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If $e>2$ and $M$, $N$ are finitely generated $R$-modules, we show that the formal power series $\\sum_{i=0}^\\infty\\text{rank}_{k}\\left(\\text{Ext}^i_R(M,N)\\otimes_R k \\right)t^i$ and $\\sum_{i=0}^\\infty\\text{rank}_{k}\\left(\\text{Tor}_i^R(M,N)\\otimes_R k \\right)t^i$ are rational, with denominator $1-et+t^2$."},"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":"1601.00930","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-01-05T18:35:02Z","cross_cats_sorted":[],"title_canon_sha256":"b5e201e7731ab9f0e7c0529cab7055384c0c1919d9c9e449a39f30a3ecb7698c","abstract_canon_sha256":"54c469849a304d98609fa4e64fd573063a4c06f8f1a9eac9fbd95b113d35b9a7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:21.285231Z","signature_b64":"98mDD2IGrSu0+i84CVHHGREYBdGQRwjM3tAL3Ga0qGTubnLk6olT/uj32jKrZmI6vfvjNOnp1owzI+ODr2cmDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7e9daa599167d2284f943e63c2600725b209c89a83752e311ca4ee8d89caec6","last_reissued_at":"2026-05-18T01:23:21.284614Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:21.284614Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cohomology of finite modules over short Gorenstein rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Liana Sega, Melissa Menning","submitted_at":"2016-01-05T18:35:02Z","abstract_excerpt":"Let $R$ be a Gorenstein local ring with maximal ideal $\\mathfrak{m}$ satisfying $\\mathfrak{m}^3=0\\ne\\mathfrak{m}^2$. Set $k=R/\\mathfrak{m}$ and $e=\\text{rank}_{k}(\\mathfrak{m}/\\mathfrak{m}^2)$. If $e>2$ and $M$, $N$ are finitely generated $R$-modules, we show that the formal power series $\\sum_{i=0}^\\infty\\text{rank}_{k}\\left(\\text{Ext}^i_R(M,N)\\otimes_R k \\right)t^i$ and $\\sum_{i=0}^\\infty\\text{rank}_{k}\\left(\\text{Tor}_i^R(M,N)\\otimes_R k \\right)t^i$ are rational, with denominator $1-et+t^2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.00930","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":"1601.00930","created_at":"2026-05-18T01:23:21.284734+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.00930v1","created_at":"2026-05-18T01:23:21.284734+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.00930","created_at":"2026-05-18T01:23:21.284734+00:00"},{"alias_kind":"pith_short_12","alias_value":"U7U5VJMZCZ6S","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_16","alias_value":"U7U5VJMZCZ6SFBHZ","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_8","alias_value":"U7U5VJMZ","created_at":"2026-05-18T12:30:46.583412+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/U7U5VJMZCZ6SFBHZIPTDYJQAOJ","json":"https://pith.science/pith/U7U5VJMZCZ6SFBHZIPTDYJQAOJ.json","graph_json":"https://pith.science/api/pith-number/U7U5VJMZCZ6SFBHZIPTDYJQAOJ/graph.json","events_json":"https://pith.science/api/pith-number/U7U5VJMZCZ6SFBHZIPTDYJQAOJ/events.json","paper":"https://pith.science/paper/U7U5VJMZ"},"agent_actions":{"view_html":"https://pith.science/pith/U7U5VJMZCZ6SFBHZIPTDYJQAOJ","download_json":"https://pith.science/pith/U7U5VJMZCZ6SFBHZIPTDYJQAOJ.json","view_paper":"https://pith.science/paper/U7U5VJMZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.00930&json=true","fetch_graph":"https://pith.science/api/pith-number/U7U5VJMZCZ6SFBHZIPTDYJQAOJ/graph.json","fetch_events":"https://pith.science/api/pith-number/U7U5VJMZCZ6SFBHZIPTDYJQAOJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/U7U5VJMZCZ6SFBHZIPTDYJQAOJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/U7U5VJMZCZ6SFBHZIPTDYJQAOJ/action/storage_attestation","attest_author":"https://pith.science/pith/U7U5VJMZCZ6SFBHZIPTDYJQAOJ/action/author_attestation","sign_citation":"https://pith.science/pith/U7U5VJMZCZ6SFBHZIPTDYJQAOJ/action/citation_signature","submit_replication":"https://pith.science/pith/U7U5VJMZCZ6SFBHZIPTDYJQAOJ/action/replication_record"}},"created_at":"2026-05-18T01:23:21.284734+00:00","updated_at":"2026-05-18T01:23:21.284734+00:00"}