Split-by-nilpotent extensions algebras and stratifying systems

TL;DR

This paper investigates conditions under which stratifying systems in module categories of artin algebras can be transferred via change of rings functors in split-by-nilpotent extensions, linking their stratified structures.

Contribution

It establishes necessary and sufficient conditions for lifting stratifying systems through change of rings functors in split-by-nilpotent extensions.

Findings

Characterization of when stratifying systems can be lifted via functors G and F.

Analysis of relationships between filtered module categories.

Conditions for restricting stratifying systems from Γ to Λ.

Abstract

Let $\Gamma$ and $\Lambda$ be artin algebras such that $\Gamma$ is a split-by-nilpotent extension of $\Lambda$ by a two sided ideal $I$ of $\Gamma.$ Consider the so-called change of rings functors $G:={}_\Gamma\Gamma_\Lambda\otimes_\Lambda -$ and $F:={}_\Lambda \Lambda_\Gamma\otimes_\Gamma -.$ In this paper, we find the necessary and sufficient conditions under which a stratifying system $(\Theta,\leq)$ in $\modu\Lambda$ can be lifted to a stratifying system $(G\Theta,\leq)$ in $\modu\,(\Gamma).$ Furthermore, by using the functors $F$ and $G,$ we study the relationship between their filtered categories of modules and some connections with their corresponding standardly stratified algebras are stated. Finally, a sufficient condition is given for stratifying systems in $\modu\,(\Gamma)$ in such a way that they can be restricted, through the functor $F,$ to stratifying systems in $\modu\,(\Lambda).$