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arXiv · 1808.01051

Lech's inequality, the St\"{u}ckrad--Vogel conjecture, and uniform behavior of Koszul homology

Patricia Klein·Linquan Ma·Pham Hung Quy·Ilya Smirnov·Yongwei Yao
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Abstract

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 the lengths of Koszul homology modules.

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Patricia Klein, Linquan Ma, Pham Hung Quy, Ilya Smirnov, Yongwei Yao. 2018-08-03. Lech's inequality, the St\"{u}ckrad--Vogel conjecture, and uniform behavior of Koszul homology. https://arxiv.org/abs/1808.01051

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