Given an $A$-coring $(C,\Delta,\epsilon)$ (a comonoid in the category of $A$-$A$-bimodules for a $k$-algebra $A$) (example: any $k$-algebra) a semi-grouplike element in $A$ is any $g\in C$ such that
A grouplike (or group-like) element is a semi-grouplike one such that $\epsilon(g) = 1$.
Proposition. An $A$-coring $(C,\Delta,\epsilon)$ has a grouplike element iff $A$ is a right or left $C$-comodule.
Proof. Given a grouplike element $g\in C$, one defines a right coaction $\rho = \rho_g : A \to A\otimes_A C\cong C$ by the formula
it is clear that this is a map of $A$-bimodules. Now $(\rho\otimes id_C)(1_A\otimes_A ga) = g\otimes_A 1_A\otimes ga = g\otimes ga$, while $(id\otimes \Delta_C)(1_A\otimes ga) = 1_A \otimes_A g\otimes ga = g\otimes ga$ hence the coassociativity and similarly for the counit.
Conversely, let $(A,\rho)$ be a right $C$-comodule. Then one checks that $\rho(1_A)\in A\otimes_A C \cong C$ is a grouplike.
For the left comodules the story is similar, e.g. $\rho(a) = ag$.
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 $G : A\text{-}\mathsf{CoRing} \to \mathsf{Set}$ is representable: $G \cong \mathrm{Hom}_{A\text{-}\mathsf{CoRing}}(A,-)$.
Proof.
We can assign each $g \in GC$ the map $\rho_g$ by $1_A \mapsto g$. It is easily checked that this is indeed a map of coalgebras, and the inverse is to take the image of $1_A$. The assignments are immediately checked to be natural and to be inverses of each other.
Every coalgebra is special case of a coring.
The grouplike elements in a $k$-Hopf algebra form a group. Indeed, a Hopf algebra $H$ is just a group over the category of coalgebras over $k$ (the monoidal structure $\otimes$ is even the categorical product), so the set of grouplike elements $\mathrm{Hom}_{k\text{-}\mathsf{CoAlg}}(k,H)$ is canonically a group.
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 $g$, the tensor algebra $\Omega C = \oplus_i \Omega^i C$ of the coring $C$, where $\Omega^i C = C\otimes_A \ldots \otimes_A C$ ($i$ times) over $A$ can be equipped with a differential $d$ of degree $+1$ in a canonical way making it a differential graded algebra:
in degree $0$, one defines
and in higher degree
In fact, by a result in
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 $1$-$1$ correspondence with the comodules over the corresponding coring with a group-like element.
A special case of this construction is when $g = 1\otimes_R 1$ and $C$ is the Sweedler coring for a $k$-algebra extension $R\to S$. The dga obtained is the classical Amitsur complex $\Omega(S/R)$ for that extension; for this reason the complex $\Omega C = \Omega(C,g)$ above for any coring $C$ and semi-grouplike $g$ 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.