Uniform Lech's inequality

Abstract

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d\ge 2$. We prove that if $e({\hat{R}}_{red})>1$, then the classical Lech's inequality can be improved uniformly for all $\mathfrak{m}$-primary ideals, that is, there exists $\epsilon >0$ such that $e(I)\le d!(e(R)-\epsilon )\ell (R/I)$ for all $\mathfrak{m}$-primary ideals $I\subseteq R$. We also obtain partial results towards improvements of Lech's inequality when we fix the number of generators of $I$.

arXiv comment11 pages, final version