{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:6FPZTYEQ35JHWBFMZCR62DLLL6","short_pith_number":"pith:6FPZTYEQ","canonical_record":{"source":{"id":"0912.0291","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-12-01T22:46:53Z","cross_cats_sorted":[],"title_canon_sha256":"e22e672e0ad019414643df3c40964910c3ef37e95da287fbdb348a96ac171033","abstract_canon_sha256":"e6465964225cf4afdf22d3273b6e958a71509e5f8d1ff424efbb0c310bd80570"},"schema_version":"1.0"},"canonical_sha256":"f15f99e090df527b04acc8a3ed0d6b5f88c09d4ddc78f38eb45045ffb9de171d","source":{"kind":"arxiv","id":"0912.0291","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.0291","created_at":"2026-05-18T04:20:38Z"},{"alias_kind":"arxiv_version","alias_value":"0912.0291v4","created_at":"2026-05-18T04:20:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.0291","created_at":"2026-05-18T04:20:38Z"},{"alias_kind":"pith_short_12","alias_value":"6FPZTYEQ35JH","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"6FPZTYEQ35JHWBFM","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"6FPZTYEQ","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:6FPZTYEQ35JHWBFMZCR62DLLL6","target":"record","payload":{"canonical_record":{"source":{"id":"0912.0291","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-12-01T22:46:53Z","cross_cats_sorted":[],"title_canon_sha256":"e22e672e0ad019414643df3c40964910c3ef37e95da287fbdb348a96ac171033","abstract_canon_sha256":"e6465964225cf4afdf22d3273b6e958a71509e5f8d1ff424efbb0c310bd80570"},"schema_version":"1.0"},"canonical_sha256":"f15f99e090df527b04acc8a3ed0d6b5f88c09d4ddc78f38eb45045ffb9de171d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:20:38.820980Z","signature_b64":"9QQK+YbjDxzUJzLDbn0YrCGeZ+9AqWorBtuSSWJr+6eURxa6L8/vqGDkzplaTYx7UkxavPYdfeowuNwYcUVjBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f15f99e090df527b04acc8a3ed0d6b5f88c09d4ddc78f38eb45045ffb9de171d","last_reissued_at":"2026-05-18T04:20:38.820169Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:20:38.820169Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0912.0291","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:20:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c8vgTFTPNF+zq82DFSiFXqnUgpTsV6fw3/TcuHh81F8sGLGw7kryYkA6iDNJYBADUS7DiaTp+QHz6xgPSmTwDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T00:41:35.759060Z"},"content_sha256":"b4bb5d81c7b6d19c23f04d0dbfa92b1c2512a1b909074bd5c422a6603bd1ee9c","schema_version":"1.0","event_id":"sha256:b4bb5d81c7b6d19c23f04d0dbfa92b1c2512a1b909074bd5c422a6603bd1ee9c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:6FPZTYEQ35JHWBFMZCR62DLLL6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Galois Theory of Hopf Galois Extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Dorota Marciniak, Marcin Szamotulski","submitted_at":"2009-12-01T22:46:53Z","abstract_excerpt":"We introduce Galois Theory for Hopf-Galois Extensions proving existence of a Galois connection between subalgebras of an H-comodule algebra and generalised quotients of the Hopf algebra H. Moreover, we show that these quotients Q which define Q-Galois extension are the closed elements of our Galois connection. We generalise important results of Hopf--Galois Theory of M. Masuoka and H.-J. Schneider by showing that there is a bijective correspondence between right ideals coideals and right coideal subalgebras of any finite dimensional Hopf algebra and we reformulate the still open problem in the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.0291","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:20:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mzidiRA469wu74CXPchKJq+rtWRggkiOlmC4fbvyFIsfGZhfKqMTc7W6IyJ5kex7xpaZL/cadvQxiwfGC34yAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T00:41:35.759750Z"},"content_sha256":"75d31f51e2f92958ab91ec56095715e530595938ba24b9041bd10dfdf5c08ee7","schema_version":"1.0","event_id":"sha256:75d31f51e2f92958ab91ec56095715e530595938ba24b9041bd10dfdf5c08ee7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6FPZTYEQ35JHWBFMZCR62DLLL6/bundle.json","state_url":"https://pith.science/pith/6FPZTYEQ35JHWBFMZCR62DLLL6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6FPZTYEQ35JHWBFMZCR62DLLL6/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":"2026-06-11T00:41:35Z","links":{"resolver":"https://pith.science/pith/6FPZTYEQ35JHWBFMZCR62DLLL6","bundle":"https://pith.science/pith/6FPZTYEQ35JHWBFMZCR62DLLL6/bundle.json","state":"https://pith.science/pith/6FPZTYEQ35JHWBFMZCR62DLLL6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6FPZTYEQ35JHWBFMZCR62DLLL6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:6FPZTYEQ35JHWBFMZCR62DLLL6","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":"e6465964225cf4afdf22d3273b6e958a71509e5f8d1ff424efbb0c310bd80570","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-12-01T22:46:53Z","title_canon_sha256":"e22e672e0ad019414643df3c40964910c3ef37e95da287fbdb348a96ac171033"},"schema_version":"1.0","source":{"id":"0912.0291","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.0291","created_at":"2026-05-18T04:20:38Z"},{"alias_kind":"arxiv_version","alias_value":"0912.0291v4","created_at":"2026-05-18T04:20:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.0291","created_at":"2026-05-18T04:20:38Z"},{"alias_kind":"pith_short_12","alias_value":"6FPZTYEQ35JH","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"6FPZTYEQ35JHWBFM","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"6FPZTYEQ","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:75d31f51e2f92958ab91ec56095715e530595938ba24b9041bd10dfdf5c08ee7","target":"graph","created_at":"2026-05-18T04:20:38Z","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 introduce Galois Theory for Hopf-Galois Extensions proving existence of a Galois connection between subalgebras of an H-comodule algebra and generalised quotients of the Hopf algebra H. Moreover, we show that these quotients Q which define Q-Galois extension are the closed elements of our Galois connection. We generalise important results of Hopf--Galois Theory of M. Masuoka and H.-J. Schneider by showing that there is a bijective correspondence between right ideals coideals and right coideal subalgebras of any finite dimensional Hopf algebra and we reformulate the still open problem in the","authors_text":"Dorota Marciniak, Marcin Szamotulski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-12-01T22:46:53Z","title":"Galois Theory of Hopf Galois Extensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.0291","kind":"arxiv","version":4},"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:b4bb5d81c7b6d19c23f04d0dbfa92b1c2512a1b909074bd5c422a6603bd1ee9c","target":"record","created_at":"2026-05-18T04:20:38Z","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":"e6465964225cf4afdf22d3273b6e958a71509e5f8d1ff424efbb0c310bd80570","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-12-01T22:46:53Z","title_canon_sha256":"e22e672e0ad019414643df3c40964910c3ef37e95da287fbdb348a96ac171033"},"schema_version":"1.0","source":{"id":"0912.0291","kind":"arxiv","version":4}},"canonical_sha256":"f15f99e090df527b04acc8a3ed0d6b5f88c09d4ddc78f38eb45045ffb9de171d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f15f99e090df527b04acc8a3ed0d6b5f88c09d4ddc78f38eb45045ffb9de171d","first_computed_at":"2026-05-18T04:20:38.820169Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:20:38.820169Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9QQK+YbjDxzUJzLDbn0YrCGeZ+9AqWorBtuSSWJr+6eURxa6L8/vqGDkzplaTYx7UkxavPYdfeowuNwYcUVjBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:20:38.820980Z","signed_message":"canonical_sha256_bytes"},"source_id":"0912.0291","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b4bb5d81c7b6d19c23f04d0dbfa92b1c2512a1b909074bd5c422a6603bd1ee9c","sha256:75d31f51e2f92958ab91ec56095715e530595938ba24b9041bd10dfdf5c08ee7"],"state_sha256":"655b55d00ec763fdd578429783a8437e41d54cbe78c8725a68299a5c1e0ec5cf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GrGShhF40dyWTwIlCqsQ+Hf8XRelg6o2g9dLlXwz1zH+Y2eQg0xjx+8vDbS2hv7Shp9zhtX7pjwuLkiA/1+RBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T00:41:35.763855Z","bundle_sha256":"687e47b88443d3f0b89fa09e78bdd90f4e41b54eb2550097371a9c326c97dcf5"}}