On the set of grouplikes of a coring

TL;DR

This paper investigates the structure of grouplike elements in corings over rings, exploring their automorphisms, actions, and conditions under which Galois grouplike elements form a group, leading to an exact sequence involving units of subalgebras.

Contribution

It provides new conditions under which the set of Galois grouplike elements forms a group and establishes an exact sequence relating units of the base ring and coinvariants.

Findings

Conditions for $ ext{galois}( ext{C})$ to be a group.

An exact sequence involving units of $A$ and coinvariants.

Analysis of actions of automorphisms on grouplike elements.

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 $g \in \galois{\coring{C}}$.