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

TL;DR

This paper establishes bounds on the ratios of lengths to multiplicities in Noetherian local rings, extending classical inequalities and confirming conjectures, with applications to the uniform behavior of Koszul homology modules.

Contribution

It extends Lech's inequality, confirms the St"uckrad--Vogel conjecture, and analyzes the uniform behavior of Koszul homology in local rings.

Findings

Lower bound for length/multiplicity ratios depending on ring dimension

Upper bound confirming St"uckrad--Vogel conjecture

Results on uniform behavior of Koszul homology modules

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.