{"paper":{"title":"The Auslander-Reiten Translation in Submodule Categories","license":"","headline":"","cross_cats":["math.CT"],"primary_cat":"math.RT","authors_text":"Claus Michael Ringel, Markus Schmidmeier","submitted_at":"2005-04-14T17:54:52Z","abstract_excerpt":"Let $\\Lambda$ be an artin algebra and $S(\\Lambda)$ the category of all embeddings $(A\\subseteq B)$ where $B$ is a finitely generated $\\Lambda$-module and $A$ is a submodule of $B$. Then $S(\\Lambda)$ is an exact Krull-Schmidt category which has Auslander-Reiten sequences. In this manuscript we show that the Auslander-Reiten translation in $S(\\Lambda)$ can be computed within the category of $\\Lambda$-modules by using our construction of minimal monomorphisms. If in addition $\\Lambda$ is uniserial then any nonprojective indecomposable object in $\\Cal S(\\Lambda)$ is invariant under the sixth power"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0504301","kind":"arxiv","version":2},"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"}