$\mathrm{\Delta }$-filtrations and projective resolutions for the Auslander-Dlab-Ringel algebra

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 $\mathrm{\Delta }$-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.