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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1610.04998","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2016-10-17T08:05:43Z","cross_cats_sorted":[],"title_canon_sha256":"cc7501e8ed8972b4d362605f00e242bf4aa0d53e3c9bb586eb8bc5ca2c6bbd61","abstract_canon_sha256":"e0189f2142e4dbdab7b4a9b5ce6577952a48e90759fb8572f987f43ed532db97"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:55.561167Z","signature_b64":"hJ8CK5TJEYkXQX0BfzpDC00GHYwgkddlB856EQBXG5aYecVslL4Ol7WXCa5BfxA7HRhPhajvgg4ZbIW8C+OuBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7f83a737a5e4364c9e10d54c46e27d7c5317f08f4d9a0f1356e73d4009ac8075","last_reissued_at":"2026-05-18T00:32:55.560645Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:55.560645Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1610.04998","created_at":"2026-05-18T00:32:55.560740+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.04998v2","created_at":"2026-05-18T00:32:55.560740+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.04998","created_at":"2026-05-18T00:32:55.560740+00:00"},{"alias_kind":"pith_short_12","alias_value":"P6B2ON5F4Q3E","created_at":"2026-05-18T12:30:36.002864+00:00"},{"alias_kind":"pith_short_16","alias_value":"P6B2ON5F4Q3EZHQQ","created_at":"2026-05-18T12:30:36.002864+00:00"},{"alias_kind":"pith_short_8","alias_value":"P6B2ON5F","created_at":"2026-05-18T12:30:36.002864+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/P6B2ON5F4Q3EZHQQ2VGENYT5PR","json":"https://pith.science/pith/P6B2ON5F4Q3EZHQQ2VGENYT5PR.json","graph_json":"https://pith.science/api/pith-number/P6B2ON5F4Q3EZHQQ2VGENYT5PR/graph.json","events_json":"https://pith.science/api/pith-number/P6B2ON5F4Q3EZHQQ2VGENYT5PR/events.json","paper":"https://pith.science/paper/P6B2ON5F"},"agent_actions":{"view_html":"https://pith.science/pith/P6B2ON5F4Q3EZHQQ2VGENYT5PR","download_json":"https://pith.science/pith/P6B2ON5F4Q3EZHQQ2VGENYT5PR.json","view_paper":"https://pith.science/paper/P6B2ON5F","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.04998&json=true","fetch_graph":"https://pith.science/api/pith-number/P6B2ON5F4Q3EZHQQ2VGENYT5PR/graph.json","fetch_events":"https://pith.science/api/pith-number/P6B2ON5F4Q3EZHQQ2VGENYT5PR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P6B2ON5F4Q3EZHQQ2VGENYT5PR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P6B2ON5F4Q3EZHQQ2VGENYT5PR/action/storage_attestation","attest_author":"https://pith.science/pith/P6B2ON5F4Q3EZHQQ2VGENYT5PR/action/author_attestation","sign_citation":"https://pith.science/pith/P6B2ON5F4Q3EZHQQ2VGENYT5PR/action/citation_signature","submit_replication":"https://pith.science/pith/P6B2ON5F4Q3EZHQQ2VGENYT5PR/action/replication_record"}},"created_at":"2026-05-18T00:32:55.560740+00:00","updated_at":"2026-05-18T00:32:55.560740+00:00"}