# $\Delta$-filtrations and projective resolutions for the Auslander-Dlab-Ringel algebra

## TL;DR

This paper investigates the structure of modules over the ADR algebra, a special class of quasihereditary algebra, providing new insights into $ ext{Δ}$-filtrations, projective covers, and counterexamples to existing claims.

## Contribution

It introduces a detailed study of $ ext{Δ}$-filtrations and projective resolutions for ADR algebras, including a counterexample to a previous conjecture.

## Key findings

- Characterization of $ ext{Δ}$-filtrations for RUSQ algebras
- Determination of projective covers for specific R_A-modules
- Counterexample to a claim by Auslander-Platzeck-Todorov

## Abstract

The ADR algebra $R_A$ of an Artin algebra $A$ is a right ultra strongly quasihereditary algebra (RUSQ algebra). In this paper we study the $\Delta$-filtrations of modules over RUSQ algebras and determine the projective covers of a certain class of $R_A$-modules. As an application, we give a counterexample to a claim by Auslander-Platzeck-Todorov, concerning projective resolutions over the ADR algebra.