Dominant Auslander-Gorenstein algebras and mixed cluster tilting

TL;DR

This paper introduces dominant Auslander-Gorenstein algebras and mixed cluster tilting modules, establishing a correspondence between them and exploring their properties and examples, including trivial extension algebras and iterated SGC-extensions.

Contribution

It generalizes higher Auslander algebras and cluster tilting modules, establishing a new Auslander type correspondence and analyzing their properties and examples.

Findings

Dominant Auslander-Gorenstein algebras are introduced and characterized.

A bijective correspondence between these algebras and mixed cluster tilting modules is established.

Every trivial extension of a d-representation-finite algebra admits a mixed cluster tilting module.

Abstract

We introduce the class of dominant Auslander-Gorenstein algebras as a generalisation of higher Auslander algebras and minimal Auslander-Gorenstein algebras, and give their basic properties. We also introduce mixed (pre)cluster tilting modules as a generalisation of (pre)cluster tilting modules, and establish an Auslander type correspondence by showing that dominant Auslander-Gorenstein (respectively, Auslander-regular) algebras correspond bijectively with mixed precluster (respectively, cluster) tilting modules. We show that every trivial extension algebra $T(A)$ of a $d$-representation-finite algebra A admits a mixed cluster tilting module and show that this can be seen as a generalisation of the well known result that $d$-representation-finite algebras are fractionally Calabi-Yau. We show that iterated SGC-extensions of a gendo-symmetric dominant Auslander-Gorenstein algebra admit mixed precluster tilting modules.