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Assume that 2 is invertible in R if G is of type B_n,C_n,F_4,G_2 and 3 is invertible in R if G is of type G_2. We prove that the congruence kernel of E(R), defined as the kernel of the natural homomorphism between the profinite completion of E(R) and the congruence completion of E(R) with "},"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":"1305.0057","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-04-30T23:56:45Z","cross_cats_sorted":[],"title_canon_sha256":"dded81dd0fe2a5df012166271c0a8486606e45169268418dd77b316eb09c3c06","abstract_canon_sha256":"73ff9997df96d0e025fead466eaaaabbea9c5d588d590e030aaca31cedf42331"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:42.126210Z","signature_b64":"rPDLlzSkxa0o9Ngpbwl7i+lqjbY0ANCP+tRNoUxAzWwAPFsW7xSK6iN6ikn3t3iFd/KqyvZ3pfnuG9YpwifQDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a112cab52347e48c8f2b785ff3e240342899117c858e251d71d7713d0f7019fa","last_reissued_at":"2026-05-17T23:54:42.125538Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:42.125538Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the congruence kernel of isotropic groups over rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"A. Stavrova","submitted_at":"2013-04-30T23:56:45Z","abstract_excerpt":"Let R be a connected noetherian commutative ring, and let G be a simply connected reductive group over R of isotropic rank ge 2. The elementary subgroup E(R) of G(R) is the subgroup generated by the R-points U_P^+(R) and U_P^-(R) of the unipotent radicals of two opposite parabolic subgroups P^+ and P^- of G. Assume that 2 is invertible in R if G is of type B_n,C_n,F_4,G_2 and 3 is invertible in R if G is of type G_2. We prove that the congruence kernel of E(R), defined as the kernel of the natural homomorphism between the profinite completion of E(R) and the congruence completion of E(R) with "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.0057","kind":"arxiv","version":4},"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":"1305.0057","created_at":"2026-05-17T23:54:42.125633+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.0057v4","created_at":"2026-05-17T23:54:42.125633+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.0057","created_at":"2026-05-17T23:54:42.125633+00:00"},{"alias_kind":"pith_short_12","alias_value":"UEJMVNJDI7SI","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_16","alias_value":"UEJMVNJDI7SIZDZL","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_8","alias_value":"UEJMVNJD","created_at":"2026-05-18T12:28:02.375192+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/UEJMVNJDI7SIZDZLPBP7HYSAGQ","json":"https://pith.science/pith/UEJMVNJDI7SIZDZLPBP7HYSAGQ.json","graph_json":"https://pith.science/api/pith-number/UEJMVNJDI7SIZDZLPBP7HYSAGQ/graph.json","events_json":"https://pith.science/api/pith-number/UEJMVNJDI7SIZDZLPBP7HYSAGQ/events.json","paper":"https://pith.science/paper/UEJMVNJD"},"agent_actions":{"view_html":"https://pith.science/pith/UEJMVNJDI7SIZDZLPBP7HYSAGQ","download_json":"https://pith.science/pith/UEJMVNJDI7SIZDZLPBP7HYSAGQ.json","view_paper":"https://pith.science/paper/UEJMVNJD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.0057&json=true","fetch_graph":"https://pith.science/api/pith-number/UEJMVNJDI7SIZDZLPBP7HYSAGQ/graph.json","fetch_events":"https://pith.science/api/pith-number/UEJMVNJDI7SIZDZLPBP7HYSAGQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UEJMVNJDI7SIZDZLPBP7HYSAGQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UEJMVNJDI7SIZDZLPBP7HYSAGQ/action/storage_attestation","attest_author":"https://pith.science/pith/UEJMVNJDI7SIZDZLPBP7HYSAGQ/action/author_attestation","sign_citation":"https://pith.science/pith/UEJMVNJDI7SIZDZLPBP7HYSAGQ/action/citation_signature","submit_replication":"https://pith.science/pith/UEJMVNJDI7SIZDZLPBP7HYSAGQ/action/replication_record"}},"created_at":"2026-05-17T23:54:42.125633+00:00","updated_at":"2026-05-17T23:54:42.125633+00:00"}