Split-by-nilpotent extensions algebras and stratifying systems

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 $(\Theta ,\le )$ in $\moduΛ$ can be lifted to a stratifying system $(G\Theta ,\le )$ in $\modu\,(Γ).$ 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 $\modu\,(Γ)$ in such a way that they can be restricted, through the functor $F,$ to stratifying systems in $\modu\,(Λ).$