A Note on Coseparable Coalgebras

TL;DR

This paper explores the relationship between coseparable coalgebras and corings, demonstrating that a coalgebra can be viewed as a coring over its dual and characterizing coseparability via the existence of counits.

Contribution

It establishes a novel connection between coseparable coalgebras and corings, providing conditions for the existence of counits in the associated coring.

Findings

A coalgebra over a commutative ring can be viewed as a coring over its dual.

Coseparability of a coalgebra is equivalent to the coring having a counit.

The paper characterizes when the coring has a left or right counit.

Abstract

Given a coalgebra $C$ over a commutative ring $R,$ we show that $C$ can be considered as a (not necessarily counital) $C^{\ast op}$-coring. Moreover, we show that this coring has a left (right) counity if and only if $C$ is coseparable as an $R$-coalgebra.