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Let $(\\mathcal F_A,\\mathcal C_A)$ be the cotorsion pair in $A\\mathsf{-Mod}$ in which $\\mathcal F_A$ is the class of all left $A$-modules whose underlying $R$-modules belong to $\\mathcal F$. Assuming that the $\\mathcal F$-resolution dimension of every left $R$-module is finite and the class $\\mathcal F$ is preserved by the coinduction functor $\\operatorname{Hom}_R(A,-)$, we show that $\\mathcal C_A$ is the class of all direct summands of left $A$-mod"},"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":"2006.01778","kind":"arxiv","version":7},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2020-06-02T16:59:51Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"51f3bc1130c71ed77a6491e27355e9ac11782409c2921ced4698f2c695e8b339","abstract_canon_sha256":"bf511ca8cf7988c75f475815374640ed17739578ade941fd2ddff66a8f19e376"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:01:54.535264Z","signature_b64":"VTDxG8oepwbF/oqZxfZUE7PEHH/Bm2ltZUev+7cdkRtDkMmPUSbWVxju4NIrqZRbzUPU/Phi4Q2y4EBgGEZqCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d8c51a14bc3191d2793d88ebf20b0c429b4bdad37fc7d857392ac86faa726a6","last_reissued_at":"2026-07-05T06:01:54.534924Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:01:54.534924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An explicit self-dual construction of complete cotorsion pairs in the relative context","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.RA","authors_text":"Leonid Positselski","submitted_at":"2020-06-02T16:59:51Z","abstract_excerpt":"Let $R\\to A$ be a homomorphism of associative rings, and let $(\\mathcal F,\\mathcal C)$ be a hereditary complete cotorsion pair in $R\\mathsf{-Mod}$. 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Assuming that the $\\mathcal F$-resolution dimension of every left $R$-module is finite and the class $\\mathcal F$ is preserved by the coinduction functor $\\operatorname{Hom}_R(A,-)$, we show that $\\mathcal C_A$ is the class of all direct summands of left $A$-mod"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.01778","kind":"arxiv","version":7},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2006.01778/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2006.01778","created_at":"2026-07-05T06:01:54.534979+00:00"},{"alias_kind":"arxiv_version","alias_value":"2006.01778v7","created_at":"2026-07-05T06:01:54.534979+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.01778","created_at":"2026-07-05T06:01:54.534979+00:00"},{"alias_kind":"pith_short_12","alias_value":"BWGFDIKLYMMR","created_at":"2026-07-05T06:01:54.534979+00:00"},{"alias_kind":"pith_short_16","alias_value":"BWGFDIKLYMMR2J4T","created_at":"2026-07-05T06:01:54.534979+00:00"},{"alias_kind":"pith_short_8","alias_value":"BWGFDIKL","created_at":"2026-07-05T06:01:54.534979+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2507.15425","citing_title":"A relative version of Bass' theorem about finite-dimensional algebras","ref_index":12,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BWGFDIKLYMMR2J4T3CHL6IFQYQ","json":"https://pith.science/pith/BWGFDIKLYMMR2J4T3CHL6IFQYQ.json","graph_json":"https://pith.science/api/pith-number/BWGFDIKLYMMR2J4T3CHL6IFQYQ/graph.json","events_json":"https://pith.science/api/pith-number/BWGFDIKLYMMR2J4T3CHL6IFQYQ/events.json","paper":"https://pith.science/paper/BWGFDIKL"},"agent_actions":{"view_html":"https://pith.science/pith/BWGFDIKLYMMR2J4T3CHL6IFQYQ","download_json":"https://pith.science/pith/BWGFDIKLYMMR2J4T3CHL6IFQYQ.json","view_paper":"https://pith.science/paper/BWGFDIKL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2006.01778&json=true","fetch_graph":"https://pith.science/api/pith-number/BWGFDIKLYMMR2J4T3CHL6IFQYQ/graph.json","fetch_events":"https://pith.science/api/pith-number/BWGFDIKLYMMR2J4T3CHL6IFQYQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BWGFDIKLYMMR2J4T3CHL6IFQYQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BWGFDIKLYMMR2J4T3CHL6IFQYQ/action/storage_attestation","attest_author":"https://pith.science/pith/BWGFDIKLYMMR2J4T3CHL6IFQYQ/action/author_attestation","sign_citation":"https://pith.science/pith/BWGFDIKLYMMR2J4T3CHL6IFQYQ/action/citation_signature","submit_replication":"https://pith.science/pith/BWGFDIKLYMMR2J4T3CHL6IFQYQ/action/replication_record"}},"created_at":"2026-07-05T06:01:54.534979+00:00","updated_at":"2026-07-05T06:01:54.534979+00:00"}