{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:7VGM552A2UJZBS6QF5OIT5WFYP","short_pith_number":"pith:7VGM552A","canonical_record":{"source":{"id":"1508.01290","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-08-06T06:46:52Z","cross_cats_sorted":[],"title_canon_sha256":"d1221c14fa3f5fdfabdb78259321b137df405ffbfa1ebc0c42becb9fd3b0cd2e","abstract_canon_sha256":"448d76dc8b8f89a710d999a63460b42b9b262887ab47987a8d5c30bcdd7fadba"},"schema_version":"1.0"},"canonical_sha256":"fd4ccef740d51390cbd02f5c89f6c5c3cf5edb789bca00122caaaf64cf2356b5","source":{"kind":"arxiv","id":"1508.01290","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.01290","created_at":"2026-05-18T00:17:55Z"},{"alias_kind":"arxiv_version","alias_value":"1508.01290v2","created_at":"2026-05-18T00:17:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.01290","created_at":"2026-05-18T00:17:55Z"},{"alias_kind":"pith_short_12","alias_value":"7VGM552A2UJZ","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"7VGM552A2UJZBS6Q","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"7VGM552A","created_at":"2026-05-18T12:29:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:7VGM552A2UJZBS6QF5OIT5WFYP","target":"record","payload":{"canonical_record":{"source":{"id":"1508.01290","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-08-06T06:46:52Z","cross_cats_sorted":[],"title_canon_sha256":"d1221c14fa3f5fdfabdb78259321b137df405ffbfa1ebc0c42becb9fd3b0cd2e","abstract_canon_sha256":"448d76dc8b8f89a710d999a63460b42b9b262887ab47987a8d5c30bcdd7fadba"},"schema_version":"1.0"},"canonical_sha256":"fd4ccef740d51390cbd02f5c89f6c5c3cf5edb789bca00122caaaf64cf2356b5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:55.490412Z","signature_b64":"2P8MMeVwHvBrHuFQ55ypyjK9vzOkAC+vKHrz08oZfKhRA8SN/rT/wK7qSLAMTLkeysPUobXxYiHTHvRW9EXuCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd4ccef740d51390cbd02f5c89f6c5c3cf5edb789bca00122caaaf64cf2356b5","last_reissued_at":"2026-05-18T00:17:55.489763Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:55.489763Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.01290","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:17:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FrB8rCTBzBeB9WECYITtqkKWdfB1FZ4P8hnpnDAdm3D2edciadFKfSEF/0a7TUSvIHsC9dWBpBab7/Kz7aczCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T06:36:02.193967Z"},"content_sha256":"bd5eda9427c255d9a6ed357f14b0f02c4ed50557b28070f43a23d44a818df26f","schema_version":"1.0","event_id":"sha256:bd5eda9427c255d9a6ed357f14b0f02c4ed50557b28070f43a23d44a818df26f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:7VGM552A2UJZBS6QF5OIT5WFYP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Toric rings, inseparability and rigidity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Dancheng Lu, J\\\"urgen Herzog, Mina Bigdeli","submitted_at":"2015-08-06T06:46:52Z","abstract_excerpt":"This article provides the basic algebraic background on infinitesimal deformations and presents the proof of the well-known fact that the non-trivial infinitesimal deformations of a $K$-algebra $R$ are parameterized by the elements of cotangent module $T^1(R)$ of $R$. In this article we focus on deformations of toric rings, and give an explicit description of $T^1(R)$ in the case that $R$ is a toric ring.\n  In particular, we are interested in unobstructed deformations which preserve the toric structure. Such deformations we call separations. Toric rings which do not admit any separation are ca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.01290","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:17:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KKc/4KdlR4rVc0Jp/jh2uG12Lu237dx1gK7mPoT+VYYja/aXj7gHjpICNUSIw8uU5SsAqSvpQ0SRnORg0SheCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T06:36:02.194638Z"},"content_sha256":"7b72076f370c055ebfba6a883101d0856aacf94025f15cceb0c670167cec8745","schema_version":"1.0","event_id":"sha256:7b72076f370c055ebfba6a883101d0856aacf94025f15cceb0c670167cec8745"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7VGM552A2UJZBS6QF5OIT5WFYP/bundle.json","state_url":"https://pith.science/pith/7VGM552A2UJZBS6QF5OIT5WFYP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7VGM552A2UJZBS6QF5OIT5WFYP/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":"2026-06-05T06:36:02Z","links":{"resolver":"https://pith.science/pith/7VGM552A2UJZBS6QF5OIT5WFYP","bundle":"https://pith.science/pith/7VGM552A2UJZBS6QF5OIT5WFYP/bundle.json","state":"https://pith.science/pith/7VGM552A2UJZBS6QF5OIT5WFYP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7VGM552A2UJZBS6QF5OIT5WFYP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:7VGM552A2UJZBS6QF5OIT5WFYP","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":"448d76dc8b8f89a710d999a63460b42b9b262887ab47987a8d5c30bcdd7fadba","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-08-06T06:46:52Z","title_canon_sha256":"d1221c14fa3f5fdfabdb78259321b137df405ffbfa1ebc0c42becb9fd3b0cd2e"},"schema_version":"1.0","source":{"id":"1508.01290","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.01290","created_at":"2026-05-18T00:17:55Z"},{"alias_kind":"arxiv_version","alias_value":"1508.01290v2","created_at":"2026-05-18T00:17:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.01290","created_at":"2026-05-18T00:17:55Z"},{"alias_kind":"pith_short_12","alias_value":"7VGM552A2UJZ","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"7VGM552A2UJZBS6Q","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"7VGM552A","created_at":"2026-05-18T12:29:10Z"}],"graph_snapshots":[{"event_id":"sha256:7b72076f370c055ebfba6a883101d0856aacf94025f15cceb0c670167cec8745","target":"graph","created_at":"2026-05-18T00:17:55Z","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":"This article provides the basic algebraic background on infinitesimal deformations and presents the proof of the well-known fact that the non-trivial infinitesimal deformations of a $K$-algebra $R$ are parameterized by the elements of cotangent module $T^1(R)$ of $R$. In this article we focus on deformations of toric rings, and give an explicit description of $T^1(R)$ in the case that $R$ is a toric ring.\n  In particular, we are interested in unobstructed deformations which preserve the toric structure. Such deformations we call separations. Toric rings which do not admit any separation are ca","authors_text":"Dancheng Lu, J\\\"urgen Herzog, Mina Bigdeli","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-08-06T06:46:52Z","title":"Toric rings, inseparability and rigidity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.01290","kind":"arxiv","version":2},"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:bd5eda9427c255d9a6ed357f14b0f02c4ed50557b28070f43a23d44a818df26f","target":"record","created_at":"2026-05-18T00:17:55Z","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":"448d76dc8b8f89a710d999a63460b42b9b262887ab47987a8d5c30bcdd7fadba","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-08-06T06:46:52Z","title_canon_sha256":"d1221c14fa3f5fdfabdb78259321b137df405ffbfa1ebc0c42becb9fd3b0cd2e"},"schema_version":"1.0","source":{"id":"1508.01290","kind":"arxiv","version":2}},"canonical_sha256":"fd4ccef740d51390cbd02f5c89f6c5c3cf5edb789bca00122caaaf64cf2356b5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fd4ccef740d51390cbd02f5c89f6c5c3cf5edb789bca00122caaaf64cf2356b5","first_computed_at":"2026-05-18T00:17:55.489763Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:17:55.489763Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2P8MMeVwHvBrHuFQ55ypyjK9vzOkAC+vKHrz08oZfKhRA8SN/rT/wK7qSLAMTLkeysPUobXxYiHTHvRW9EXuCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:17:55.490412Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.01290","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bd5eda9427c255d9a6ed357f14b0f02c4ed50557b28070f43a23d44a818df26f","sha256:7b72076f370c055ebfba6a883101d0856aacf94025f15cceb0c670167cec8745"],"state_sha256":"a830a72a7892b70b90695389577036cf959d973ec7457060d589dba14221c8b2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gs/AjBUTeWlmBEgAuCjCRKhZxIzUly6dWC3Lvs7msuTlsqG6u7UPqInLZUeqQKQ4hcDFByQ2B4EZJQX9w9wvCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T06:36:02.198496Z","bundle_sha256":"04027c9bc409be26ac5fee87c99862715a42d1442357e578904e3f666e9dfeeb"}}