{"paper":{"title":"Homology pro stability for Tor-unital pro rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.KT","authors_text":"Ryomei Iwasa","submitted_at":"2016-10-17T08:05:43Z","abstract_excerpt":"Let $\\{A_m\\}$ be a pro system of associative commutative, not necessarily unital, rings. Assume that the pro systems $\\{\\mathrm{Tor}^{\\mathbb{Z}\\ltimes A_m}_i(\\mathbb{Z},\\mathbb{Z})\\}_m$ vanish for all $i>0$. Then we prove that the sequence \\[\n  \\{H_l(\\mathrm{GL}_n(A_m))\\}_m \\to \\{H_l(\\mathrm{GL}_{n+1}(A_m))\\}_m \\to \\{H_l(\\mathrm{GL}_{n+2}(A_m)\\}_m \\to \\cdots \\] stabilizes up to pro isomorphisms for $n$ large enough than $l$ and the stable range of $A_m$'s."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.04998","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"}