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

Time: 9h00, Wednesday, October 3, 2018
: Rom 611-612, Building A6, Institute of Mathematics

Abstract: (Join work with Patricia Klein, Linquan Ma, Ilya Smirnov and Yongwei Yao)
Let $(R,fm)$ be a Noetherian local ring, and let $M$ be a finitely generated $R$-module of dimension $d$. We prove that the set $left{frac{l(M/IM)}{e(I, M)} right}_{sqrt{I}=fm}$ 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.