{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:VIP3MNGR36RFQAXGARYSBWTXXO","short_pith_number":"pith:VIP3MNGR","canonical_record":{"source":{"id":"1208.1008","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-08-05T13:27:45Z","cross_cats_sorted":[],"title_canon_sha256":"21af4bab66ec5ba2854cf6c5f50744b3c9348d803859a821de03296e7cc3832b","abstract_canon_sha256":"b873e25faa2f295d718f98be3e230fba8f07032983f973789a9412a459eccaab"},"schema_version":"1.0"},"canonical_sha256":"aa1fb634d1dfa25802e6047120da77bb918d61e738d32c0cd6ef5e2d0374fb2c","source":{"kind":"arxiv","id":"1208.1008","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.1008","created_at":"2026-05-18T03:49:20Z"},{"alias_kind":"arxiv_version","alias_value":"1208.1008v1","created_at":"2026-05-18T03:49:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.1008","created_at":"2026-05-18T03:49:20Z"},{"alias_kind":"pith_short_12","alias_value":"VIP3MNGR36RF","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"VIP3MNGR36RFQAXG","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"VIP3MNGR","created_at":"2026-05-18T12:27:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:VIP3MNGR36RFQAXGARYSBWTXXO","target":"record","payload":{"canonical_record":{"source":{"id":"1208.1008","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-08-05T13:27:45Z","cross_cats_sorted":[],"title_canon_sha256":"21af4bab66ec5ba2854cf6c5f50744b3c9348d803859a821de03296e7cc3832b","abstract_canon_sha256":"b873e25faa2f295d718f98be3e230fba8f07032983f973789a9412a459eccaab"},"schema_version":"1.0"},"canonical_sha256":"aa1fb634d1dfa25802e6047120da77bb918d61e738d32c0cd6ef5e2d0374fb2c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:20.810347Z","signature_b64":"KsO5U109kz7tFRz+rsPMqSVFurq20pp2hXIBAhKz8mSQ/zytWa+G1RT6XZgPswF+Cz8woQ29ETPweblEz7nvBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa1fb634d1dfa25802e6047120da77bb918d61e738d32c0cd6ef5e2d0374fb2c","last_reissued_at":"2026-05-18T03:49:20.809908Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:20.809908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1208.1008","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:49:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hy1FCwF/h+5AIiz1A9/DBBVqdzBP6Qo4uwUV0ZBx749l4waYqFNFebOecDxqLUcJtHmZsqG2F65bbws2OwgEDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T01:58:49.432661Z"},"content_sha256":"899749dd9e7a34c16b3a9252f13f92c4316d26c25461432722f7b5543fc1f8b6","schema_version":"1.0","event_id":"sha256:899749dd9e7a34c16b3a9252f13f92c4316d26c25461432722f7b5543fc1f8b6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:VIP3MNGR36RFQAXGARYSBWTXXO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Groups with many finitary cohomology functors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"P. H. Kropholler","submitted_at":"2012-08-05T13:27:45Z","abstract_excerpt":"For a group G we consider the set of natural numbers n for which the nth cohomology functor of G commutes with filtered colimit systems of coefficient modules. We find that for the large class of hierarchically decomposable groups there is a dichotomy: this set of natural numbers is either finite or cofinite. Moreover any finite or cofinite set is shown to arise in suitable examples. A survey of recent literature on this topic, especially the work of Martin Hamilton, is included."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.1008","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:49:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WDN7xgOdzXQiF2tNRvqKUajNQlSLyCYn/4U9rDM+RZtcXKkLBGt8eAVxHegHuo+QqOuMB0VCNffLcgxorhZDDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T01:58:49.433012Z"},"content_sha256":"02aa2505a60b43097d8491021c063b5dd3c828480890e8c067954e033e2990f1","schema_version":"1.0","event_id":"sha256:02aa2505a60b43097d8491021c063b5dd3c828480890e8c067954e033e2990f1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VIP3MNGR36RFQAXGARYSBWTXXO/bundle.json","state_url":"https://pith.science/pith/VIP3MNGR36RFQAXGARYSBWTXXO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VIP3MNGR36RFQAXGARYSBWTXXO/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":"2026-06-02T01:58:49Z","links":{"resolver":"https://pith.science/pith/VIP3MNGR36RFQAXGARYSBWTXXO","bundle":"https://pith.science/pith/VIP3MNGR36RFQAXGARYSBWTXXO/bundle.json","state":"https://pith.science/pith/VIP3MNGR36RFQAXGARYSBWTXXO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VIP3MNGR36RFQAXGARYSBWTXXO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:VIP3MNGR36RFQAXGARYSBWTXXO","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":"b873e25faa2f295d718f98be3e230fba8f07032983f973789a9412a459eccaab","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-08-05T13:27:45Z","title_canon_sha256":"21af4bab66ec5ba2854cf6c5f50744b3c9348d803859a821de03296e7cc3832b"},"schema_version":"1.0","source":{"id":"1208.1008","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.1008","created_at":"2026-05-18T03:49:20Z"},{"alias_kind":"arxiv_version","alias_value":"1208.1008v1","created_at":"2026-05-18T03:49:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.1008","created_at":"2026-05-18T03:49:20Z"},{"alias_kind":"pith_short_12","alias_value":"VIP3MNGR36RF","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"VIP3MNGR36RFQAXG","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"VIP3MNGR","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:02aa2505a60b43097d8491021c063b5dd3c828480890e8c067954e033e2990f1","target":"graph","created_at":"2026-05-18T03:49:20Z","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":"For a group G we consider the set of natural numbers n for which the nth cohomology functor of G commutes with filtered colimit systems of coefficient modules. We find that for the large class of hierarchically decomposable groups there is a dichotomy: this set of natural numbers is either finite or cofinite. Moreover any finite or cofinite set is shown to arise in suitable examples. A survey of recent literature on this topic, especially the work of Martin Hamilton, is included.","authors_text":"P. H. Kropholler","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-08-05T13:27:45Z","title":"Groups with many finitary cohomology functors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.1008","kind":"arxiv","version":1},"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":"sha256:899749dd9e7a34c16b3a9252f13f92c4316d26c25461432722f7b5543fc1f8b6","target":"record","created_at":"2026-05-18T03:49:20Z","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":"b873e25faa2f295d718f98be3e230fba8f07032983f973789a9412a459eccaab","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-08-05T13:27:45Z","title_canon_sha256":"21af4bab66ec5ba2854cf6c5f50744b3c9348d803859a821de03296e7cc3832b"},"schema_version":"1.0","source":{"id":"1208.1008","kind":"arxiv","version":1}},"canonical_sha256":"aa1fb634d1dfa25802e6047120da77bb918d61e738d32c0cd6ef5e2d0374fb2c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aa1fb634d1dfa25802e6047120da77bb918d61e738d32c0cd6ef5e2d0374fb2c","first_computed_at":"2026-05-18T03:49:20.809908Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:49:20.809908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KsO5U109kz7tFRz+rsPMqSVFurq20pp2hXIBAhKz8mSQ/zytWa+G1RT6XZgPswF+Cz8woQ29ETPweblEz7nvBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:49:20.810347Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.1008","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:899749dd9e7a34c16b3a9252f13f92c4316d26c25461432722f7b5543fc1f8b6","sha256:02aa2505a60b43097d8491021c063b5dd3c828480890e8c067954e033e2990f1"],"state_sha256":"ad2644fb49ba3b08ef7e6a12bcbc970eb9a9288180b00a92f162c775f25bf885"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ku+qbdBRuzHa6L5eljQYSM1VvhJH3Ugwc0jIuYoCqDeWbjL5/vLnYHFTETWTG+rS3G36iIAta4G1oBHq0LR8Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T01:58:49.434962Z","bundle_sha256":"4f738d6f157c63ea42aa3b3c844342f365a9157a213fc57a8eaefbf6400fa18d"}}