A generalization of an inequality of Lech relating multiplicity and colength

TL;DR

This paper explores generalizations of Lech's inequality connecting multiplicity and colength of ideals in local rings, proving a specific case in dimension three and proposing new related formulas.

Contribution

It introduces new conjectures and proves a case of the generalized inequality in three dimensions, extending understanding of multiplicity and colength relations.

Findings

Proved a generalized Lech inequality in dimension three.

Established a weaker version of the conjecture in all dimensions.

Proposed a new Lech-type formula relating multiplicity and number of generators.

Abstract

We study conjectured generalizations of a formula of Lech which relates the multiplicity of a finite colength ideal in an equicharacteristic local ring to its colength, and prove one of these generalizations involving the multiplicity of the maximal ideal times the finite colength ideal. We also propose a Lech-type formula that relates multiplicity and the number of generators. We prove the conjecture in dimension three and establish a weaker result in full generality.