{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:J3ZMV6FHR7I6MRSTUBTB2H6FKX","short_pith_number":"pith:J3ZMV6FH","schema_version":"1.0","canonical_sha256":"4ef2caf8a78fd1e64653a0661d1fc555c53d794b6293cab1dc7bcc704358c322","source":{"kind":"arxiv","id":"1107.3056","version":1},"attestation_state":"computed","paper":{"title":"Multiple Commutator Formulas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"R. Hazrat, Z. Zhang","submitted_at":"2011-07-15T12:29:25Z","abstract_excerpt":"Let A be a quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. Let I_i, i=0,...,m, be two-sided ideals of A, \\GL_n(A,I_i) the principal congruence subgroup of level I_i in GL_n(A) and E_n(A,I_i) be the relative elementary subgroup of level I_i. We prove a multiple commutator formula\n  [E_n(A,I_0),\\GL_n(A,I_1),& \\GL_n(A, I_2),..., \\GL_n(A, I_m)] = [E_n(A,I_0),E_n(A,I_1),E_n(A, I_2),..., E_n(A, I_m)],\n  which is a broad generalization of the standard commutator formulas."},"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":"1107.3056","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-07-15T12:29:25Z","cross_cats_sorted":[],"title_canon_sha256":"1125a325a8c18ee3f5a31c91e7965e65f2f3815ee128e012f6585998446bf75a","abstract_canon_sha256":"7ed2f0436a362db71cb58310db1b50ff69367cc441ce7d00c01ed2b444449f88"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:14.838298Z","signature_b64":"MvnnSiOTt2x5Rexo2NPH/0qY3OZGIwWsdERXhIfeiocsdbEb+lDzCAhqPo55+0H38EoQ90zWbbJvxuOwq92KBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ef2caf8a78fd1e64653a0661d1fc555c53d794b6293cab1dc7bcc704358c322","last_reissued_at":"2026-05-18T04:18:14.837676Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:14.837676Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiple Commutator Formulas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"R. Hazrat, Z. Zhang","submitted_at":"2011-07-15T12:29:25Z","abstract_excerpt":"Let A be a quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. Let I_i, i=0,...,m, be two-sided ideals of A, \\GL_n(A,I_i) the principal congruence subgroup of level I_i in GL_n(A) and E_n(A,I_i) be the relative elementary subgroup of level I_i. We prove a multiple commutator formula\n  [E_n(A,I_0),\\GL_n(A,I_1),& \\GL_n(A, I_2),..., \\GL_n(A, I_m)] = [E_n(A,I_0),E_n(A,I_1),E_n(A, I_2),..., E_n(A, I_m)],\n  which is a broad generalization of the standard commutator formulas."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3056","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1107.3056","created_at":"2026-05-18T04:18:14.837775+00:00"},{"alias_kind":"arxiv_version","alias_value":"1107.3056v1","created_at":"2026-05-18T04:18:14.837775+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.3056","created_at":"2026-05-18T04:18:14.837775+00:00"},{"alias_kind":"pith_short_12","alias_value":"J3ZMV6FHR7I6","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_16","alias_value":"J3ZMV6FHR7I6MRST","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_8","alias_value":"J3ZMV6FH","created_at":"2026-05-18T12:26:32.869790+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/J3ZMV6FHR7I6MRSTUBTB2H6FKX","json":"https://pith.science/pith/J3ZMV6FHR7I6MRSTUBTB2H6FKX.json","graph_json":"https://pith.science/api/pith-number/J3ZMV6FHR7I6MRSTUBTB2H6FKX/graph.json","events_json":"https://pith.science/api/pith-number/J3ZMV6FHR7I6MRSTUBTB2H6FKX/events.json","paper":"https://pith.science/paper/J3ZMV6FH"},"agent_actions":{"view_html":"https://pith.science/pith/J3ZMV6FHR7I6MRSTUBTB2H6FKX","download_json":"https://pith.science/pith/J3ZMV6FHR7I6MRSTUBTB2H6FKX.json","view_paper":"https://pith.science/paper/J3ZMV6FH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1107.3056&json=true","fetch_graph":"https://pith.science/api/pith-number/J3ZMV6FHR7I6MRSTUBTB2H6FKX/graph.json","fetch_events":"https://pith.science/api/pith-number/J3ZMV6FHR7I6MRSTUBTB2H6FKX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J3ZMV6FHR7I6MRSTUBTB2H6FKX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J3ZMV6FHR7I6MRSTUBTB2H6FKX/action/storage_attestation","attest_author":"https://pith.science/pith/J3ZMV6FHR7I6MRSTUBTB2H6FKX/action/author_attestation","sign_citation":"https://pith.science/pith/J3ZMV6FHR7I6MRSTUBTB2H6FKX/action/citation_signature","submit_replication":"https://pith.science/pith/J3ZMV6FHR7I6MRSTUBTB2H6FKX/action/replication_record"}},"created_at":"2026-05-18T04:18:14.837775+00:00","updated_at":"2026-05-18T04:18:14.837775+00:00"}