Uniform Lech's inequality

2022/03/13 by Ma, Linquan, Smirnov, Ilya
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2203.06739

Abstract

Let (R,\mathfrakm) be a Noetherian local ring of dimension d≥ 2. We prove that if e(\widehatR_red)>1, then the classical Lech's inequality can be improved uniformly for all \mathfrakm-primary ideals, that is, there exists ε>0 such that e(I)≤ d!(e(R)-ε)ℓ(R/I) for all \mathfrakm-primary ideals I⊆ R. We also obtain partial results towards improvements of Lech's inequality when we fix the number of generators of I.

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