Lech's inequality, the Stückrad--Vogel conjecture, and uniform behavior of Koszul homology

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 $\hat{M}$ 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 Stückrad--Vogel in the affirmative. As an application, we obtain results on uniform behavior of the lengths of Koszul homology modules.

24 pages, minor changes. To appear in Advances in Mathematics