On the set of grouplikes of a coring

Abstract

We focus our attention to the set $\gl{\coring{C}}$ of grouplike elements of a coring $\coring{C}$ over a ring $A$. We do some observations on the actions of the groups $U(A)$ and $\aut{\coring{C}}$ of units of $A$ and of automorphisms of corings of $\coring{C}$, respectively, on $\gl{\coring{C}}$, and on the subset $\galois{\coring{C}}$ of all Galois grouplike elements. Among them, we give conditions on $\coring{C}$ under which $\galois{\coring{C}}$ is a group, in such a way that there is an exact sequence of groups $\{1\} \to U(A^{g}) \to U(A) \to \galois{\coring{C}} \to \{1\},$ where $A^g$ is the subalgebra of coinvariants for some $g \in \galois{\coring{C}}$.