{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:X3B2KWDYXSHPWT25TBSL3I2DSV","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":"e375b25073f510844b1d368fc393d306d5b21764ae3200bacad74470bcdc4a7c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-01-05T06:52:06Z","title_canon_sha256":"90a9d63114bbfe99e64b0e58853402ba6455520f76a71deb778b06bb69f2ccfc"},"schema_version":"1.0","source":{"id":"1601.00748","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.00748","created_at":"2026-05-18T00:43:21Z"},{"alias_kind":"arxiv_version","alias_value":"1601.00748v3","created_at":"2026-05-18T00:43:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.00748","created_at":"2026-05-18T00:43:21Z"},{"alias_kind":"pith_short_12","alias_value":"X3B2KWDYXSHP","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"X3B2KWDYXSHPWT25","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"X3B2KWDY","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:6087c9e7052faa597fd3fbfb72249dbb8472d167b09d604ef7b99a7292fbc9ac","target":"graph","created_at":"2026-05-18T00:43:21Z","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 study the flatness and the projectivity of Hopf algebras, defined over a Dedekind ring, over their Hopf subalgebras. We give a criterion for the faithful flatness and use it to show the faithful flatness of an arbitrary flat Hopf algebra upon its finite normal Hopf subalgebra. For the projectivity of a projective Hopf algebras we need some finiteness condition in terms of the module of integral. In particular we show the the module of integral has rank one.","authors_text":"Nguyen Dai Duong, Nguyen Huy Hung, Phung Ho Hai","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-01-05T06:52:06Z","title":"On the flatness and the projectivity over Hopf subalgebras of Hopf algebras over discrete valuation rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.00748","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":"sha256:2355df68471e80e5514bfa1fdd71cfc4bf86bf97caf28a585a2963b2da2440e5","target":"record","created_at":"2026-05-18T00:43:21Z","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":"e375b25073f510844b1d368fc393d306d5b21764ae3200bacad74470bcdc4a7c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-01-05T06:52:06Z","title_canon_sha256":"90a9d63114bbfe99e64b0e58853402ba6455520f76a71deb778b06bb69f2ccfc"},"schema_version":"1.0","source":{"id":"1601.00748","kind":"arxiv","version":3}},"canonical_sha256":"bec3a55878bc8efb4f5d9864bda343957d4a998e5d6359d3a2186411b280b11d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bec3a55878bc8efb4f5d9864bda343957d4a998e5d6359d3a2186411b280b11d","first_computed_at":"2026-05-18T00:43:21.232498Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:43:21.232498Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XBLArCEz/E8AA3vUecMWlkP3r5++8iPhstM3uNbf7DyQOnfx63QNMP8TWdzUWpXZSOTtjzemO2bITt7JmQg8CA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:43:21.233198Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.00748","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2355df68471e80e5514bfa1fdd71cfc4bf86bf97caf28a585a2963b2da2440e5","sha256:6087c9e7052faa597fd3fbfb72249dbb8472d167b09d604ef7b99a7292fbc9ac"],"state_sha256":"6bf46ba7c3526534277d1872c2c57d6cc5ca112c61b3e394045aa29d6bd83fa8"}