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

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.