Commutative Algebra Seminar

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 in the affirmative. 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. Speaker(s): Linquan Ma (Purdue University)