# Tilting modules and dominant dimension with respect to injective modules

## TL;DR

This paper explores the connection between tilting modules with finite projective dimension and dominant dimension relative to injective modules, extending previous results and characterizing certain algebra classes.

## Contribution

It generalizes existing results on tilting modules and dominant dimension, providing new characterizations of almost $n$-Auslander-Gorenstein and almost $n$-Auslander algebras.

## Key findings

- Characterization of almost $n$-Auslander-Gorenstein algebras
- Conditions for almost $1$-Auslander algebras to be strongly quasi-hereditary
- Extension of previous results on tilting modules and dominant dimension

## Abstract

In this paper, we study a relationship between tilting modules with finite projective dimension and dominant dimension with respect to injective modules as a generalization of results of Crawley-Boevey-Sauter, Nguyen-Reiten-Todorov-Zhu and Pressland-Sauter. Moreover, we give characterizations of almost $n$-Auslander-Gorenstein algebras and almost $n$-Auslander algebras by the existence of tilting modules. As an application, we describe a sufficient condition for almost $1$-Auslander algebras to be strongly quasi-hereditary by comparing such tilting modules and characteristic tilting modules.