The stable Auslander-Reiten components of certain monomorphism categories. (English) Zbl 07838102
Summary: Let A be an Artin algebra and let \(\operatorname{Gprj}\)-\(\Lambda\) denote the class of all the finitely generated Gorenstein projective \(\Lambda \)-modules. In this paper, we study the components of the stable Auslander-Reiten quiver of a certain subcategory of the monomorphism category \({\mathcal{S}}(\operatorname{Gprj}\)-\(\Lambda)\) containing boundary vertices. We describe the shape of such components. It is shown that certain components are linked to the orbits of an auto-equivalence on the stable category \(\underline{\operatorname{Gprj}}\)-\(\Lambda\). In particular, for the finite components, we show that under certain mild conditions, their cardinalities are divisible by 3. We see that this three-periodicity phenomenon reoccurs several times in the paper.
MSC:
| 16G20 | Representations of quivers and partially ordered sets |
| 16E65 | Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) |
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 16G10 | Representations of associative Artinian rings |
Keywords:
monomorphism category; almost split sequence; Auslander-Reiten quiver; Gorenstein projective moduleReferences:
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