{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:72X7YL6KS3A2L3WVKRIWQKOYHP","short_pith_number":"pith:72X7YL6K","canonical_record":{"source":{"id":"1711.07964","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-11-21T18:43:32Z","cross_cats_sorted":["math.AC","math.AT","math.CO"],"title_canon_sha256":"22631e353a8568ad3643d84942adffffd4a6044ff93b48016ba4c4f36c9b0a5c","abstract_canon_sha256":"419951d0273e1a3d5f9953d1b9fa7f9d2d693da352216468a2ce765266a7ae24"},"schema_version":"1.0"},"canonical_sha256":"feaffc2fca96c1a5eed554516829d83bea6e4eccd53d1976d0a0fad14c82eb8f","source":{"kind":"arxiv","id":"1711.07964","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.07964","created_at":"2026-05-18T00:29:56Z"},{"alias_kind":"arxiv_version","alias_value":"1711.07964v1","created_at":"2026-05-18T00:29:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.07964","created_at":"2026-05-18T00:29:56Z"},{"alias_kind":"pith_short_12","alias_value":"72X7YL6KS3A2","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_16","alias_value":"72X7YL6KS3A2L3WV","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_8","alias_value":"72X7YL6K","created_at":"2026-05-18T12:31:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:72X7YL6KS3A2L3WVKRIWQKOYHP","target":"record","payload":{"canonical_record":{"source":{"id":"1711.07964","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-11-21T18:43:32Z","cross_cats_sorted":["math.AC","math.AT","math.CO"],"title_canon_sha256":"22631e353a8568ad3643d84942adffffd4a6044ff93b48016ba4c4f36c9b0a5c","abstract_canon_sha256":"419951d0273e1a3d5f9953d1b9fa7f9d2d693da352216468a2ce765266a7ae24"},"schema_version":"1.0"},"canonical_sha256":"feaffc2fca96c1a5eed554516829d83bea6e4eccd53d1976d0a0fad14c82eb8f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:56.063121Z","signature_b64":"PMR0ru3Ut1wWsWWjWWweT629MXR3ZytRhNK3sKIUXbANbfkyihhm7S2Oew00I67WtfR96SD0YokcCP/oj8cXDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"feaffc2fca96c1a5eed554516829d83bea6e4eccd53d1976d0a0fad14c82eb8f","last_reissued_at":"2026-05-18T00:29:56.062615Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:56.062615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1711.07964","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":"2026-05-18T00:29:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"enav0FaTsZySRYzUHER3Ggw1MLUd3gzFF+0YowQM4LDMuqPIQDu0ua44XKC/3Mx6R42EZOzlWZVDgQa412KaBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T16:40:29.104155Z"},"content_sha256":"05e9487a206202563e1059feb569796ddeba0b2b887b98bdddf0ca4063191521","schema_version":"1.0","event_id":"sha256:05e9487a206202563e1059feb569796ddeba0b2b887b98bdddf0ca4063191521"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:72X7YL6KS3A2L3WVKRIWQKOYHP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Local cohomology and the multi-graded regularity of FI$^m$-modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.AT","math.CO"],"primary_cat":"math.RT","authors_text":"Eric Ramos, Liping Li","submitted_at":"2017-11-21T18:43:32Z","abstract_excerpt":"We develop a local cohomology theory for FI$^m$-modules, and show that it in many ways mimics the classical theory for multi-graded modules over a polynomial ring. In particular, we define an invariant of FI$^m$-modules using this local cohomology theory which closely resembles an invariant of multi-graded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.07964","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":"2026-05-18T00:29:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cNYp2+u0kbNC/+d9ZGqrrp+v225TgAyyEHh9m5ghUdpTAAHXI6wya5yFEtiO9UuqbMfln37uKXZaJuecrAxhDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T16:40:29.104901Z"},"content_sha256":"cffecf5b099220899d12b88a47e96f5de2816fe7cb77cbfceea20e77b31284ac","schema_version":"1.0","event_id":"sha256:cffecf5b099220899d12b88a47e96f5de2816fe7cb77cbfceea20e77b31284ac"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/72X7YL6KS3A2L3WVKRIWQKOYHP/bundle.json","state_url":"https://pith.science/pith/72X7YL6KS3A2L3WVKRIWQKOYHP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/72X7YL6KS3A2L3WVKRIWQKOYHP/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":"2026-05-25T16:40:29Z","links":{"resolver":"https://pith.science/pith/72X7YL6KS3A2L3WVKRIWQKOYHP","bundle":"https://pith.science/pith/72X7YL6KS3A2L3WVKRIWQKOYHP/bundle.json","state":"https://pith.science/pith/72X7YL6KS3A2L3WVKRIWQKOYHP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/72X7YL6KS3A2L3WVKRIWQKOYHP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:72X7YL6KS3A2L3WVKRIWQKOYHP","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":"419951d0273e1a3d5f9953d1b9fa7f9d2d693da352216468a2ce765266a7ae24","cross_cats_sorted":["math.AC","math.AT","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-11-21T18:43:32Z","title_canon_sha256":"22631e353a8568ad3643d84942adffffd4a6044ff93b48016ba4c4f36c9b0a5c"},"schema_version":"1.0","source":{"id":"1711.07964","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.07964","created_at":"2026-05-18T00:29:56Z"},{"alias_kind":"arxiv_version","alias_value":"1711.07964v1","created_at":"2026-05-18T00:29:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.07964","created_at":"2026-05-18T00:29:56Z"},{"alias_kind":"pith_short_12","alias_value":"72X7YL6KS3A2","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_16","alias_value":"72X7YL6KS3A2L3WV","created_at":"2026-05-18T12:31:03Z"},{"alias_kind":"pith_short_8","alias_value":"72X7YL6K","created_at":"2026-05-18T12:31:03Z"}],"graph_snapshots":[{"event_id":"sha256:cffecf5b099220899d12b88a47e96f5de2816fe7cb77cbfceea20e77b31284ac","target":"graph","created_at":"2026-05-18T00:29:56Z","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 develop a local cohomology theory for FI$^m$-modules, and show that it in many ways mimics the classical theory for multi-graded modules over a polynomial ring. In particular, we define an invariant of FI$^m$-modules using this local cohomology theory which closely resembles an invariant of multi-graded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith.","authors_text":"Eric Ramos, Liping Li","cross_cats":["math.AC","math.AT","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-11-21T18:43:32Z","title":"Local cohomology and the multi-graded regularity of FI$^m$-modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.07964","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":"sha256:05e9487a206202563e1059feb569796ddeba0b2b887b98bdddf0ca4063191521","target":"record","created_at":"2026-05-18T00:29:56Z","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":"419951d0273e1a3d5f9953d1b9fa7f9d2d693da352216468a2ce765266a7ae24","cross_cats_sorted":["math.AC","math.AT","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-11-21T18:43:32Z","title_canon_sha256":"22631e353a8568ad3643d84942adffffd4a6044ff93b48016ba4c4f36c9b0a5c"},"schema_version":"1.0","source":{"id":"1711.07964","kind":"arxiv","version":1}},"canonical_sha256":"feaffc2fca96c1a5eed554516829d83bea6e4eccd53d1976d0a0fad14c82eb8f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"feaffc2fca96c1a5eed554516829d83bea6e4eccd53d1976d0a0fad14c82eb8f","first_computed_at":"2026-05-18T00:29:56.062615Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:29:56.062615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PMR0ru3Ut1wWsWWjWWweT629MXR3ZytRhNK3sKIUXbANbfkyihhm7S2Oew00I67WtfR96SD0YokcCP/oj8cXDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:29:56.063121Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.07964","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:05e9487a206202563e1059feb569796ddeba0b2b887b98bdddf0ca4063191521","sha256:cffecf5b099220899d12b88a47e96f5de2816fe7cb77cbfceea20e77b31284ac"],"state_sha256":"2d32fb439de76ef902e2d0afe785d381bb059af4f818d259f3f6261c6c6cd6c6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qf1EesQ5ZlZDwqfRbReONkSUdPccHCutGpIjcfAEnJnkOqGa9t0o0Ttq9jNWLUZIFUwcF3rZeuYJ5lfWTskhAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T16:40:29.108880Z","bundle_sha256":"1f20f2e32efd912ddd22fed2e58eef2bf102846726dc0485de41a51e3d5f0607"}}