On the composition of three irreducible morphisms in the bounded homotopy category. (English) Zbl 1521.16013
Let \(A\) be an Artin algebra over a fixed commutative Artin ring. We denote by \(\mathbf{K}^b(\mathrm{proj}\,A)\) the bounded homotopy category of finitely generated projective right \(A\)-modules.
In the paper, the authors study when the composition of three irreducible morphisms between indecomposable complexes in \(\mathbf{K}^b(\mathrm{proj}\,A)\) is a non-zero morphism in the fourth power of the Jacobson radical of \(\mathbf{K}^b(\mathrm{proj}\,A)\) under the assumption that \(A\) is of finite global dimension (Theorem A).
As an application of Theorem A they prove that the composition of three irreducible morphisms between indecomposable complexes in the bounded homotopy category of a gentle Nakayama algebra, not selfinjective, whose ordinary quiver is an oriented cycle, belongs to the fourth power of the Jacobson radical of \(\mathbf{K}^b(\mathrm{proj}\,A)\) if and only if it vanishes (Theorem B).
In the paper, the authors study when the composition of three irreducible morphisms between indecomposable complexes in \(\mathbf{K}^b(\mathrm{proj}\,A)\) is a non-zero morphism in the fourth power of the Jacobson radical of \(\mathbf{K}^b(\mathrm{proj}\,A)\) under the assumption that \(A\) is of finite global dimension (Theorem A).
As an application of Theorem A they prove that the composition of three irreducible morphisms between indecomposable complexes in the bounded homotopy category of a gentle Nakayama algebra, not selfinjective, whose ordinary quiver is an oriented cycle, belongs to the fourth power of the Jacobson radical of \(\mathbf{K}^b(\mathrm{proj}\,A)\) if and only if it vanishes (Theorem B).
Reviewer: Piotr Malicki (Toruń)
MSC:
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 16G20 | Representations of quivers and partially ordered sets |
| 16E10 | Homological dimension in associative algebras |
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