Abstract
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.
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Acknowledgements
The author would like to thank her supervisor Petter Andreas Bergh for helpful discussions and comments. She would also thank an anonymous referee for careful reading and suggestions which led to significant improvement of the paper.
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Open access funding provided by NTNU Norwegian University of Science and Technology (incl St. Olavs Hospital - Trondheim University Hospital).
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Presented by: Michela Varagnolo
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Haugland, J. Auslander–Reiten Triangles and Grothendieck Groups of Triangulated Categories. Algebr Represent Theor 25, 1379–1387 (2022). https://doi.org/10.1007/s10468-021-10071-9
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DOI: https://doi.org/10.1007/s10468-021-10071-9