{"paper":{"title":"On irreducible morphisms and Auslander-Reiten triangles in the stable category of modules over repetitive algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Hern\\'an Giraldo, Jos\\'e A. V\\'elez-Marulanda, Yohny Calder\\'on-Henao","submitted_at":"2019-08-08T03:16:45Z","abstract_excerpt":"Let $\\mathbf{k}$ be an algebraically closed field, let $\\Lambda$ be a finite dimensional $\\mathbf{k}$-algebra, and let $\\widehat{\\Lambda}$ be the repetitive algebra of $\\Lambda$. For the stable category of finitely generated left $\\widehat{\\Lambda}$-modules $\\widehat{\\Lambda}$-\\underline{mod}, we show that the irreducible morphisms fall into three canonical forms: (i) all the component morphisms are split monomorphisms; (ii) all of them are split epimorphisms; (iii) there is exactly one irreducible component. We next use this fact in order to describe the shape of the Auslander-Reiten triangle"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.02912","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.02912/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}