{"paper":{"title":"On modules $M$ with $\\tau(M) \\cong \\nu \\Omega^{d+2}(M)$ for isolated singularities of Krull dimension $d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Rene Marczinzik","submitted_at":"2019-02-21T18:15:28Z","abstract_excerpt":"A classical formula for the Auslander-Reiten translate $\\tau$ says that $\\tau(M)\\cong \\nu \\Omega^2(M)$ for every indecomposable module $M$ of a selfinjective Artin algebra. We generalise this by showing that for a $2d$-periodic isolated singularity $A$ of Krull dimension $d$, we have for the Auslander-Reiten translate of an indecomposable non-projective Cohen-Macaulay $A$-module $M$, $\\tau(M)\\cong \\nu \\Omega^{d+2}(M)$ if and only if $Ext_A^{d+1}(M,A)=Ext_A^{d+2}(M,A)=0$. We give several applications for Artin algebras."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.08169","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":""},"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"}