{"paper":{"title":"Separability for relative extensions of object unital strongly groupoid graded rings","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.RA","authors_text":"Patrik Lundstr\\\"om, Zaqueu Cristiano","submitted_at":"2026-05-18T07:41:57Z","abstract_excerpt":"We prove that if $R$ is a ring that is object unital and strongly graded by a groupoid $\\Gamma$, and if $\\Delta$ is a wide subgroupoid of $\\Gamma$, then $R/R_\\Delta$ is separable if and only if, for each $e \\in \\Gamma_0$, there exist $f \\in [e]$ and $r \\in C_{R_0}(R_\\Lambda) := \\{ x \\in R_0 \\mid xy = yx \\text{ for all } y \\in R_\\Lambda \\}$ with ${\\rm tr}_{\\Gamma/\\Delta}^f(r) = 1_{R_f}$. Here, $\\Gamma_0$ denotes the set of objects of $\\Gamma$, $[e]$ the connected component of $\\Gamma_0$ containing $e$, $\\Lambda$ the isotropy groupoid of $\\Delta$, and ${\\rm tr}_{\\Gamma/\\Delta}^f$ the relative tr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.17987","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/2605.17987/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T23:33:35.556830Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"de728dba4080e730434e2397c11e2370d5484a0bdeeb89cdbbd017e97bbd4d88"},"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"}