Lech’s inequality and the Stuckrad-Vogel conjecture

Patricia Klein – University of Kentucky

Abstract: Let $(R, m)$ be a Noetherian local ring, and let $M$ be a finitely generated $R$-module of dimension $d$. Let $e(I,M)$ denote the Hilbert-Samuel multiplicity of $M$ on the ideal $I$. Lech’s inequality states that the set ${\ell(R/I)/e(I,R)}$, as $I$ runs through all $m$-primary ideals, is bounded below by $1/d!e(m,R)$. Stuckrad and Vogel showed that this set is not in general bounded above. However, they conjectured that whenever the completion of $M$ is equidimensional that ${\ell(M/IM)/e(I,M)}$ will indeed be bounded above. We prove this conjecture. This talk is based on joint work with Linquan Ma, Pham Hung Quy, Ilya Smirnov, and Yongwei Yao.