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

Abstract

Let (R,m) be a Noetherian local ring, and let M be a finitely generated R-module of dimension d. We prove that the set {l(M/IM)e(I,M)}I=m is bounded below by 1/d!e(R‾) where R‾=R/Ann(M). Moreover, when Mˆ 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.