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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"},"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":"1705.05033","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-05-14T22:46:23Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"a6b6d35195243f5139f71be4a74e11d698971b13e5cec2b4a7f6b3b6d0c24ef8","abstract_canon_sha256":"be893d27026c7f40f55b49c7c87d6625f9a05be087aedf2008f01151aac32398"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:32.047119Z","signature_b64":"A1BU3NesUV4LWdk5H8DlTs/TVF12W5pmvJuDHRni7RF3IclFeZuk6Jg1hwnCwh3qOEDBo3pnwkW1nlmHw1/cCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aacd86d3fec824da20187188c7aa4188ae5194b48a8dae41a6ec0a4f66b90fce","last_reissued_at":"2026-05-18T00:33:32.046625Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:32.046625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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