Relative Auslander--Gorenstein Pairs

TL;DR

This paper introduces and studies relative Auslander--Gorenstein pairs, unifying and extending existing homological algebra results through the use of self-orthogonal modules and tilting-cotilting modules.

Contribution

It generalizes known results by characterizing relative Auslander pairs via tilting-cotilting modules and explores their module categories, providing explicit examples.

Findings

Characterization of relative Auslander pairs using tilting-cotilting modules.

Unification of existing results on Auslander--Gorenstein algebras.

Identification of module category pieces via endomorphism algebras.

Abstract

In this paper, we introduce and study relative Auslander--Gorenstein pairs. This consists of a finite-dimensional Gorenstein algebra together with a self-orthogonal module that provides a further homological feature of the algebra in terms of relative dominant dimension. These pairs will be called relative Auslander pairs whenever the algebra in question has finite global dimension. We characterize relative Auslander pairs by the existence and uniqueness of tilting-cotilting modules having higher values relative dominant and codominant dimension with respect to the self-orthogonal module. The same characterisation remains valid for relative Auslander--Gorenstein pairs if the self-orthogonal module has injective or projective dimension at most one. Our relative approach generalises and unifies the known results from the literature, for instance, the characterization of minimal Auslander--Gorenstein algebras. As an application of our methods, we prove that for any relative Auslander pair pieces of the module category of the endomorphism algebra of the self-orthogonal module can be identified with pieces of the module category of the endomorphism algebra of the unique tilting-cotilting module associated with the relative Auslander pair. We provide explicit examples of relative Auslander pairs.