{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:FQO5MTYT6AHF22ON575JB27Y3Q","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":"f70ecee0ba0a22c2ae8215cea678cbeb59dabffbe8db5d917f4667d19e9c1775","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-03-01T16:02:31Z","title_canon_sha256":"952bb7c1d9bfa53535958b50a8e4540d56d3241367825f8752fb6924c757a2d8"},"schema_version":"1.0","source":{"id":"1603.00334","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.00334","created_at":"2026-05-18T01:19:45Z"},{"alias_kind":"arxiv_version","alias_value":"1603.00334v1","created_at":"2026-05-18T01:19:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.00334","created_at":"2026-05-18T01:19:45Z"},{"alias_kind":"pith_short_12","alias_value":"FQO5MTYT6AHF","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FQO5MTYT6AHF22ON","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FQO5MTYT","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:a684c0af2ade01cac66a735ec3be0d5961ebcb2c3f3c95e5a435d8b19194cbc3","target":"graph","created_at":"2026-05-18T01:19:45Z","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":"In this note we introduce and study basic properties of two types of modules over a commutative noetherian ring $R$ of positive prime characteristic. The first is the category of modules of finite $F$-type. These objects include reflexive ideals representing torsion elements in the divisor class group of $R$. The second class is what we call $F$-abundant modules. These include, for example, the ring $R$ itself and the canonical module when $R$ has positive splitting dimension. We prove various facts about these two categories and how they are related, for example that $\\mathrm{Hom}_R(M,N)$ is ","authors_text":"Hailong Dao, Tony Se","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-03-01T16:02:31Z","title":"Finite F-type and F-abundant modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.00334","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":"sha256:71588be12509d4c92fdd0c006af56dc8a67f683a47f7fe83667b265e70f42954","target":"record","created_at":"2026-05-18T01:19:45Z","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":"f70ecee0ba0a22c2ae8215cea678cbeb59dabffbe8db5d917f4667d19e9c1775","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-03-01T16:02:31Z","title_canon_sha256":"952bb7c1d9bfa53535958b50a8e4540d56d3241367825f8752fb6924c757a2d8"},"schema_version":"1.0","source":{"id":"1603.00334","kind":"arxiv","version":1}},"canonical_sha256":"2c1dd64f13f00e5d69cdeffa90ebf8dc19c2070fe03666af2eda7a457ed63046","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2c1dd64f13f00e5d69cdeffa90ebf8dc19c2070fe03666af2eda7a457ed63046","first_computed_at":"2026-05-18T01:19:45.326757Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:45.326757Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bMhejyqyFiTp5ooqCD8doJv0Z2nyS63lqwx9Fnqda94EzTErnILc2kMhIUm0EmptgtEUyX9BjLaZtwYid/k9DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:45.327268Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.00334","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:71588be12509d4c92fdd0c006af56dc8a67f683a47f7fe83667b265e70f42954","sha256:a684c0af2ade01cac66a735ec3be0d5961ebcb2c3f3c95e5a435d8b19194cbc3"],"state_sha256":"0bac98bfad065a51d6e1f1d43f14a2dd2e865e2d065a8bf8a1797e72841cdc6f"}