{"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"}