{"paper":{"title":"Graded components of local cohomology modules over polynomial rings","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Tony J. Puthenpurakal","submitted_at":"2024-11-20T07:39:04Z","abstract_excerpt":"Let $K$ be a field and let $R = K[X_1, \\ldots, X_m]$ with $m \\geq 2$. Give $R$ the standard grading. Let $I$ be a homogeneous ideal of height $g$. Assume $1 \\leq g \\leq m -1$. Suppose $H^i_I(R) \\neq 0$ for some $i \\geq 0$. We show\n  (1) $H^i_I(R)_n \\neq 0$ for all $n \\leq -m$.\n  (2) if Supp $H^i_I(R) \\neq \\{ (X_1, \\ldots, X_m)\\}$ then $H^i_I(R)_n \\neq 0$ for all $n \\in \\mathbb{Z}$. Furthermore if char $K = 0$ then $\\dim_K H^i_I(R)_n$ is infinite for all $n \\in \\mathbb{Z}$.\n  (3) $\\dim_K H^g_I(R)_n$ is infinite for all $n \\in \\mathbb{Z}$.\n  In fact we prove our results for $\\mathcal{T}(R)$ wher"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.13090","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/2411.13090/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"}