{"paper":{"title":"Length of local cohomology of powers of ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Hailong Dao, Jonathan Monta\\~no","submitted_at":"2017-05-14T22:46:23Z","abstract_excerpt":"Let $R$ be a polynomial ring over a field $k$ with irrelevant ideal $\\frak m$ and dimension $d$. Let $I$ be a homogeneous ideal in $R$. We study the asymptotic behavior of the length of the modules $H^{i}_{\\frak m}(R/I^n)$ for $n\\gg 0$. We show that for a fixed number $\\alpha \\in \\mathbb Z$, $\\limsup_{n\\rightarrow \\infty}\\frac{\\lambda(H^{i}_{\\frak m}(R/I^n)_{\\geq -\\alpha n})}{n^d}<\\infty.$ Combining this with recent strong vanishing results gives that $\\limsup_{n\\rightarrow \\infty}\\frac{\\lambda(H^{i}_{\\frak m}{R/I^n})}{n^d}<\\infty$ in many situations. We also establish that the actual limit ex"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.05033","kind":"arxiv","version":3},"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"}