Auslander–Reiten Triangles and Grothendieck Groups of Triangulated Categories
Peer reviewed, Journal article
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Date
2021Metadata
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- Institutt for matematiske fag [2473]
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Original version
10.1007/s10468-021-10071-9Abstract
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.