{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:2R5D4UZT6BG5GGLWSQGS2ERB3N","short_pith_number":"pith:2R5D4UZT","schema_version":"1.0","canonical_sha256":"d47a3e5333f04dd31976940d2d1221db62ea7b7cc9ccd7712ddc2f2a14662685","source":{"kind":"arxiv","id":"1206.4748","version":2},"attestation_state":"computed","paper":{"title":"A generalized Koszul theory and its relation to the classical theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Liping Li","submitted_at":"2012-06-21T00:31:01Z","abstract_excerpt":"Let $A = \\bigoplus_{i \\geqslant 0} A_i$ be a graded locally finite $k$-algebra such that $A_0$ is an arbitrary finite-dimensional algebra satisfying a certain splitting condition. In this paper we develop a generalized Koszul theory preserving many classical results. Moreover, we define a quotient graded algebra $\\bar{A} = \\bigoplus_{i \\geqslant 0} \\bar{A}_i$ and show that $A$ is a generalized Koszul algebra if and only if $\\bar{A}$ is a classical Koszul algebra and a projective $A_0$-module. We also describe an application of this theory to the extension algebras of standard modules of standa"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1206.4748","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-06-21T00:31:01Z","cross_cats_sorted":[],"title_canon_sha256":"f13e2ebaaa9b0a2c18f2ebf55112d657b535d90c79ecf70d699fa3ae3b75af4d","abstract_canon_sha256":"b79cc0fa94d450b5c781331d1e8b2bda294a16be8cb3cbbf043c7c10875a8ad5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:26.388311Z","signature_b64":"oo94x1/33Wu88UlUEup4LUSDYtxSAuUmLIq17VM3YKopLS16LytE8njp93kS7aFgBZ2oVxhAqPUCh2p+g+ROAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d47a3e5333f04dd31976940d2d1221db62ea7b7cc9ccd7712ddc2f2a14662685","last_reissued_at":"2026-05-18T03:05:26.387755Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:26.387755Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A generalized Koszul theory and its relation to the classical theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Liping Li","submitted_at":"2012-06-21T00:31:01Z","abstract_excerpt":"Let $A = \\bigoplus_{i \\geqslant 0} A_i$ be a graded locally finite $k$-algebra such that $A_0$ is an arbitrary finite-dimensional algebra satisfying a certain splitting condition. In this paper we develop a generalized Koszul theory preserving many classical results. Moreover, we define a quotient graded algebra $\\bar{A} = \\bigoplus_{i \\geqslant 0} \\bar{A}_i$ and show that $A$ is a generalized Koszul algebra if and only if $\\bar{A}$ is a classical Koszul algebra and a projective $A_0$-module. We also describe an application of this theory to the extension algebras of standard modules of standa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4748","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1206.4748","created_at":"2026-05-18T03:05:26.387835+00:00"},{"alias_kind":"arxiv_version","alias_value":"1206.4748v2","created_at":"2026-05-18T03:05:26.387835+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.4748","created_at":"2026-05-18T03:05:26.387835+00:00"},{"alias_kind":"pith_short_12","alias_value":"2R5D4UZT6BG5","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_16","alias_value":"2R5D4UZT6BG5GGLW","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_8","alias_value":"2R5D4UZT","created_at":"2026-05-18T12:26:50.516681+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2R5D4UZT6BG5GGLWSQGS2ERB3N","json":"https://pith.science/pith/2R5D4UZT6BG5GGLWSQGS2ERB3N.json","graph_json":"https://pith.science/api/pith-number/2R5D4UZT6BG5GGLWSQGS2ERB3N/graph.json","events_json":"https://pith.science/api/pith-number/2R5D4UZT6BG5GGLWSQGS2ERB3N/events.json","paper":"https://pith.science/paper/2R5D4UZT"},"agent_actions":{"view_html":"https://pith.science/pith/2R5D4UZT6BG5GGLWSQGS2ERB3N","download_json":"https://pith.science/pith/2R5D4UZT6BG5GGLWSQGS2ERB3N.json","view_paper":"https://pith.science/paper/2R5D4UZT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1206.4748&json=true","fetch_graph":"https://pith.science/api/pith-number/2R5D4UZT6BG5GGLWSQGS2ERB3N/graph.json","fetch_events":"https://pith.science/api/pith-number/2R5D4UZT6BG5GGLWSQGS2ERB3N/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2R5D4UZT6BG5GGLWSQGS2ERB3N/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2R5D4UZT6BG5GGLWSQGS2ERB3N/action/storage_attestation","attest_author":"https://pith.science/pith/2R5D4UZT6BG5GGLWSQGS2ERB3N/action/author_attestation","sign_citation":"https://pith.science/pith/2R5D4UZT6BG5GGLWSQGS2ERB3N/action/citation_signature","submit_replication":"https://pith.science/pith/2R5D4UZT6BG5GGLWSQGS2ERB3N/action/replication_record"}},"created_at":"2026-05-18T03:05:26.387835+00:00","updated_at":"2026-05-18T03:05:26.387835+00:00"}