{"paper":{"title":"A study of the length function of generalized fractions of modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Marcel Morales, Pham Hung Quy","submitted_at":"2014-05-28T13:54:48Z","abstract_excerpt":"Let $(R, \\frak m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module of dimension $d$. Let $\\underline{x} = x_1, ..., x_d$ be a system of parameters of $M$ and $\\underline{n} = (n_1, ..., n_d)$ a $d$-tuple of positive integers. In this paper we study the length of generalized fractions $M (1/(x_1, ..., x_d, 1))$ which was introduced by Sharp and Hamieh in \\cite{ShH85}. First, we study the growth of the function $J_{\\underline{x}, M}(\\underline{n}) = \\ell(M (1/(x_1^{n_1}, ..., x_d^{n_d}, 1))) - n_1...n_d e(\\underline{x};M)$. Then we give an explicit calculation for the function"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.7240","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":""},"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"}