HOẠT ĐỘNG TRONG TUẦN

Thời gian: 9h00, Thứ 4, Ngày 3/10/2018,
Địa điểm: Phòng 611-612 Tầng 6 Nhà A6
Tóm tắt: (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