TItle: Lech's inequality and a conjecture of Stuckrad-Vogel
 
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 ${l(M/IM)/e(I,M)}$, when $I$ runs through all $m$-primary ideals, is bounded below by $1/d!e(R)$. Moreover, when the completion of $M$ is equidimensional, this set is bounded above by a finite constant depending only on $M$. This extends a classical inequality of Lech and answers a question of Stuckrad-Vogel. Our main tool is to use Vasconcelos's homological degree. The talk is based on joint work with Patricia Klein, Pham Hung Quy, Ilya Smirnov, and Yongwei Yao.