{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:PJNYHE5JF3LUGJ53AI3ATU5UN6","short_pith_number":"pith:PJNYHE5J","schema_version":"1.0","canonical_sha256":"7a5b8393a92ed74327bb023609d3b46f893116224ae0762940e35df4c69dfae9","source":{"kind":"arxiv","id":"1709.10160","version":1},"attestation_state":"computed","paper":{"title":"$\\rm{FP}_{n}$-injective and $\\rm{FP}_{n}$-flat covers and preenvelopes, and Gorenstein AC-flat covers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.AC","authors_text":"Alina Iacob, Daniel Bravo, Sergio Estrada","submitted_at":"2017-09-28T20:39:10Z","abstract_excerpt":"We prove that, for any $n \\geq 2$, the classes of $\\rm{FP}_{n}$-injective modules and of $\\rm{FP}_n$-flat modules are both covering and preenveloping over any ring $R$. This includes the case of $\\rm{FP}_{\\infty}$-injective and $\\rm{FP}_{\\infty}$-flat modules (i.e. absolutely clean and, respectively, level modules). Then we consider a generalization of the class of (strongly) Gorenstein flat modules - the (strongly) Gorenstein AC-flat modules (cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module). We prove that some of the properties of Gor"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1709.10160","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-09-28T20:39:10Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"6bfa9bff47fdfa64eb5ae32c0926a30dfc22ed84e5bb402b4b7831198c50b4dd","abstract_canon_sha256":"c3a7406766fd17910470e38366e442c899d10cecb2751c0895a1b47ffe3dc770"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:34:04.160867Z","signature_b64":"B0QV2mryalvdRadSCyFKoaTH00T8KEPjjwrUmnNmEprrUuEek+lEnd29BWeOWRcaQFrW/wbpEqVSsIinoGjQDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7a5b8393a92ed74327bb023609d3b46f893116224ae0762940e35df4c69dfae9","last_reissued_at":"2026-05-18T00:34:04.160135Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:34:04.160135Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$\\rm{FP}_{n}$-injective and $\\rm{FP}_{n}$-flat covers and preenvelopes, and Gorenstein AC-flat covers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.AC","authors_text":"Alina Iacob, Daniel Bravo, Sergio Estrada","submitted_at":"2017-09-28T20:39:10Z","abstract_excerpt":"We prove that, for any $n \\geq 2$, the classes of $\\rm{FP}_{n}$-injective modules and of $\\rm{FP}_n$-flat modules are both covering and preenveloping over any ring $R$. This includes the case of $\\rm{FP}_{\\infty}$-injective and $\\rm{FP}_{\\infty}$-flat modules (i.e. absolutely clean and, respectively, level modules). Then we consider a generalization of the class of (strongly) Gorenstein flat modules - the (strongly) Gorenstein AC-flat modules (cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module). We prove that some of the properties of Gor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.10160","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1709.10160","created_at":"2026-05-18T00:34:04.160264+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.10160v1","created_at":"2026-05-18T00:34:04.160264+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.10160","created_at":"2026-05-18T00:34:04.160264+00:00"},{"alias_kind":"pith_short_12","alias_value":"PJNYHE5JF3LU","created_at":"2026-05-18T12:31:37.085036+00:00"},{"alias_kind":"pith_short_16","alias_value":"PJNYHE5JF3LUGJ53","created_at":"2026-05-18T12:31:37.085036+00:00"},{"alias_kind":"pith_short_8","alias_value":"PJNYHE5J","created_at":"2026-05-18T12:31:37.085036+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PJNYHE5JF3LUGJ53AI3ATU5UN6","json":"https://pith.science/pith/PJNYHE5JF3LUGJ53AI3ATU5UN6.json","graph_json":"https://pith.science/api/pith-number/PJNYHE5JF3LUGJ53AI3ATU5UN6/graph.json","events_json":"https://pith.science/api/pith-number/PJNYHE5JF3LUGJ53AI3ATU5UN6/events.json","paper":"https://pith.science/paper/PJNYHE5J"},"agent_actions":{"view_html":"https://pith.science/pith/PJNYHE5JF3LUGJ53AI3ATU5UN6","download_json":"https://pith.science/pith/PJNYHE5JF3LUGJ53AI3ATU5UN6.json","view_paper":"https://pith.science/paper/PJNYHE5J","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.10160&json=true","fetch_graph":"https://pith.science/api/pith-number/PJNYHE5JF3LUGJ53AI3ATU5UN6/graph.json","fetch_events":"https://pith.science/api/pith-number/PJNYHE5JF3LUGJ53AI3ATU5UN6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PJNYHE5JF3LUGJ53AI3ATU5UN6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PJNYHE5JF3LUGJ53AI3ATU5UN6/action/storage_attestation","attest_author":"https://pith.science/pith/PJNYHE5JF3LUGJ53AI3ATU5UN6/action/author_attestation","sign_citation":"https://pith.science/pith/PJNYHE5JF3LUGJ53AI3ATU5UN6/action/citation_signature","submit_replication":"https://pith.science/pith/PJNYHE5JF3LUGJ53AI3ATU5UN6/action/replication_record"}},"created_at":"2026-05-18T00:34:04.160264+00:00","updated_at":"2026-05-18T00:34:04.160264+00:00"}