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This can be applied to study the big and the little finitistic dimension of $R$. We show that $\\Findim R<\\infty$ if and only if the following dimensions are finite for some cotorsion pair $(\\mathcal A, \\mathcal B)$ in $\\mathrm{Mod} R$: the relative projective dimension of $\\A$ with respect to itself, and"},"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":"0808.1585","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2008-08-11T21:16:56Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"a891cb231d0647459f22358209b59ad3c927a1da29a124fb40476d7b3f5084c8","abstract_canon_sha256":"6b3c81eb28aaff99f12e46dba9f02bfea4ec08d85925be68f22d15996e71e5c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:13:34.318064Z","signature_b64":"KvcP2Ipw7DOEHKnaxFCrrjZaVLpvC2xyO4d6Y2iMNCjWbnOisShtrVphoSXkVA9u39dm3egep9s4EHjAa2NkBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c4a83ff595a837de9b010826b6bc4d09827f08334668d6993bb4494ca543a4c0","last_reissued_at":"2026-07-04T15:13:34.317646Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:13:34.317646Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Homological Dimensions in Cotorsion Pairs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.RA","authors_text":"Lidia Angeleri Hugel, Octavio Mendoza Hernandez","submitted_at":"2008-08-11T21:16:56Z","abstract_excerpt":"Two classes $\\mathcal A$ and $\\mathcal B$ of modules over a ring $R$ are said to form a cotorsion pair $(\\mathcal A, \\mathcal B)$ if $\\mathcal A={\\rm Ker Ext}^1_R(-,\\mathcal B)$ and $\\mathcal B={\\rm Ker Ext}^1_R(\\mathcal A,-)$. 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