{"paper":{"title":"Completeness of the ring of polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Anders Thorup","submitted_at":"2013-12-19T12:28:01Z","abstract_excerpt":"Let $k$ be an uncountable field. We prove that the polynomial ring $R:=k[X_1,\\dots,X_n]$ in $n\\ge 2$ variables over $k$ is complete in its adic topology. In addition we prove that also the localization $R_{\\goth m}$ at a maximal ideal $\\goth m\\subset R$ is adically complete. The first result settles an old conjecture of C. U. Jensen, the second a conjecture of L. Gruson. Our proofs are based on a result of Gruson stating (in two variables) that $R_{\\goth m}$ is adically complete when $R=k[X_1,X_2]$ and $\\goth m=(X_1,X_2)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.5509","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"}