{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:ERWO5T26PH6Q3G3IP7BP4RMV4N","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":"ac22e2b80de4d36ddf40d7ced040283a9074d6834ba6ab7c24c2b04507f4aaa3","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.KT","submitted_at":"2015-04-14T13:54:54Z","title_canon_sha256":"7c9e800e0c156538af1ef38e4086db5cde88a9bff45359d2915858ed2c5ec069"},"schema_version":"1.0","source":{"id":"1504.03548","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.03548","created_at":"2026-05-18T01:04:41Z"},{"alias_kind":"arxiv_version","alias_value":"1504.03548v3","created_at":"2026-05-18T01:04:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.03548","created_at":"2026-05-18T01:04:41Z"},{"alias_kind":"pith_short_12","alias_value":"ERWO5T26PH6Q","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"ERWO5T26PH6Q3G3I","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"ERWO5T26","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:b1fe36d841ab6e2f36ade6ca77e1ef3f370bdd2ac8fb8df3ebffd3c7217e21b5","target":"graph","created_at":"2026-05-18T01:04:41Z","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":"Koszul pairs were introduced in [arXiv:1011.4243] as an instrument for the study of Koszul rings. In this paper, we continue the enquiry of such pairs, focusing on the description of the second component, as a follow-up of the study in [arXiv:1605.05458]. As such, we introduce Koszul corings and prove several equivalent characterizations for them. As applications, in the case of locally finite $R$-rings, we show that a graded $R$-ring is Koszul if and only if its left (or right) graded dual coring is Koszul. Finally, for finite graded posets, we obtain that the respective incidence ring is Kos","authors_text":"Adrian Manea, Drago\\c{s} \\c{S}tefan","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.KT","submitted_at":"2015-04-14T13:54:54Z","title":"Further Properties and Applications of Koszul Pairs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03548","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:c75a9217e0e585956480588589601cf503548625bc24e52dd779b1f5ccaa9c89","target":"record","created_at":"2026-05-18T01:04:41Z","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":"ac22e2b80de4d36ddf40d7ced040283a9074d6834ba6ab7c24c2b04507f4aaa3","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.KT","submitted_at":"2015-04-14T13:54:54Z","title_canon_sha256":"7c9e800e0c156538af1ef38e4086db5cde88a9bff45359d2915858ed2c5ec069"},"schema_version":"1.0","source":{"id":"1504.03548","kind":"arxiv","version":3}},"canonical_sha256":"246ceecf5e79fd0d9b687fc2fe4595e366560ebd77b22d5a62bc023daa89ddd3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"246ceecf5e79fd0d9b687fc2fe4595e366560ebd77b22d5a62bc023daa89ddd3","first_computed_at":"2026-05-18T01:04:41.749948Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:41.749948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ioYMg9St37rMRYqzX76b6sZLFCZ7WwR7OTWfmYZ3/XedIX7AtBL5pfcBOQ4cx0lhH2SENiL6PrqzAg4cEIEyAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:41.750431Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.03548","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c75a9217e0e585956480588589601cf503548625bc24e52dd779b1f5ccaa9c89","sha256:b1fe36d841ab6e2f36ade6ca77e1ef3f370bdd2ac8fb8df3ebffd3c7217e21b5"],"state_sha256":"a109da6bf514eaa6bac96bdd75e12b3d8ec8ef00571299a135c22bb9dbc02bc1"}