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We show that, if $G \\in \\widehat{{\\scriptstyle\\bf L}'{\\scriptstyle\\bf H}_R}\\mathfrak{F}$ is of type $\\operatorname{FP}_\\infty$ over $R$, then there is some $n$ such that $H_R^n(G,R [[ G ]]) \\neq 0$, and deduce that torsion-free soluble pro-$p$ groups of type $\\operatorname{FP}_\\infty$ over $\\mathbb{Z}_p$ have finite rank, thus answering the torsio"},"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":"1412.1876","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-12-05T01:46:55Z","cross_cats_sorted":[],"title_canon_sha256":"44390808f470f49b8eabc098e70267c4af741559d7eea59b2980c825cf157589","abstract_canon_sha256":"e7447b57998f40ec26b507a2314b4d1bed6de3920d14224c58ec480d08164bb8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:05.859115Z","signature_b64":"dWsgwjXZW4GdjExmCg1XYryIWya6IrTmdtuPvTmEoxNSLHacbXGIkM1ieyUiPt4eDACRCW4AiQSkNVmn31EsAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8cbe942df4602b5594c3cbe8701187d44e3816ed5b86d8185ba96ea85e859f3","last_reissued_at":"2026-05-18T02:32:05.858766Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:05.858766Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Profinite Groups of Type $\\operatorname{FP}_\\infty$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Ged Corob Cook","submitted_at":"2014-12-05T01:46:55Z","abstract_excerpt":"Suppose $R$ is a profinite ring. We construct a large class of profinite groups $\\widehat{{\\scriptstyle\\bf L}'{\\scriptstyle\\bf H}_R}\\mathfrak{F}$, including all soluble profinite groups and profinite groups of finite cohomological dimension over $R$. We show that, if $G \\in \\widehat{{\\scriptstyle\\bf L}'{\\scriptstyle\\bf H}_R}\\mathfrak{F}$ is of type $\\operatorname{FP}_\\infty$ over $R$, then there is some $n$ such that $H_R^n(G,R [[ G ]]) \\neq 0$, and deduce that torsion-free soluble pro-$p$ groups of type $\\operatorname{FP}_\\infty$ over $\\mathbb{Z}_p$ have finite rank, thus answering the torsio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.1876","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":"1412.1876","created_at":"2026-05-18T02:32:05.858817+00:00"},{"alias_kind":"arxiv_version","alias_value":"1412.1876v1","created_at":"2026-05-18T02:32:05.858817+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.1876","created_at":"2026-05-18T02:32:05.858817+00:00"},{"alias_kind":"pith_short_12","alias_value":"VDF6SQW7IYBL","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_16","alias_value":"VDF6SQW7IYBLKWKM","created_at":"2026-05-18T12:28:52.271510+00:00"},{"alias_kind":"pith_short_8","alias_value":"VDF6SQW7","created_at":"2026-05-18T12:28:52.271510+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/VDF6SQW7IYBLKWKMHS7IOAIYPV","json":"https://pith.science/pith/VDF6SQW7IYBLKWKMHS7IOAIYPV.json","graph_json":"https://pith.science/api/pith-number/VDF6SQW7IYBLKWKMHS7IOAIYPV/graph.json","events_json":"https://pith.science/api/pith-number/VDF6SQW7IYBLKWKMHS7IOAIYPV/events.json","paper":"https://pith.science/paper/VDF6SQW7"},"agent_actions":{"view_html":"https://pith.science/pith/VDF6SQW7IYBLKWKMHS7IOAIYPV","download_json":"https://pith.science/pith/VDF6SQW7IYBLKWKMHS7IOAIYPV.json","view_paper":"https://pith.science/paper/VDF6SQW7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1412.1876&json=true","fetch_graph":"https://pith.science/api/pith-number/VDF6SQW7IYBLKWKMHS7IOAIYPV/graph.json","fetch_events":"https://pith.science/api/pith-number/VDF6SQW7IYBLKWKMHS7IOAIYPV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VDF6SQW7IYBLKWKMHS7IOAIYPV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VDF6SQW7IYBLKWKMHS7IOAIYPV/action/storage_attestation","attest_author":"https://pith.science/pith/VDF6SQW7IYBLKWKMHS7IOAIYPV/action/author_attestation","sign_citation":"https://pith.science/pith/VDF6SQW7IYBLKWKMHS7IOAIYPV/action/citation_signature","submit_replication":"https://pith.science/pith/VDF6SQW7IYBLKWKMHS7IOAIYPV/action/replication_record"}},"created_at":"2026-05-18T02:32:05.858817+00:00","updated_at":"2026-05-18T02:32:05.858817+00:00"}