nLab grouplike element




General definition in a coring

Given an AA-coring (C,Δ,ϵ)(C,\Delta,\epsilon) (a comonoid in the category of AA-AA-bimodules for a kk-algebra AA) (example: any kk-algebra) a semi-grouplike element in AA is any gCg\in C such that

Δ(g)=gg. \Delta(g) = g\otimes g \,.

A grouplike (or group-like) element is a semi-grouplike one such that ϵ(g)=1\epsilon(g) = 1.

Proposition. An AA-coring (C,Δ,ϵ)(C,\Delta,\epsilon) has a grouplike element iff AA is a right or left CC-comodule.

Proof. Given a grouplike element gCg\in C, one defines a right coaction ρ=ρ g:AA ACC\rho = \rho_g : A \to A\otimes_A C\cong C by the formula

ρ(a)=1 A Aga=ga \rho(a) = 1_A \otimes_A ga = ga

it is clear that this is a map of AA-bimodules. Now (ρid C)(1 A Aga)=g A1 Aga=gga(\rho\otimes id_C)(1_A\otimes_A ga) = g\otimes_A 1_A\otimes ga = g\otimes ga, while (idΔ C)(1 Aga)=1 A Agga=gga(id\otimes \Delta_C)(1_A\otimes ga) = 1_A \otimes_A g\otimes ga = g\otimes ga hence the coassociativity and similarly for the counit.

Conversely, let (A,ρ)(A,\rho) be a right CC-comodule. Then one checks that ρ(1 A)A ACC\rho(1_A)\in A\otimes_A C \cong C is a grouplike.

For the left comodules the story is similar, e.g. ρ(a)=ag\rho(a) = ag.

Note that coring homomorphisms send grouplike elements to grouplike elements. This can be assembled into a functor, and moreover, this functor is representable.


The functor G:A-CoRingSetG : A\text{-}\mathsf{CoRing} \to \mathsf{Set} is representable: GHom A-CoRing(A,)G \cong \mathrm{Hom}_{A\text{-}\mathsf{CoRing}}(A,-).


We can assign each gGCg \in GC the map ρ g\rho_g by 1 Ag1_A \mapsto g. It is easily checked that this is indeed a map of coalgebras, and the inverse is to take the image of 1 A1_A. The assignments are immediately checked to be natural and to be inverses of each other.

Special case: grouplike elements in coalgebras

Every coalgebra is special case of a coring.

The grouplike elements in a kk-Hopf algebra form a group. Indeed, a Hopf algebra HH is just a group over the category of coalgebras over kk (the monoidal structure \otimes is even the categorical product), so the set of grouplike elements Hom k-CoAlg(k,H)\mathrm{Hom}_{k\text{-}\mathsf{CoAlg}}(k,H) is canonically a group.

Relation to differential graded algebras

For corings with a (sometimes semi-)grouplike element one can define many useful notions which do not exist for general corings.

For example, given a semi-grouplike element gg, the tensor algebra ΩC= iΩ iC\Omega C = \oplus_i \Omega^i C of the coring CC, where Ω iC=C A AC\Omega^i C = C\otimes_A \ldots \otimes_A C (ii times) over AA can be equipped with a differential dd of degree +1+1 in a canonical way making it a differential graded algebra:

in degree 00, one defines

da=gaag d a = g a - a g

and in higher degree

d(c 1c n)=gc 1c n+(1) n+1c 1c ng+ i=1 nc 1c i1Δ(c i)c i+1c n. d(c_1\otimes\ldots\otimes c_n) = g\otimes c_1\otimes\ldots\otimes c_n + (-1)^{n+1}c_1\otimes\ldots\otimes c_n\otimes g + \sum_{i=1}^n c_1\otimes\ldots\otimes c_{i-1}\otimes \Delta(c_i)\otimes c_{i+1}\otimes\ldots \otimes c_n\,.

In fact, by a result in

  • A. V. Roiter, Matrix problems and representations of BOCS’s; in Lec. Notes. Math. 831, 288–324 (1980)

semi-free differential graded algebras are in bijective correspondence with corings with a group-like element. Moreover flat connections for a semi-free dga are in 11-11 correspondence with the comodules over the corresponding coring with a group-like element.

A special case of this construction is when g=1 R1g = 1\otimes_R 1 and CC is the Sweedler coring for a kk-algebra extension RSR\to S. The dga obtained is the classical Amitsur complex Ω(S/R)\Omega(S/R) for that extension; for this reason the complex ΩC=Ω(C,g)\Omega C = \Omega(C,g) above for any coring CC and semi-grouplike gg is sometimes said to be an Amitsur complex.

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

  • C. Menini, D. Ştefan, Descent theory and Amitsur cohomology of triples, J. Algebra 266 (2003), no. 1, 261–304.

  • T. Brzeziński, Flat connections and comodules, math.QA/0608170

  • T. Brzeziński, Galois structures, Warszawa 2007/8 course, part III, pdf, ps

Last revised on June 4, 2024 at 12:00:42. See the history of this page for a list of all contributions to it.