{"paper":{"title":"Galois closure data for extensions of rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.AC","authors_text":"Owen Biesel","submitted_at":"2016-01-27T14:32:45Z","abstract_excerpt":"To generalize the notion of Galois closure for separable field extensions, we devise a notion of $G$-closure for algebras of commutative rings $R\\to A$, where $A$ is locally free of rank $n$ as an $R$-module and $G$ is a subgroup of $\\mathrm{S}_n$. A $G$-closure datum for $A$ over $R$ is an $R$-algebra homomorphism $\\varphi: (A^{\\otimes n})^{G}\\to R$ satisfying certain properties, and we associate to a closure datum $\\varphi$ a closure algebra $A^{\\otimes n}\\otimes_{(A^{\\otimes n})^G} R$. This construction reproduces the normal closure of a finite separable field extension if $G$ is the corres"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.07389","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"}