Standardly stratified lower triangular $\mathbb{K}$-algebras with enough idempotents

TL;DR

This paper characterizes when a lower triangular matrix algebra built from two basic algebras with enough idempotents is standardly stratified or locally bounded, and explores related projective dimension properties.

Contribution

It provides necessary and sufficient conditions for the stratification and boundedness of such triangular algebras based on their components.

Findings

Characterization of standardly stratified lower triangular algebras.

Conditions for local boundedness in these algebras.

Analysis of projective dimensions across module categories.

Abstract

In this paper we study the lower triangular matrix $\mathbb{K}$-algebra $\Lambda:=\left[\begin{smallmatrix} T & 0 \\ M & U \end{smallmatrix}\right],$ where $U$ and $T$ are basic $\mathbb{K}$-algebras with enough idempotents and $M$ is an $U$-$T$-bimodule where $\mathbb{K}$ acts centrally. Moreover, we characterise in terms of $U,$ $T$ and $M$ when, on one hand, the lower triangular matrix $\mathbb{K}$-algebra $\Lambda$ is standardly stratified in the sense of the paper "A generalization theory of standardly stratified algebras I: Standardly stratified ringois"; and on another hand, when $\Lambda$ is locally bounded in the sense of the paper "Locally finite generated modules over rings with enough idempotents". Finally, it is also studied several properties relating the projective dimensions in the categories of finitely generated modules $\mathrm{mod}(U)$, $\mathrm{mod}(T)$ and $\mathrm{mod}(\Lambda).$