{"paper":{"title":"Cohomology Vanishing theorems over some rings containing nilpotents","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Tony J. Puthenpurakal","submitted_at":"2025-04-18T09:08:45Z","abstract_excerpt":"(1) Let $(A,\\mathfrak{m})$ be complete Noetherian local ring of dimension $d$ and let $P$ be a prime ideal with $G_P(A) = \\bigoplus_{n \\geq 0}P^n/P^{n+1}$ a domain. Fix $r \\geq 1$. If $J$ is a homogeneous ideal of $G_{P^r}(A)$ with $\\text{dim} \\ G_{P^r}(A)/J > 0$ then the local cohomology module $H^d_J(G_{P^r}(A)) = 0$.\n  (2) Let $A = K[[X_1, \\ldots,X_d]]$ and let $\\mathfrak{m} = (X_1, \\ldots, X_d)$. Assume $K$ is separably closed. Fix $r \\geq 1$. Let $J$ be a homogeneous ideal of $G_{\\mathfrak{m}^r}(A)$. We show that local cohomology modules $H^{j}_J(G_{\\mathfrak{m}^r}(A)) = 0$ for $j \\geq d "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.13566","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.13566/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}