Auslander–Reiten Triangles and Grothendieck Groups of Triangulated Categories
We prove that if the Auslander–Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull–Schmidt triangulated category with a (co)generator, then the category has only finitely many isomorphism classes of indecomposable objects up to translation. This gives a triangula...
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| Veröffentlicht in: | Algebras and representation theory 2022-12, Vol.25 (6), p.1379-1387 |
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| container_title | Algebras and representation theory |
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| creator | Haugland, Johanne |
| description | We prove that if the Auslander–Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull–Schmidt triangulated category with a (co)generator, then the category has only finitely many isomorphism classes of indecomposable objects up to translation. This gives a triangulated converse to a theorem of Butler and Auslander–Reiten on the relations for Grothendieck groups. Our approach has applications in the context of Frobenius categories. |
| doi_str_mv | 10.1007/s10468-021-10071-9 |
| format | Article |
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| identifier | ISSN: 1386-923X |
| ispartof | Algebras and representation theory, 2022-12, Vol.25 (6), p.1379-1387 |
| issn | 1386-923X 1572-9079 |
| language | eng |
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| source | SpringerLink Journals - AutoHoldings |
| subjects | Associative Rings and Algebras Commutative Rings and Algebras Isomorphism Mathematics Mathematics and Statistics Non-associative Rings and Algebras |
| title | Auslander–Reiten Triangles and Grothendieck Groups of Triangulated Categories |
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