{"paper":{"title":"Lech's inequality, the St\\\"{u}ckrad--Vogel conjecture, and uniform behavior of Koszul homology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Ilya Smirnov, Linquan Ma, Patricia Klein, Pham Hung Quy, Yongwei Yao","submitted_at":"2018-08-03T00:09:58Z","abstract_excerpt":"Let $(R,\\mathfrak{m})$ be a Noetherian local ring, and let $M$ be a finitely generated $R$-module of dimension $d$. We prove that the set $\\left\\{\\frac{l(M/IM)}{e(I, M)} \\right\\}_{\\sqrt{I}=\\mathfrak{m}}$ is bounded below by ${1}/{d!e(\\overline{R})}$ where $\\overline{R}=R/Ann(M)$. Moreover, when $\\widehat{M}$ is equidimensional, this set is bounded above by a finite constant depending only on $M$. The lower bound extends a classical inequality of Lech, and the upper bound answers a question of St\\\"{u}ckrad--Vogel in the affirmative. As an application, we obtain results on uniform behavior of th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.01051","kind":"arxiv","version":2},"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"}