism; by restriction this makes DD an AA-AA-bimodule by restriction. Denote also by p:D ADD BDp:D\otimes_A D\to D\otimes_B D the canonical projection of bimodules induced by α\alpha.

(ii) γ: AC A AD A\gamma : {}_A C_A\to {}_A D_A is a map of AA-AA-bimodules, that is

commutes where the vertical arrows are the combined bimodule actions

(iii) γ\gamma commutes with counit αϵ C=ϵ Dγ\alpha \circ \epsilon_C = \epsilon_D\circ \gamma

(iv) p(γ Aγ)Δ C=Δ Dγp\circ (\gamma\otimes_A\gamma)\circ \Delta_C = \Delta_D\circ \gamma, or diagramatically,

Map bcbbγ(c)bb\otimes c\otimes b'\mapsto b\gamma(c)b', B AC ABDB\otimes_A C\otimes_A B\to D from the base extension of CC to DD is by construction a map of BB-bimodules (externally we just use the BB-actions on BB and on DD which are compatible by action axioms) and the conditions (iii),(iv) express that this map of BB-bimodules is a morphism of BB-corings.

Morphism (α,γ)(\alpha,\gamma) factorizes into a morphism of AA-corings AC A AB AC AB A{}_A C_A\to {}_A B\otimes_A C\otimes_A B_A, c1c1c\mapsto 1\otimes c\otimes 1 into the base extension coring (determined by α\alpha, AA, BB and CC), followed by a morphism of BB-corings BB AC AB B BD B{}_B B\otimes_A C\otimes_A B_B\to {}_B D_B.

Every morphism (α,γ)(\alpha,\gamma) as above induces an induction functor C D{}^C\mathcal{M}\to{}^D\mathcal{M}, among the categories of (say, left) comodules, MB AMM\mapsto B\otimes_A M, fid Bff\mapsto id_B\otimes f with, which for CC-coaction ρ M:MC AM,mm (1)m (0)\rho^M: M\to C\otimes_A M, m\mapsto \sum m_{(-1)}\otimes m_{(0)} gives DD-comodule structure

B AMD BB AMD AM,bmbγ(m (1))m (0). B\otimes_A M \to D\otimes_B B\otimes_A M\cong D\otimes_A M,\,\,\, b\otimes m \mapsto b\gamma(m_{(-1)}) \otimes m_{(0)}.

Similarly, one defines the induction functor for right comodules, and in particular coaction M ABM AB BDM ADM'\otimes_A B\to M'\otimes_A B\otimes_B D\cong M'\otimes_A D, mbm (0)γ(m (1))bm'\otimes b\mapsto m_{(0)}\otimes\gamma(m_{(1)})b. In particular, one can start with Δ C\Delta_C and induce C ABC ADC\otimes_A B\to C\otimes_A D via cbc (1)γ(c (2))bc\otimes b\mapsto c_{(1)}\otimes \gamma(c_{(2)})b.

Under the assumption that the morphism (α,γ)(\alpha,\gamma) is a pure morphism of corings (left hand version), which is for example satisfied automatically when CC is flat as a right AA-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 NN is a left DD-comodule, consider the cotensor product written in two ways as the equalizer

where p:D AND BNp:D\otimes_A N\to D\otimes_B N is the canonical projection.

Comodules over a coring

Given an AA-coring C=( AC A,Δ,ϵ)C = ({}_A C_A, \Delta,\epsilon), a left CC-comodule MM is a left AA-module with a map of left AA-modules ν:MM A AC A\nu: M\to M\otimes_A {}_A C_A (left CC-coaction), such that the composition MρM ACρC(M AC) ACM\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 MρM ACMΔ CM A(C AC)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 CC-comodules f:(M,ρ M)(N,ρ N)f:(M,\rho^M)\to(N,\rho^N) is a morphism of underlying AA-modules f:MNf:M\to N such that (M Af)ρ M=ρ Nf(M\otimes_A f)\circ\rho^M = \rho^N\circ f. Thus there is a category of left CC-comodules equipped with a forgetful functor to the category of left AA-modules.

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

Analogously, one considers right CC-comodules as right AA-comodules with right CC-coactions.

Examples

Canonical (Sweedler) coring

The classical example of a coring is the canonical or Sweedler coring corresponding to an extension RSR\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,SR,S and an SS-RR-bimodule, finitely generated as right projective RR-module M= SM RM = {}_S M_R. Clearly, M *=Hom R(M,R)M^* = Hom_R(M,R) is an RR-SS-bimodule. Let MM be given as a direct summand of a free module F= iR iF = \oplus_i R_i; the decomposition F=MLF = M\oplus L induces projection p:FMp:F\to M, ir i ir ix i\sum_i r_i\mapsto \sum_i r_i x_i, a section s:MFs: M\hookrightarrow F, s(m)= ix i *(m) iR is(m) = \sum_i x_i^*(m)\in\oplus_i R_i, and dual basis x 1,,x nMx_1,\ldots,x_n\in M, x 1 *,,x n *Hom R(M,R)x_1^*,\ldots,x_n^*\in Hom_R(M,R) characterized by m= ix i *(m)x im = \sum_i x_i^*(m) x_i for all mMm\in M. Then define an RR-coring CC over RR as RR-bimodule M * SMM^*\otimes_S M with comultiplication

Δ:M * SM(M * SM) R(M * SM) \Delta : M^*\otimes_S M\to (M^*\otimes_S M)\otimes_R(M^*\otimes_S M)
Δ(fm)= i=1 nfx i *x im \Delta(f\otimes m) = \sum_{i=1}^n f\otimes x_i^*\otimes x_i\otimes m

and ϵ:M * SM,fmf(m)R\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 coring?s.

References

The notion of an AA-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 AA-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.

  • 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:

category: algebra

Last revised on February 28, 2021 at 11:38:24. See the history of this page for a list of all contributions to it.