TY - JOUR

T1 - The Auslander-Reiten quiver of perfect complexes for a self-injective algebra

AU - Webb, Peter

N1 - Publisher Copyright:
© 2023 Elsevier B.V.

PY - 2024/5

Y1 - 2024/5

N2 - We consider the homotopy category of perfect complexes for a finite dimensional self-injective algebra over a field, identifying many aspects of perfect complexes according to their position in the Auslander-Reiten quiver. Short complexes lie close to the rim. We characterize the position in the quiver of complexes of lengths 1, 2 and 3, as well as rigid complexes and truncated projective resolutions. We describe completely the quiver components that contain projective modules (complexes of length 1). We obtain relationships between the homology of complexes at different places in the quiver, deducing that every self-injective algebra of radical length at least 3 has indecomposable perfect complexes with arbitrarily large homology in any given degree. We show that homology stabilizes, in a certain sense, away from the rim of the quiver.

AB - We consider the homotopy category of perfect complexes for a finite dimensional self-injective algebra over a field, identifying many aspects of perfect complexes according to their position in the Auslander-Reiten quiver. Short complexes lie close to the rim. We characterize the position in the quiver of complexes of lengths 1, 2 and 3, as well as rigid complexes and truncated projective resolutions. We describe completely the quiver components that contain projective modules (complexes of length 1). We obtain relationships between the homology of complexes at different places in the quiver, deducing that every self-injective algebra of radical length at least 3 has indecomposable perfect complexes with arbitrarily large homology in any given degree. We show that homology stabilizes, in a certain sense, away from the rim of the quiver.

KW - Auslander-Reiten triangle

KW - Perfect complex

KW - Self-injective algebra

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U2 - 10.1016/j.jpaa.2023.107599

DO - 10.1016/j.jpaa.2023.107599

M3 - Article

AN - SCOPUS:85182007438

SN - 0022-4049

VL - 228

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 5

M1 - 107599

ER -