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For a non-free maximal Cohen-Macaulay (=MCM) $A$-module $M$ and an integer $i\\ge 1$, it is well known that the functions $n \\mapsto \\ell(Tor_i^A(M,N/I^{n+1}N))$ and $n \\mapsto \\ell(Ext^i_A(M,N/I^{n+1}N))$ are of polynomial types of degrees $r_i^{I,N}(M)$ and $s_{I,N}^i(M)$, respectively. We prove that $r_i^{I,N}(M)\\le t-1$ and $s^i_{I,N}(M)\\le t-1$ and when $I$ is the maximal ideal $\\mathfrak{m}$, both the inequalities become equalities. We also"},"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":"2606.18245","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AC","submitted_at":"2026-06-16T17:59:03Z","cross_cats_sorted":[],"title_canon_sha256":"8ea2007e353931f28557d3c45423d6b997eafdc4dcbd44715694b2b9c758ebd9","abstract_canon_sha256":"cb00a6acebbb755e892e09e3c644d09887c9755c9c89f27cd2c9f727fb5b50d6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:10:52.410878Z","signature_b64":"y7Q4YDZ+BexgpD115SoVtnx9+wlOt3KlyVEcWniWBCDSuBDCds8g6dydLGjHXUXnhWG5+0c/2rcxeAah7qtmBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1613a8b0f0658d4fe6f30feadec33bd2c101d861b9635c3f0b68b0992fc37a1c","last_reissued_at":"2026-06-19T16:10:52.410521Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:10:52.410521Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Derived functors and Hilbert polynomials over Gorenstein rings","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Satyabrata Paul, Tony J. Puthenpurakal","submitted_at":"2026-06-16T17:59:03Z","abstract_excerpt":"Let $(A,\\mathfrak{m},k)$ be a Gorenstein ring of dimension $d\\ge 1$, $N$ a perfect module of dimension $t\\ge 1$ and $I$ an ideal of definition of $N$. For a non-free maximal Cohen-Macaulay (=MCM) $A$-module $M$ and an integer $i\\ge 1$, it is well known that the functions $n \\mapsto \\ell(Tor_i^A(M,N/I^{n+1}N))$ and $n \\mapsto \\ell(Ext^i_A(M,N/I^{n+1}N))$ are of polynomial types of degrees $r_i^{I,N}(M)$ and $s_{I,N}^i(M)$, respectively. We prove that $r_i^{I,N}(M)\\le t-1$ and $s^i_{I,N}(M)\\le t-1$ and when $I$ is the maximal ideal $\\mathfrak{m}$, both the inequalities become equalities. 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