{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:UWZLM2XOQADVR4ZM3FJ4VKGS66","short_pith_number":"pith:UWZLM2XO","canonical_record":{"source":{"id":"1004.5475","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-04-30T09:04:51Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"235a2f84bc890d6f480fed3d14474f4fcfd6ef2b46e1f4b2b557325877857e74","abstract_canon_sha256":"235e01e77a8c546c1b313188d6f920fd2c30a3d4ac94e54f219a5f58b13c1312"},"schema_version":"1.0"},"canonical_sha256":"a5b2b66aee800758f32cd953caa8d2f7b650b3bc972d03d269b469bc446a99c7","source":{"kind":"arxiv","id":"1004.5475","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.5475","created_at":"2026-05-18T02:24:10Z"},{"alias_kind":"arxiv_version","alias_value":"1004.5475v3","created_at":"2026-05-18T02:24:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.5475","created_at":"2026-05-18T02:24:10Z"},{"alias_kind":"pith_short_12","alias_value":"UWZLM2XOQADV","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"UWZLM2XOQADVR4ZM","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"UWZLM2XO","created_at":"2026-05-18T12:26:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:UWZLM2XOQADVR4ZM3FJ4VKGS66","target":"record","payload":{"canonical_record":{"source":{"id":"1004.5475","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-04-30T09:04:51Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"235a2f84bc890d6f480fed3d14474f4fcfd6ef2b46e1f4b2b557325877857e74","abstract_canon_sha256":"235e01e77a8c546c1b313188d6f920fd2c30a3d4ac94e54f219a5f58b13c1312"},"schema_version":"1.0"},"canonical_sha256":"a5b2b66aee800758f32cd953caa8d2f7b650b3bc972d03d269b469bc446a99c7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:10.010618Z","signature_b64":"dG4ddf5sjlsYMg9ZbV/XlcsGThWmGC8652o+yVonFbTP7wyfyBLn5tp5k6bGpyAQ6cl1nPdGNOEg9or3cOmyAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5b2b66aee800758f32cd953caa8d2f7b650b3bc972d03d269b469bc446a99c7","last_reissued_at":"2026-05-18T02:24:10.010045Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:10.010045Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1004.5475","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:24:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n1vluXRGRp/g9mg6DJ0bpLmX+jl2kc6ovR9CsmCLfIBGGzXdJsoZ496p8gZjB7ANOUj6w8NQ23RyKkprI8soCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:38:01.436489Z"},"content_sha256":"054175b4f4d723117a9b3c876594f4fd367b269a879a1392cfc3adee45b29260","schema_version":"1.0","event_id":"sha256:054175b4f4d723117a9b3c876594f4fd367b269a879a1392cfc3adee45b29260"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:UWZLM2XOQADVR4ZM3FJ4VKGS66","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Maximal rigid subcategories in 2-Calabi-Yau triangulated categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.RT","authors_text":"Bin Zhu, Yu Zhou","submitted_at":"2010-04-30T09:04:51Z","abstract_excerpt":"We study the maximal rigid subcategories in $2-$CY triangulated categories and their endomorphism algebras. Cluster tilting subcategories are obviously maximal rigid; we prove that the converse is true if the $2-$CY triangulated categories admit a cluster tilting subcategory. As a generalization of a result of [KR], we prove that any maximal rigid subcategory is Gorenstein with Gorenstein dimension at most 1. Similar as cluster tilting subcategory, one can mutate maximal rigid subcategories at any indecomposable object. If two maximal rigid objects are reachable via mutations, then their endom"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.5475","kind":"arxiv","version":3},"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":"2026-05-18T02:24:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uGq0r68FEruTcB6YezDAkcBzpxZwaaqmc2ULyTpUbjAtOwO+4bOPkyBn2bsIGcYTWS8AWKBVCQnFtqp6lPKMBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:38:01.437329Z"},"content_sha256":"58ffc8366c07057fb88d482113b6cf8890b2b1c770c68531a0c9e6f21107d625","schema_version":"1.0","event_id":"sha256:58ffc8366c07057fb88d482113b6cf8890b2b1c770c68531a0c9e6f21107d625"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UWZLM2XOQADVR4ZM3FJ4VKGS66/bundle.json","state_url":"https://pith.science/pith/UWZLM2XOQADVR4ZM3FJ4VKGS66/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UWZLM2XOQADVR4ZM3FJ4VKGS66/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":"2026-06-11T06:38:01Z","links":{"resolver":"https://pith.science/pith/UWZLM2XOQADVR4ZM3FJ4VKGS66","bundle":"https://pith.science/pith/UWZLM2XOQADVR4ZM3FJ4VKGS66/bundle.json","state":"https://pith.science/pith/UWZLM2XOQADVR4ZM3FJ4VKGS66/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UWZLM2XOQADVR4ZM3FJ4VKGS66/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:UWZLM2XOQADVR4ZM3FJ4VKGS66","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":"235e01e77a8c546c1b313188d6f920fd2c30a3d4ac94e54f219a5f58b13c1312","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-04-30T09:04:51Z","title_canon_sha256":"235a2f84bc890d6f480fed3d14474f4fcfd6ef2b46e1f4b2b557325877857e74"},"schema_version":"1.0","source":{"id":"1004.5475","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.5475","created_at":"2026-05-18T02:24:10Z"},{"alias_kind":"arxiv_version","alias_value":"1004.5475v3","created_at":"2026-05-18T02:24:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.5475","created_at":"2026-05-18T02:24:10Z"},{"alias_kind":"pith_short_12","alias_value":"UWZLM2XOQADV","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"UWZLM2XOQADVR4ZM","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"UWZLM2XO","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:58ffc8366c07057fb88d482113b6cf8890b2b1c770c68531a0c9e6f21107d625","target":"graph","created_at":"2026-05-18T02:24:10Z","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 study the maximal rigid subcategories in $2-$CY triangulated categories and their endomorphism algebras. Cluster tilting subcategories are obviously maximal rigid; we prove that the converse is true if the $2-$CY triangulated categories admit a cluster tilting subcategory. As a generalization of a result of [KR], we prove that any maximal rigid subcategory is Gorenstein with Gorenstein dimension at most 1. Similar as cluster tilting subcategory, one can mutate maximal rigid subcategories at any indecomposable object. If two maximal rigid objects are reachable via mutations, then their endom","authors_text":"Bin Zhu, Yu Zhou","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-04-30T09:04:51Z","title":"Maximal rigid subcategories in 2-Calabi-Yau triangulated categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.5475","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":"sha256:054175b4f4d723117a9b3c876594f4fd367b269a879a1392cfc3adee45b29260","target":"record","created_at":"2026-05-18T02:24:10Z","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":"235e01e77a8c546c1b313188d6f920fd2c30a3d4ac94e54f219a5f58b13c1312","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-04-30T09:04:51Z","title_canon_sha256":"235a2f84bc890d6f480fed3d14474f4fcfd6ef2b46e1f4b2b557325877857e74"},"schema_version":"1.0","source":{"id":"1004.5475","kind":"arxiv","version":3}},"canonical_sha256":"a5b2b66aee800758f32cd953caa8d2f7b650b3bc972d03d269b469bc446a99c7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a5b2b66aee800758f32cd953caa8d2f7b650b3bc972d03d269b469bc446a99c7","first_computed_at":"2026-05-18T02:24:10.010045Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:24:10.010045Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dG4ddf5sjlsYMg9ZbV/XlcsGThWmGC8652o+yVonFbTP7wyfyBLn5tp5k6bGpyAQ6cl1nPdGNOEg9or3cOmyAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:24:10.010618Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.5475","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:054175b4f4d723117a9b3c876594f4fd367b269a879a1392cfc3adee45b29260","sha256:58ffc8366c07057fb88d482113b6cf8890b2b1c770c68531a0c9e6f21107d625"],"state_sha256":"9ebdf6543f44319414514a9ae3402b96b74e112e8f7ff91ffe85de943d9112a9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mG0GDzpeqvxdXd8dC9yqp2SyYf5QwtPzwz8sV+25D3d9nJYOImKNtPos2joXPOu/XJm/RMVqKK6tKtq9bFt+BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T06:38:01.441023Z","bundle_sha256":"56831ea98f58e5b4e7844fe90d2dfbe4f4b8c3a66946a2a4ef866b549643f9bd"}}