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For instance, if R is noetherian, then given R-modules M and M' such that M is Matlis reflexive and M' is mini-max (e.g., noetherian or artinian), we prove that Ext^i_R(M,M'), Ext^i_R(M',M), and Tor_i^R(M,M') are Matlis reflexive over R for all i\\geq 0 and that Ext^i_R(M,M')^\\vee\\cong Tor_i^R(M,M'^\\vee) and Ext^i_R(M',M)^\\vee\\cong"},"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":"1208.5534","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-08-28T01:43:49Z","cross_cats_sorted":[],"title_canon_sha256":"a40b547ecc4fedec28fc96fbfc2d69fcba62496536c86a080884e6bdbf38ddf8","abstract_canon_sha256":"44c1873ef05385fb29ff4caefae0ab16f4d1c3a413a5cdae386a821d08dc6018"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:58.439064Z","signature_b64":"E27cYz74+/ZyxrOqrQ+/kFPodlUMCO7s1I5kie3RXAhptQGeh5rnj0Lj8003BJJxYSaThABon1uJksDj9+lFAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c28b39dc46215b61c38785241e36cd2e940493b2b295cebe9e0f3bb8c93798b6","last_reissued_at":"2026-05-18T03:46:58.438382Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:58.438382Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Homology of artinian and mini-max modules, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Bethany Kubik, Micah Leamer, Sean Sather-Wagstaff","submitted_at":"2012-08-28T01:43:49Z","abstract_excerpt":"Let R be a commutative ring, and let L and L' be R-modules. 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