Equivalence of categories, Gruson-Jensen duality, and applications

TL;DR

This paper investigates conditions under which categories of left and right comodules over a coalgebra are symmetric, establishing dualities between their associated finitely presented module categories.

Contribution

It characterizes when categories of comodules over a coalgebra are symmetric, linking them to dualities between finitely presented module categories.

Findings

Identifies conditions for symmetry between comodule categories

Establishes duality between finitely presented module categories

Provides applications of Gruson-Jensen duality in coalgebra contexts

Abstract

For coalgebras $C$ over a field, we study when the categories ${}^C\Mm$ of left $C$-comodules and $\Mm^C$ of right $C$-comodules are symmetric categories, in the sense that there is a duality between the categories of finitely presented unitary left $R$-modules and finitely presented unitary left $L$-modules, where $R$ and $L$ are the functor rings associated to the finitely accessible categories ${}^C\Mm$ and $\Mm^C$.