Similarity, Codepth Two Bicomodules and QF Bimodules

TL;DR

This paper investigates the properties of similar quasi-finite comodules over coalgebras, introduces the concept of codepth two bicomodules, and explores their implications for ring extensions and endomorphism rings.

Contribution

It establishes the equivalence of certain coalgebra and bimodule properties, introduces codepth two bicomodules, and connects similarity with depth conditions in ring theory.

Findings

Similar quasi-finite comodules have strongly equivalent coendomorphism coalgebras.

Left and right depth two are equivalent for QF extensions.

Characterization of depth three in terms of separability and depth two.

Abstract

For any k-coalgebra C it is shown that similar quasi-finite C-comodules have strongly equivalent coendomorphism coalgebras; (the converse is in general not true). As an application we give a general result about codepth two coalgebra homomorphisms. Also a notion of codepth two bicomodule is introduced. The last section applies similarity to an endomorphism ring theorem for quasi-Frobenius (QF) bimodules and then to finite depth ring extensions. For QF extensions, we establish that left and right depth two are equivalent notions as well as a converse endomorphism theorem, and characterize depth three in terms of separability and depth two.