# Special tilting modules for algebras with positive dominant dimension

## TL;DR

This paper investigates special tilting and cotilting modules in algebras with positive dominant dimension, revealing their properties, characterizations of certain algebra classes, and their connections via Morita-Tachikawa correspondence.

## Contribution

It introduces and analyzes specific tilting and cotilting modules generated by projective-injectives, characterizes minimal d-Auslander-Gorenstein and d-Auslander algebras, and explores their relationships through Morita-Tachikawa correspondence.

## Key findings

- Endomorphism algebras of these modules have global dimension at most that of the original algebra.
- Minimal d-Auslander-Gorenstein and d-Auslander algebras are characterized by the coincidence of special tilting and cotilting modules.
- Algebras with dominant dimension ≥ 2 can be expressed as endomorphism algebras via Morita-Tachikawa correspondence.

## Abstract

We study certain special tilting and cotilting modules for an algebra with positive dominant dimension, each of which is generated or cogenerated (and usually both) by projective-injectives. These modules have various interesting properties, for example that their endomorphism algebras always have global dimension at most that of the original algebra. We characterise minimal d-Auslander-Gorenstein algebras and d-Auslander algebras via the property that these special tilting and cotilting modules coincide. By the Morita-Tachikawa correspondence, any algebra of dominant dimension at least 2 may be expressed (essentially uniquely) as the endomorphism algebra of a generator-cogenerator for another algebra, and we also study our special tilting and cotilting modules from this point of view, via the theory of recollements and intermediate extension functors.