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

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 ${\left\{\frac{l(M\mathrm{/}IM)}{e(I,M)}\right\}}_{\sqrt{I}=\mathfrak{m}}$ is bounded below by $1\mathrm{/}d!e(\bar{R})$ where $\bar{R}=R\mathrm{/}Ann(M)$. 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.