{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:TQL76WSRNMWY6ESD6VXV52HEQK","short_pith_number":"pith:TQL76WSR","canonical_record":{"source":{"id":"1110.0383","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-10-03T15:25:00Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"960493d0f736432613ca3dbf6fb7cae4bd05a2a2ee2bbfa0deda92f9e5a81a4e","abstract_canon_sha256":"719d24a416486cf31d912314c795982bb396b39eefcc30ea3c1c2fa9b9802242"},"schema_version":"1.0"},"canonical_sha256":"9c17ff5a516b2d8f1243f56f5ee8e482a761aaaec35348691a19b515ce364f5a","source":{"kind":"arxiv","id":"1110.0383","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.0383","created_at":"2026-05-18T03:19:35Z"},{"alias_kind":"arxiv_version","alias_value":"1110.0383v3","created_at":"2026-05-18T03:19:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.0383","created_at":"2026-05-18T03:19:35Z"},{"alias_kind":"pith_short_12","alias_value":"TQL76WSRNMWY","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"TQL76WSRNMWY6ESD","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"TQL76WSR","created_at":"2026-05-18T12:26:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:TQL76WSRNMWY6ESD6VXV52HEQK","target":"record","payload":{"canonical_record":{"source":{"id":"1110.0383","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-10-03T15:25:00Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"960493d0f736432613ca3dbf6fb7cae4bd05a2a2ee2bbfa0deda92f9e5a81a4e","abstract_canon_sha256":"719d24a416486cf31d912314c795982bb396b39eefcc30ea3c1c2fa9b9802242"},"schema_version":"1.0"},"canonical_sha256":"9c17ff5a516b2d8f1243f56f5ee8e482a761aaaec35348691a19b515ce364f5a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:35.518279Z","signature_b64":"e5g67wDSfrx2d2MVIjRF2VmPnTm7VgUZNwgk8Xo/D+53r66E49pP/o8UtnhecP7swjoi4nAUT82CgPvErTeeDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c17ff5a516b2d8f1243f56f5ee8e482a761aaaec35348691a19b515ce364f5a","last_reissued_at":"2026-05-18T03:19:35.517808Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:35.517808Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.0383","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":"2026-05-18T03:19:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5/nObm+kjbCwj4Zw05+SrHOBxFUwN/1/JAnH9z/PpxAyAe+hzUNx2FlX6rGGGOL6xe6hMi3ggPiGN5po47IMBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T04:49:08.793456Z"},"content_sha256":"43a67b5270740f2cbde4da846e85f7eceb886765c3cbcc36272dced26664da34","schema_version":"1.0","event_id":"sha256:43a67b5270740f2cbde4da846e85f7eceb886765c3cbcc36272dced26664da34"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:TQL76WSRNMWY6ESD6VXV52HEQK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The eventual shape of Betti tables of powers of ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Amir Bagheri, Huy Tai Ha, Marc Chardin","submitted_at":"2011-10-03T15:25:00Z","abstract_excerpt":"Let $G$ be a finitely generated abelian group, and let $S = A[x_1, ..., x_n]$ be a $G$-graded polynomial ring over a commutative ring $A$. Let $I_1, ..., I_s$ be $G$-homogeneous ideals in $S$, and let $M$ be a finitely generated $G$-graded $S$-module. We show that, when $A$ is Noetherian, the nonzero $G$-graded Betti numbers of $MI_1^{t_1} ... I_s^{t_s}$ exhibit an asymptotic linear behavior as the $t_i$s get large."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.0383","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":"2026-05-18T03:19:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q23RfJcKT+dkGmwMYLcU7vMb+Pz1iuM3auIvOxe8J2eDNbNHjIyQjoc83nStZkN+quViQjybwzQjYb2NBu8EBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T04:49:08.793797Z"},"content_sha256":"5746f3d7d9fe36cc84d06a2b3bae631e4b322a80c8227db4ddfc3fb598801f98","schema_version":"1.0","event_id":"sha256:5746f3d7d9fe36cc84d06a2b3bae631e4b322a80c8227db4ddfc3fb598801f98"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TQL76WSRNMWY6ESD6VXV52HEQK/bundle.json","state_url":"https://pith.science/pith/TQL76WSRNMWY6ESD6VXV52HEQK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TQL76WSRNMWY6ESD6VXV52HEQK/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":"2026-06-03T04:49:08Z","links":{"resolver":"https://pith.science/pith/TQL76WSRNMWY6ESD6VXV52HEQK","bundle":"https://pith.science/pith/TQL76WSRNMWY6ESD6VXV52HEQK/bundle.json","state":"https://pith.science/pith/TQL76WSRNMWY6ESD6VXV52HEQK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TQL76WSRNMWY6ESD6VXV52HEQK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:TQL76WSRNMWY6ESD6VXV52HEQK","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":"719d24a416486cf31d912314c795982bb396b39eefcc30ea3c1c2fa9b9802242","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-10-03T15:25:00Z","title_canon_sha256":"960493d0f736432613ca3dbf6fb7cae4bd05a2a2ee2bbfa0deda92f9e5a81a4e"},"schema_version":"1.0","source":{"id":"1110.0383","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.0383","created_at":"2026-05-18T03:19:35Z"},{"alias_kind":"arxiv_version","alias_value":"1110.0383v3","created_at":"2026-05-18T03:19:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.0383","created_at":"2026-05-18T03:19:35Z"},{"alias_kind":"pith_short_12","alias_value":"TQL76WSRNMWY","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"TQL76WSRNMWY6ESD","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"TQL76WSR","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:5746f3d7d9fe36cc84d06a2b3bae631e4b322a80c8227db4ddfc3fb598801f98","target":"graph","created_at":"2026-05-18T03:19:35Z","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":"Let $G$ be a finitely generated abelian group, and let $S = A[x_1, ..., x_n]$ be a $G$-graded polynomial ring over a commutative ring $A$. Let $I_1, ..., I_s$ be $G$-homogeneous ideals in $S$, and let $M$ be a finitely generated $G$-graded $S$-module. We show that, when $A$ is Noetherian, the nonzero $G$-graded Betti numbers of $MI_1^{t_1} ... I_s^{t_s}$ exhibit an asymptotic linear behavior as the $t_i$s get large.","authors_text":"Amir Bagheri, Huy Tai Ha, Marc Chardin","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-10-03T15:25:00Z","title":"The eventual shape of Betti tables of powers of ideals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.0383","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":"sha256:43a67b5270740f2cbde4da846e85f7eceb886765c3cbcc36272dced26664da34","target":"record","created_at":"2026-05-18T03:19:35Z","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":"719d24a416486cf31d912314c795982bb396b39eefcc30ea3c1c2fa9b9802242","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-10-03T15:25:00Z","title_canon_sha256":"960493d0f736432613ca3dbf6fb7cae4bd05a2a2ee2bbfa0deda92f9e5a81a4e"},"schema_version":"1.0","source":{"id":"1110.0383","kind":"arxiv","version":3}},"canonical_sha256":"9c17ff5a516b2d8f1243f56f5ee8e482a761aaaec35348691a19b515ce364f5a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9c17ff5a516b2d8f1243f56f5ee8e482a761aaaec35348691a19b515ce364f5a","first_computed_at":"2026-05-18T03:19:35.517808Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:19:35.517808Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"e5g67wDSfrx2d2MVIjRF2VmPnTm7VgUZNwgk8Xo/D+53r66E49pP/o8UtnhecP7swjoi4nAUT82CgPvErTeeDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:19:35.518279Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.0383","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:43a67b5270740f2cbde4da846e85f7eceb886765c3cbcc36272dced26664da34","sha256:5746f3d7d9fe36cc84d06a2b3bae631e4b322a80c8227db4ddfc3fb598801f98"],"state_sha256":"5336748062cf7b02c45cedda057ea3c56845bf6e2c5ccf8e98d99e7e61568c1a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5MbzWvT+5cwgGn8HwNBrN5RXfAetSZXeJCMQbm6eYYitA75OwtJ2tzjMzgS8uYrccgyXqNJDW412rZeTQq/IDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T04:49:08.795714Z","bundle_sha256":"fc0cfb3d228480f722d3c86f00ea88f7ded8ca8d256e1f0198555cecbff42949"}}