{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:7EEICLLR5XXY4SS3AZLGVTJWJ6","short_pith_number":"pith:7EEICLLR","canonical_record":{"source":{"id":"1406.3731","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-06-14T13:49:48Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"24826ad3948bff1487f2352a44d3cad574441ea7dc5117422c0953e46c7f3188","abstract_canon_sha256":"2007c0c778598c2f3e9916fec14f1d292c0476bc1534ae73c3f182dd84a2f82c"},"schema_version":"1.0"},"canonical_sha256":"f908812d71edef8e4a5b06566acd364f8e7b23587a237cc6cff09725bfb9497d","source":{"kind":"arxiv","id":"1406.3731","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.3731","created_at":"2026-05-18T02:30:31Z"},{"alias_kind":"arxiv_version","alias_value":"1406.3731v3","created_at":"2026-05-18T02:30:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.3731","created_at":"2026-05-18T02:30:31Z"},{"alias_kind":"pith_short_12","alias_value":"7EEICLLR5XXY","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"7EEICLLR5XXY4SS3","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"7EEICLLR","created_at":"2026-05-18T12:28:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:7EEICLLR5XXY4SS3AZLGVTJWJ6","target":"record","payload":{"canonical_record":{"source":{"id":"1406.3731","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-06-14T13:49:48Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"24826ad3948bff1487f2352a44d3cad574441ea7dc5117422c0953e46c7f3188","abstract_canon_sha256":"2007c0c778598c2f3e9916fec14f1d292c0476bc1534ae73c3f182dd84a2f82c"},"schema_version":"1.0"},"canonical_sha256":"f908812d71edef8e4a5b06566acd364f8e7b23587a237cc6cff09725bfb9497d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:31.937703Z","signature_b64":"v2rvOoavaXgnR5qp7061N2Zl/nJMS/2vC+vsqlOBh5RrQJnX9Ha/RGCAcPbHzUu9CdMWOCQCDfIX04eKURUcBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f908812d71edef8e4a5b06566acd364f8e7b23587a237cc6cff09725bfb9497d","last_reissued_at":"2026-05-18T02:30:31.937130Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:31.937130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1406.3731","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:30:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a1IWD7e3BUve6recmkNV7XfzVJJTk1zJEjtynUk8M6uhJkimcg4lokT0uEmzvnHeyuZahz3RpXVCFuuZXGkFCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T18:55:17.053148Z"},"content_sha256":"0217e4e9f96d6c50e64a1d13fa48021bd405e5df19a02f0164184ee281ee7a07","schema_version":"1.0","event_id":"sha256:0217e4e9f96d6c50e64a1d13fa48021bd405e5df19a02f0164184ee281ee7a07"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:7EEICLLR5XXY4SS3AZLGVTJWJ6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Torsion cohomology for solvable groups of finite rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.GR","authors_text":"Karl Lorensen","submitted_at":"2014-06-14T13:49:48Z","abstract_excerpt":"We define a class $\\mathcal{U}$ of solvable groups of finite abelian section rank which includes all such groups that are virtually torsion-free as well as those that are finitely generated. Assume that $G$ is a group in $\\mathcal{U}$ and $A$ a $\\mathbb ZG$-module. If $A$ is $\\mathbb Z$-torsion-free and has finite $\\mathbb Z$-rank, we stipulate a condition on $A$ that guarantees that $H^n(G,A)$ and $H_n(G,A)$ must be finite for $n\\geq 0$. Moreover, if the underlying abelian group of $A$ is a \\v{C}ernikov group, we identify a similar condition on $A$ that ensures that $H^n(G,A)$ must be a \\v{C}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.3731","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:30:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y05LLvnbH6FJzGoXDHVuVhuBAT61oV7N3DtSjHDQZDRjg5h+nL7sHsiDW7dC6jKop4rFsEwvI8umZ3UgJQtyBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T18:55:17.053520Z"},"content_sha256":"8f2e97c379d7d2acbe684a9dfacbff7eafddec1291c7a5f3d48f4a94ad4f30e9","schema_version":"1.0","event_id":"sha256:8f2e97c379d7d2acbe684a9dfacbff7eafddec1291c7a5f3d48f4a94ad4f30e9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7EEICLLR5XXY4SS3AZLGVTJWJ6/bundle.json","state_url":"https://pith.science/pith/7EEICLLR5XXY4SS3AZLGVTJWJ6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7EEICLLR5XXY4SS3AZLGVTJWJ6/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-24T18:55:17Z","links":{"resolver":"https://pith.science/pith/7EEICLLR5XXY4SS3AZLGVTJWJ6","bundle":"https://pith.science/pith/7EEICLLR5XXY4SS3AZLGVTJWJ6/bundle.json","state":"https://pith.science/pith/7EEICLLR5XXY4SS3AZLGVTJWJ6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7EEICLLR5XXY4SS3AZLGVTJWJ6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:7EEICLLR5XXY4SS3AZLGVTJWJ6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2007c0c778598c2f3e9916fec14f1d292c0476bc1534ae73c3f182dd84a2f82c","cross_cats_sorted":["math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-06-14T13:49:48Z","title_canon_sha256":"24826ad3948bff1487f2352a44d3cad574441ea7dc5117422c0953e46c7f3188"},"schema_version":"1.0","source":{"id":"1406.3731","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.3731","created_at":"2026-05-18T02:30:31Z"},{"alias_kind":"arxiv_version","alias_value":"1406.3731v3","created_at":"2026-05-18T02:30:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.3731","created_at":"2026-05-18T02:30:31Z"},{"alias_kind":"pith_short_12","alias_value":"7EEICLLR5XXY","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"7EEICLLR5XXY4SS3","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"7EEICLLR","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:8f2e97c379d7d2acbe684a9dfacbff7eafddec1291c7a5f3d48f4a94ad4f30e9","target":"graph","created_at":"2026-05-18T02:30:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We define a class $\\mathcal{U}$ of solvable groups of finite abelian section rank which includes all such groups that are virtually torsion-free as well as those that are finitely generated. Assume that $G$ is a group in $\\mathcal{U}$ and $A$ a $\\mathbb ZG$-module. If $A$ is $\\mathbb Z$-torsion-free and has finite $\\mathbb Z$-rank, we stipulate a condition on $A$ that guarantees that $H^n(G,A)$ and $H_n(G,A)$ must be finite for $n\\geq 0$. Moreover, if the underlying abelian group of $A$ is a \\v{C}ernikov group, we identify a similar condition on $A$ that ensures that $H^n(G,A)$ must be a \\v{C}","authors_text":"Karl Lorensen","cross_cats":["math.KT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-06-14T13:49:48Z","title":"Torsion cohomology for solvable groups of finite rank"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.3731","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:0217e4e9f96d6c50e64a1d13fa48021bd405e5df19a02f0164184ee281ee7a07","target":"record","created_at":"2026-05-18T02:30:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2007c0c778598c2f3e9916fec14f1d292c0476bc1534ae73c3f182dd84a2f82c","cross_cats_sorted":["math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-06-14T13:49:48Z","title_canon_sha256":"24826ad3948bff1487f2352a44d3cad574441ea7dc5117422c0953e46c7f3188"},"schema_version":"1.0","source":{"id":"1406.3731","kind":"arxiv","version":3}},"canonical_sha256":"f908812d71edef8e4a5b06566acd364f8e7b23587a237cc6cff09725bfb9497d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f908812d71edef8e4a5b06566acd364f8e7b23587a237cc6cff09725bfb9497d","first_computed_at":"2026-05-18T02:30:31.937130Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:30:31.937130Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"v2rvOoavaXgnR5qp7061N2Zl/nJMS/2vC+vsqlOBh5RrQJnX9Ha/RGCAcPbHzUu9CdMWOCQCDfIX04eKURUcBA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:30:31.937703Z","signed_message":"canonical_sha256_bytes"},"source_id":"1406.3731","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0217e4e9f96d6c50e64a1d13fa48021bd405e5df19a02f0164184ee281ee7a07","sha256:8f2e97c379d7d2acbe684a9dfacbff7eafddec1291c7a5f3d48f4a94ad4f30e9"],"state_sha256":"e6a6d796ed1ee193990ab04e025b8430988fd48dc7369c12651ea7449d87e2cb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gnDI1HL3m4G2tEHvmzVI04+835GlFE59x3fY41KJKcJ55AVCDgWXt33JOFKrClwarOYVYIppg6s9pTRF9VmCDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T18:55:17.055428Z","bundle_sha256":"fafa3fb208ed625687a794ba03bfbbd4d8e4c003eee14237643d85c741d69ea2"}}