{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:CJWMMGXFE46VR7V3KK33TZIB5T","short_pith_number":"pith:CJWMMGXF","canonical_record":{"source":{"id":"1802.07596","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-02-21T14:55:16Z","cross_cats_sorted":[],"title_canon_sha256":"93803f29a695786f11dfee959013c0207096c68f9e87bc271c48c54e1ecb062b","abstract_canon_sha256":"032d5095d59331db1b2b78476e0c5df19175b005ae55e6f3aad9859f20b7457c"},"schema_version":"1.0"},"canonical_sha256":"126cc61ae5273d58febb52b7b9e501ecf34f912832776e78268339ef095fbba5","source":{"kind":"arxiv","id":"1802.07596","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.07596","created_at":"2026-05-18T00:22:49Z"},{"alias_kind":"arxiv_version","alias_value":"1802.07596v1","created_at":"2026-05-18T00:22:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.07596","created_at":"2026-05-18T00:22:49Z"},{"alias_kind":"pith_short_12","alias_value":"CJWMMGXFE46V","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_16","alias_value":"CJWMMGXFE46VR7V3","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_8","alias_value":"CJWMMGXF","created_at":"2026-05-18T12:32:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:CJWMMGXFE46VR7V3KK33TZIB5T","target":"record","payload":{"canonical_record":{"source":{"id":"1802.07596","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-02-21T14:55:16Z","cross_cats_sorted":[],"title_canon_sha256":"93803f29a695786f11dfee959013c0207096c68f9e87bc271c48c54e1ecb062b","abstract_canon_sha256":"032d5095d59331db1b2b78476e0c5df19175b005ae55e6f3aad9859f20b7457c"},"schema_version":"1.0"},"canonical_sha256":"126cc61ae5273d58febb52b7b9e501ecf34f912832776e78268339ef095fbba5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:49.832771Z","signature_b64":"mGAZWIfAtSUR04QcdCF9rEZ64rgauP+nPY8OydqP2vmWps0nQ4MXZoAfJNF3NGt8rxeeX//lb1RWWZgio0RvCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"126cc61ae5273d58febb52b7b9e501ecf34f912832776e78268339ef095fbba5","last_reissued_at":"2026-05-18T00:22:49.832323Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:49.832323Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1802.07596","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":"2026-05-18T00:22:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LyLtxVhYyEEk57GHqCRID8Xoax7qoGJ6d6EpwK7D41O2nYaTVZ6FRxjyLdbVkRPpiYlWaBq+9ytd1V+cHAKDCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T18:58:50.225620Z"},"content_sha256":"114ac90620a40d12166d90c25583614f198f5286804eba22385fa3c15517fe1f","schema_version":"1.0","event_id":"sha256:114ac90620a40d12166d90c25583614f198f5286804eba22385fa3c15517fe1f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:CJWMMGXFE46VR7V3KK33TZIB5T","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Maximal depth property of finitely generated modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Ahad Rahimi","submitted_at":"2018-02-21T14:55:16Z","abstract_excerpt":"Let $(R,\\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We say $M$ has maximal depth if there is an associated prime $\\mathfrak{p}$ of $M$ such that depth $M=\\dim R/\\mathfrak{p}$. In this paper, we study finitely generated modules with maximal depth. It is shown that the maximal depth property is preserved under some important module operations. Generalized Cohen--Macaulay modules with maximal depth are classified. Finally, the attached primes of $H^i_{\\mathfrak{m}}(M)$ are considered for $i<\\mathrm{dim} M$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.07596","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":"2026-05-18T00:22:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UOW+blE4qKcE0lkS2w0tXTm4dNRkrzEbIBtdsAQrx6TyxRJlCjkWtgO7aoDXpZ3Jbrc5Ds4CbuKttWC3gDqMCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T18:58:50.225975Z"},"content_sha256":"5a9d5ccfe2fb47e35c22d65911b957809f46c56e40b1567d9ac853023d585f9a","schema_version":"1.0","event_id":"sha256:5a9d5ccfe2fb47e35c22d65911b957809f46c56e40b1567d9ac853023d585f9a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CJWMMGXFE46VR7V3KK33TZIB5T/bundle.json","state_url":"https://pith.science/pith/CJWMMGXFE46VR7V3KK33TZIB5T/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CJWMMGXFE46VR7V3KK33TZIB5T/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":"2026-06-24T18:58:50Z","links":{"resolver":"https://pith.science/pith/CJWMMGXFE46VR7V3KK33TZIB5T","bundle":"https://pith.science/pith/CJWMMGXFE46VR7V3KK33TZIB5T/bundle.json","state":"https://pith.science/pith/CJWMMGXFE46VR7V3KK33TZIB5T/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CJWMMGXFE46VR7V3KK33TZIB5T/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:CJWMMGXFE46VR7V3KK33TZIB5T","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":"032d5095d59331db1b2b78476e0c5df19175b005ae55e6f3aad9859f20b7457c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-02-21T14:55:16Z","title_canon_sha256":"93803f29a695786f11dfee959013c0207096c68f9e87bc271c48c54e1ecb062b"},"schema_version":"1.0","source":{"id":"1802.07596","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.07596","created_at":"2026-05-18T00:22:49Z"},{"alias_kind":"arxiv_version","alias_value":"1802.07596v1","created_at":"2026-05-18T00:22:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.07596","created_at":"2026-05-18T00:22:49Z"},{"alias_kind":"pith_short_12","alias_value":"CJWMMGXFE46V","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_16","alias_value":"CJWMMGXFE46VR7V3","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_8","alias_value":"CJWMMGXF","created_at":"2026-05-18T12:32:16Z"}],"graph_snapshots":[{"event_id":"sha256:5a9d5ccfe2fb47e35c22d65911b957809f46c56e40b1567d9ac853023d585f9a","target":"graph","created_at":"2026-05-18T00:22:49Z","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 $(R,\\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We say $M$ has maximal depth if there is an associated prime $\\mathfrak{p}$ of $M$ such that depth $M=\\dim R/\\mathfrak{p}$. In this paper, we study finitely generated modules with maximal depth. It is shown that the maximal depth property is preserved under some important module operations. Generalized Cohen--Macaulay modules with maximal depth are classified. Finally, the attached primes of $H^i_{\\mathfrak{m}}(M)$ are considered for $i<\\mathrm{dim} M$.","authors_text":"Ahad Rahimi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-02-21T14:55:16Z","title":"Maximal depth property of finitely generated modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.07596","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":"sha256:114ac90620a40d12166d90c25583614f198f5286804eba22385fa3c15517fe1f","target":"record","created_at":"2026-05-18T00:22:49Z","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":"032d5095d59331db1b2b78476e0c5df19175b005ae55e6f3aad9859f20b7457c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-02-21T14:55:16Z","title_canon_sha256":"93803f29a695786f11dfee959013c0207096c68f9e87bc271c48c54e1ecb062b"},"schema_version":"1.0","source":{"id":"1802.07596","kind":"arxiv","version":1}},"canonical_sha256":"126cc61ae5273d58febb52b7b9e501ecf34f912832776e78268339ef095fbba5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"126cc61ae5273d58febb52b7b9e501ecf34f912832776e78268339ef095fbba5","first_computed_at":"2026-05-18T00:22:49.832323Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:49.832323Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mGAZWIfAtSUR04QcdCF9rEZ64rgauP+nPY8OydqP2vmWps0nQ4MXZoAfJNF3NGt8rxeeX//lb1RWWZgio0RvCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:49.832771Z","signed_message":"canonical_sha256_bytes"},"source_id":"1802.07596","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:114ac90620a40d12166d90c25583614f198f5286804eba22385fa3c15517fe1f","sha256:5a9d5ccfe2fb47e35c22d65911b957809f46c56e40b1567d9ac853023d585f9a"],"state_sha256":"791267005dc26096b4f9748009ef576f7afdffc43a4719eef7100485aa327648"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T+SEMUfLqOt1u6s54gR8FcpyeV+ciVVRTH7auZORLlf1X/aBp35AB+3JH3ERSZSFAIRY2SO/p9JqTqtJSDD1BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T18:58:50.227957Z","bundle_sha256":"746056a5af07d150fe0a6a3151f99d4389ab86fee42e6621213082816e02bf13"}}