{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:7MCC3AZPPAGACNE23FEMCHMK6R","short_pith_number":"pith:7MCC3AZP","canonical_record":{"source":{"id":"1012.1754","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-08T13:10:29Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"654efe57997fd7c23e3610818899ceb3e4c40521aaab2e9bcf014fd04a8b1d94","abstract_canon_sha256":"78ed5294933102a8226c01d0fa63f2c595f01bf5996840c2b0edbc294081a883"},"schema_version":"1.0"},"canonical_sha256":"fb042d832f780c01349ad948c11d8af47afcec69a42e7a3056845fb6e6b4f7f4","source":{"kind":"arxiv","id":"1012.1754","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.1754","created_at":"2026-05-18T04:33:51Z"},{"alias_kind":"arxiv_version","alias_value":"1012.1754v1","created_at":"2026-05-18T04:33:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.1754","created_at":"2026-05-18T04:33:51Z"},{"alias_kind":"pith_short_12","alias_value":"7MCC3AZPPAGA","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"7MCC3AZPPAGACNE2","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"7MCC3AZP","created_at":"2026-05-18T12:26:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:7MCC3AZPPAGACNE23FEMCHMK6R","target":"record","payload":{"canonical_record":{"source":{"id":"1012.1754","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-08T13:10:29Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"654efe57997fd7c23e3610818899ceb3e4c40521aaab2e9bcf014fd04a8b1d94","abstract_canon_sha256":"78ed5294933102a8226c01d0fa63f2c595f01bf5996840c2b0edbc294081a883"},"schema_version":"1.0"},"canonical_sha256":"fb042d832f780c01349ad948c11d8af47afcec69a42e7a3056845fb6e6b4f7f4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:33:51.311077Z","signature_b64":"4vsDP4m8YlfNqsVG6HFp9rflZTLZlSVtNdpw0+zJLbiMx6QZzH1Q3cDmrqEIgYSy2kVaiyu2PbP009kprUflCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb042d832f780c01349ad948c11d8af47afcec69a42e7a3056845fb6e6b4f7f4","last_reissued_at":"2026-05-18T04:33:51.310578Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:33:51.310578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1012.1754","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:33:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sdjCa4PvLhDx0qu0eCgMx2fr7KEUKWTJyW7LMt1asFb7Lf2RTaUpQ1UUQTkDG9Gl/EnIh/Jqmp9Vq4eE58MmBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T06:44:47.719057Z"},"content_sha256":"fa58f2992c7923966eb107cfd4550c0f2ab407c79c5951a24e5c471446acf87b","schema_version":"1.0","event_id":"sha256:fa58f2992c7923966eb107cfd4550c0f2ab407c79c5951a24e5c471446acf87b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:7MCC3AZPPAGACNE23FEMCHMK6R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ideal depth of QF extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.RA","authors_text":"Lars Kadison","submitted_at":"2010-12-08T13:10:29Z","abstract_excerpt":"A minimum depth d^I(S --> R) is assigned to a ring homomorphism S --> R and a R-R-bimodule I. The recent notion of depth of a subring d(S,R)in a paper by Boltje-Danz-Kuelshammer is recovered when I = R and S --> R is the inclusion mapping. Ideal depth gives lower bounds for d(S,R) in case of group C-algebra pair or semisimple complex algebra extensions. If R | S is a QF extension of finite depth, minimum left and right even depth are shown to coincide. If R < S is moreover a Frobenius extension with R a right S-generator, its subring depth is shown to coincide with its tower depth. In the proc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.1754","kind":"arxiv","version":1},"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":"2026-05-18T04:33:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a8WUgc5cXb6TBDStHysppUyLWmhajDZTz6ze4meyllMVlvtAVK6OreWFTezwFrWMXkM/x0+2MRKBCeh9LyfoCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T06:44:47.719814Z"},"content_sha256":"c4732c8a16d6f7e5dc04e135e8ba0dbc62e2a82a18c8950f7076c7e8f97acf7e","schema_version":"1.0","event_id":"sha256:c4732c8a16d6f7e5dc04e135e8ba0dbc62e2a82a18c8950f7076c7e8f97acf7e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7MCC3AZPPAGACNE23FEMCHMK6R/bundle.json","state_url":"https://pith.science/pith/7MCC3AZPPAGACNE23FEMCHMK6R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7MCC3AZPPAGACNE23FEMCHMK6R/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":"2026-07-01T06:44:47Z","links":{"resolver":"https://pith.science/pith/7MCC3AZPPAGACNE23FEMCHMK6R","bundle":"https://pith.science/pith/7MCC3AZPPAGACNE23FEMCHMK6R/bundle.json","state":"https://pith.science/pith/7MCC3AZPPAGACNE23FEMCHMK6R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7MCC3AZPPAGACNE23FEMCHMK6R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:7MCC3AZPPAGACNE23FEMCHMK6R","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":"78ed5294933102a8226c01d0fa63f2c595f01bf5996840c2b0edbc294081a883","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-08T13:10:29Z","title_canon_sha256":"654efe57997fd7c23e3610818899ceb3e4c40521aaab2e9bcf014fd04a8b1d94"},"schema_version":"1.0","source":{"id":"1012.1754","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.1754","created_at":"2026-05-18T04:33:51Z"},{"alias_kind":"arxiv_version","alias_value":"1012.1754v1","created_at":"2026-05-18T04:33:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.1754","created_at":"2026-05-18T04:33:51Z"},{"alias_kind":"pith_short_12","alias_value":"7MCC3AZPPAGA","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"7MCC3AZPPAGACNE2","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"7MCC3AZP","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:c4732c8a16d6f7e5dc04e135e8ba0dbc62e2a82a18c8950f7076c7e8f97acf7e","target":"graph","created_at":"2026-05-18T04:33:51Z","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":"A minimum depth d^I(S --> R) is assigned to a ring homomorphism S --> R and a R-R-bimodule I. The recent notion of depth of a subring d(S,R)in a paper by Boltje-Danz-Kuelshammer is recovered when I = R and S --> R is the inclusion mapping. Ideal depth gives lower bounds for d(S,R) in case of group C-algebra pair or semisimple complex algebra extensions. If R | S is a QF extension of finite depth, minimum left and right even depth are shown to coincide. If R < S is moreover a Frobenius extension with R a right S-generator, its subring depth is shown to coincide with its tower depth. In the proc","authors_text":"Lars Kadison","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-08T13:10:29Z","title":"Ideal depth of QF extensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.1754","kind":"arxiv","version":1},"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:fa58f2992c7923966eb107cfd4550c0f2ab407c79c5951a24e5c471446acf87b","target":"record","created_at":"2026-05-18T04:33:51Z","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":"78ed5294933102a8226c01d0fa63f2c595f01bf5996840c2b0edbc294081a883","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-08T13:10:29Z","title_canon_sha256":"654efe57997fd7c23e3610818899ceb3e4c40521aaab2e9bcf014fd04a8b1d94"},"schema_version":"1.0","source":{"id":"1012.1754","kind":"arxiv","version":1}},"canonical_sha256":"fb042d832f780c01349ad948c11d8af47afcec69a42e7a3056845fb6e6b4f7f4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fb042d832f780c01349ad948c11d8af47afcec69a42e7a3056845fb6e6b4f7f4","first_computed_at":"2026-05-18T04:33:51.310578Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:33:51.310578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4vsDP4m8YlfNqsVG6HFp9rflZTLZlSVtNdpw0+zJLbiMx6QZzH1Q3cDmrqEIgYSy2kVaiyu2PbP009kprUflCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:33:51.311077Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.1754","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fa58f2992c7923966eb107cfd4550c0f2ab407c79c5951a24e5c471446acf87b","sha256:c4732c8a16d6f7e5dc04e135e8ba0dbc62e2a82a18c8950f7076c7e8f97acf7e"],"state_sha256":"7f43c8aecf0e4c4a8ad73ddbaccd05359eeb79b0c8923e55746937def96cc501"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mOaX04ZRz7uXflJxt/twkb2OLRbIMh+OBsGim+j20OFrbvrKhVToXPZpwcG8zISHACkFvz6Ag1EfkbOacu3NAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T06:44:47.722933Z","bundle_sha256":"597596458c472d2b8786f7f013ecaaa278bdb6319064327a11611507580ce879"}}