On relative Auslander algebras

TL;DR

This paper investigates relative Auslander algebras using intermediate extension functors, establishing conditions for Morita equivalence of Gorenstein algebras via their Cohen-Macaulay Auslander algebras.

Contribution

It introduces a novel approach applying recollements and extension functors to study tilting modules and Morita equivalences in the context of relative Auslander algebras.

Findings

Existence of tilting-cotilting modules over relative Auslander algebras.

Morita equivalence of Gorenstein algebras of G-dimension 1 with finite Cohen-Macaulay type is characterized by their Cohen-Macaulay Auslander algebras.

Application of intermediate extension functors to study algebraic properties.

Abstract

Relative Auslander algebras were introduced and studied by Beligiannis. In this paper, we apply intermediate extension functors associated to certain recollements of functor categories to study them. In particular, we study the existence of tilting-cotilting modules over such algebras. As a consequence, it will be shown that two Gorenstein algebras of G-dimension 1 being of finite Cohen-Macaulay type are Morita equivalent if and only if their Cohen-Macaulay Auslander algebras are Morita equivalent.