Locally finite triangulated categories. (English) Zbl 1110.16013
Summary: A \(k\)-linear triangulated category \(\mathcal A\) is called locally finite provided
\(\sum_{X\in\text{ind\,}\mathcal A}\dim_k\operatorname{Hom}_{\mathcal A}(X,Y)<\infty\) for any indecomposable object \(Y\) in \(\mathcal A\). It has Auslander-Reiten triangles. In this paper, we show that if a (connected) triangulated category has Auslander-Reiten triangles and contains loops, then its Auslander-Reiten quiver is of the form \(\widehat L_n\): \[ \underset {n}\circ{^{\longleftarrow}_{\longrightarrow}}\underset{n-1}\circ{ ^{\longleftarrow}_{\longrightarrow}}\circ\cdots\underset {2}\circ{^{\longleftarrow}_{\longrightarrow}}\underset {1}\circ\circlearrowleft. \] By using this, we prove that the Auslander-Reiten quiver of any locally finite triangulated category \(\mathcal A\) is of the form \(\mathbb{Z}\vec\Delta/G\), where \(\Delta\) is a Dynkin diagram and \(G\) is an automorphism group of \(\mathbb{Z}\vec\Delta\). For most automorphism groups \(G\), the triangulated categories with \(\mathbb{Z}\vec\Delta/G\) as their Auslander-Reiten quivers are constructed. In particular, a triangulated category with \(\widehat L_n\) as its Auslander-Reiten quiver is constructed.
\(\sum_{X\in\text{ind\,}\mathcal A}\dim_k\operatorname{Hom}_{\mathcal A}(X,Y)<\infty\) for any indecomposable object \(Y\) in \(\mathcal A\). It has Auslander-Reiten triangles. In this paper, we show that if a (connected) triangulated category has Auslander-Reiten triangles and contains loops, then its Auslander-Reiten quiver is of the form \(\widehat L_n\): \[ \underset {n}\circ{^{\longleftarrow}_{\longrightarrow}}\underset{n-1}\circ{ ^{\longleftarrow}_{\longrightarrow}}\circ\cdots\underset {2}\circ{^{\longleftarrow}_{\longrightarrow}}\underset {1}\circ\circlearrowleft. \] By using this, we prove that the Auslander-Reiten quiver of any locally finite triangulated category \(\mathcal A\) is of the form \(\mathbb{Z}\vec\Delta/G\), where \(\Delta\) is a Dynkin diagram and \(G\) is an automorphism group of \(\mathbb{Z}\vec\Delta\). For most automorphism groups \(G\), the triangulated categories with \(\mathbb{Z}\vec\Delta/G\) as their Auslander-Reiten quivers are constructed. In particular, a triangulated category with \(\widehat L_n\) as its Auslander-Reiten quiver is constructed.
MSC:
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 16G20 | Representations of quivers and partially ordered sets |
Keywords:
locally finite triangulated categories; triangulated categories with loops; Auslander-Reiten quivers; Dynkin diagramsReferences:
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