{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:7SMVH27CVCEDPACU4JPC4GJIN4","short_pith_number":"pith:7SMVH27C","canonical_record":{"source":{"id":"1603.08133","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2016-03-26T18:29:59Z","cross_cats_sorted":["math.CT","math.RA"],"title_canon_sha256":"fb611d65339075bb5adf49c33c90260cb965e064a0aab33d1c0fcb5fc0692a44","abstract_canon_sha256":"eaaaa9967174efddde1b78d1c6d565633b7e1bacdc788da24e129c7abcf6da1c"},"schema_version":"1.0"},"canonical_sha256":"fc9953ebe2a888378054e25e2e19286f2ff1a123e2741cd774600b7304657356","source":{"kind":"arxiv","id":"1603.08133","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.08133","created_at":"2026-05-18T00:06:43Z"},{"alias_kind":"arxiv_version","alias_value":"1603.08133v4","created_at":"2026-05-18T00:06:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.08133","created_at":"2026-05-18T00:06:43Z"},{"alias_kind":"pith_short_12","alias_value":"7SMVH27CVCED","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7SMVH27CVCEDPACU","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7SMVH27C","created_at":"2026-05-18T12:30:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:7SMVH27CVCEDPACU4JPC4GJIN4","target":"record","payload":{"canonical_record":{"source":{"id":"1603.08133","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2016-03-26T18:29:59Z","cross_cats_sorted":["math.CT","math.RA"],"title_canon_sha256":"fb611d65339075bb5adf49c33c90260cb965e064a0aab33d1c0fcb5fc0692a44","abstract_canon_sha256":"eaaaa9967174efddde1b78d1c6d565633b7e1bacdc788da24e129c7abcf6da1c"},"schema_version":"1.0"},"canonical_sha256":"fc9953ebe2a888378054e25e2e19286f2ff1a123e2741cd774600b7304657356","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:43.950641Z","signature_b64":"GnJjlqqcGtXleZJZPMi8VCt+Q2ESWQeMdlOWNnIeclTLUtsUMdPcBD44GVlraDrJGdeHtbv1dJ5s+ZPfrncPAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc9953ebe2a888378054e25e2e19286f2ff1a123e2741cd774600b7304657356","last_reissued_at":"2026-05-18T00:06:43.950024Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:43.950024Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.08133","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":"2026-05-18T00:06:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wtHZs7lROcT2Anw7An9KoyMFzOU3ghYm40tViYU/ietCxtDHJdlnLXSu15Bd+UaUTjMaDRqkCZd7C0xT6E17Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T21:13:48.029102Z"},"content_sha256":"f2f7237a15a59844ff52c90fada86b55d2116b95e05bd621d4db68d4ea889522","schema_version":"1.0","event_id":"sha256:f2f7237a15a59844ff52c90fada86b55d2116b95e05bd621d4db68d4ea889522"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:7SMVH27CVCEDPACU4JPC4GJIN4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Koszul duality and the PBW theorem in symmetric tensor categories in positive characteristic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT","math.RA"],"primary_cat":"math.QA","authors_text":"Pavel Etingof","submitted_at":"2016-03-26T18:29:59Z","abstract_excerpt":"We generalize the theory of Koszul complexes and Koszul algebras (in particular, Koszul duality between symmetric and exterior algebras) to symmetric tensor categories. In characteristic $p\\ge 5$, this theory exhibits peculiar effects, not observed in the classical theory. In particular, we show that the symmetric and exterior algebras of a non-invertible simple object in the Verlinde category ${\\rm Ver}_p$ are almost Koszul (although not Koszul), and show how this gives examples of $(r,s)$-Koszul algebras with any $r,s\\ge 2$. We also develop a theory of Lie algebras in symmetric tensor catego"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.08133","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":"2026-05-18T00:06:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rTuZkHYxBhgjx9Uvoci9czPGsKMN1NMXomAPghUkS0gPrpRTo5GKn1U76qXwtRzMqOkSgkNF5t6YYHWA90frAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T21:13:48.029613Z"},"content_sha256":"fba4133f1e75f272d888b0dea8729c16f0469f430d9f9a57b695ce72348fa5af","schema_version":"1.0","event_id":"sha256:fba4133f1e75f272d888b0dea8729c16f0469f430d9f9a57b695ce72348fa5af"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7SMVH27CVCEDPACU4JPC4GJIN4/bundle.json","state_url":"https://pith.science/pith/7SMVH27CVCEDPACU4JPC4GJIN4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7SMVH27CVCEDPACU4JPC4GJIN4/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":"2026-08-08T21:13:48Z","links":{"resolver":"https://pith.science/pith/7SMVH27CVCEDPACU4JPC4GJIN4","bundle":"https://pith.science/pith/7SMVH27CVCEDPACU4JPC4GJIN4/bundle.json","state":"https://pith.science/pith/7SMVH27CVCEDPACU4JPC4GJIN4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7SMVH27CVCEDPACU4JPC4GJIN4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:7SMVH27CVCEDPACU4JPC4GJIN4","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":"eaaaa9967174efddde1b78d1c6d565633b7e1bacdc788da24e129c7abcf6da1c","cross_cats_sorted":["math.CT","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2016-03-26T18:29:59Z","title_canon_sha256":"fb611d65339075bb5adf49c33c90260cb965e064a0aab33d1c0fcb5fc0692a44"},"schema_version":"1.0","source":{"id":"1603.08133","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.08133","created_at":"2026-05-18T00:06:43Z"},{"alias_kind":"arxiv_version","alias_value":"1603.08133v4","created_at":"2026-05-18T00:06:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.08133","created_at":"2026-05-18T00:06:43Z"},{"alias_kind":"pith_short_12","alias_value":"7SMVH27CVCED","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7SMVH27CVCEDPACU","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7SMVH27C","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:fba4133f1e75f272d888b0dea8729c16f0469f430d9f9a57b695ce72348fa5af","target":"graph","created_at":"2026-05-18T00:06:43Z","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 generalize the theory of Koszul complexes and Koszul algebras (in particular, Koszul duality between symmetric and exterior algebras) to symmetric tensor categories. In characteristic $p\\ge 5$, this theory exhibits peculiar effects, not observed in the classical theory. In particular, we show that the symmetric and exterior algebras of a non-invertible simple object in the Verlinde category ${\\rm Ver}_p$ are almost Koszul (although not Koszul), and show how this gives examples of $(r,s)$-Koszul algebras with any $r,s\\ge 2$. We also develop a theory of Lie algebras in symmetric tensor catego","authors_text":"Pavel Etingof","cross_cats":["math.CT","math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2016-03-26T18:29:59Z","title":"Koszul duality and the PBW theorem in symmetric tensor categories in positive characteristic"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.08133","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:f2f7237a15a59844ff52c90fada86b55d2116b95e05bd621d4db68d4ea889522","target":"record","created_at":"2026-05-18T00:06:43Z","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":"eaaaa9967174efddde1b78d1c6d565633b7e1bacdc788da24e129c7abcf6da1c","cross_cats_sorted":["math.CT","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2016-03-26T18:29:59Z","title_canon_sha256":"fb611d65339075bb5adf49c33c90260cb965e064a0aab33d1c0fcb5fc0692a44"},"schema_version":"1.0","source":{"id":"1603.08133","kind":"arxiv","version":4}},"canonical_sha256":"fc9953ebe2a888378054e25e2e19286f2ff1a123e2741cd774600b7304657356","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fc9953ebe2a888378054e25e2e19286f2ff1a123e2741cd774600b7304657356","first_computed_at":"2026-05-18T00:06:43.950024Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:06:43.950024Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GnJjlqqcGtXleZJZPMi8VCt+Q2ESWQeMdlOWNnIeclTLUtsUMdPcBD44GVlraDrJGdeHtbv1dJ5s+ZPfrncPAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:06:43.950641Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.08133","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f2f7237a15a59844ff52c90fada86b55d2116b95e05bd621d4db68d4ea889522","sha256:fba4133f1e75f272d888b0dea8729c16f0469f430d9f9a57b695ce72348fa5af"],"state_sha256":"0e4dcf622ab84c8bf8284cf1354b785ef173240f1e67a0ea006adb373c876ada"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WLXICioygUhRf7wl1AuJtljfpiFb/DiKAA26DMdHECYJ/jNJBpcKG1XHiKEtkMRxzfK291qCLR5xfJimE3TfDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T21:13:48.034674Z","bundle_sha256":"c28021217f4ae231d983eeb2ef9f238ca1a4bda82120f6190a7273eb803676ba"}}