The Auslander-Reiten theory of the morphism category of projective modules

TL;DR

This paper explores the structure of almost split sequences in the morphism category of projective modules over Artin algebras, linking it to $ au$-tilting theory, $g$-vectors, and Auslander-Reiten theory, with explicit constructions and applications.

Contribution

It provides explicit constructions of almost split sequences in the morphism category of projective modules and establishes a connection to $g$-vectors and 0-Auslander categories.

Findings

Explicit constructions of almost split sequences in $ ext{Mor}( ext{proj} ext{-} ext{modules})$

Establishment of a link between morphism categories and $ au$-tilting theory

Injection from Morita classes to 0-Auslander exact categories

Abstract

We investigate the structure of certain almost split sequences in $\mathcal{P}(\Lambda)$, i.e., the category of morphisms between projective modules over an Artin algebra $\Lambda$. The category $\mathcal{P}(\Lambda)$ has very nice properties and is closely related to $\tau$-tilting theory, $g$-vectors, and Auslander-Reiten theory. We provide explicit constructions of certain almost split sequences ending at or starting from certain objects. Applications, such as to $g$-vectors, are given. As a byproduct, we also show that there exists an injection from Morita equivalence classes of Artin algebras to equivalence classes of 0-Auslander exact categories.