{"paper":{"title":"On the asymptotic linearity of reduction number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Dancheng Lu","submitted_at":"2016-08-20T02:29:21Z","abstract_excerpt":"Let $R$ be a standard graded Noetherian algebra over an infinite field $K$ and $M$ a finitely generated $\\mathbb{Z}$-graded $R$-module. Then for any graded ideal $I\\subseteq R_+$ of $R$, we show that there exist integers $e_1\\geq e_2$ such that $r(I^nM)=\\rho_I(M)n+e_1$ and $D(I^nM)=\\rho_I(M)n+e_2$ for $n\\gg0$. Here $r(M)$ and $D(M)$ denote the reduction number of $M$ and the maximal degree of minimal generators of $M$ respectively, and $\\rho_I(M)$ is an integer determined by both $M$ and $I$. We introduce the notion of generalized regularity function $\\Gamma$ for a standard graded algebra over"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.05769","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"}