Strongly Lech-independent ideals and Lech's conjecture

TL;DR

This paper introduces strongly Lech-independent ideals to generalize Lech-independent ideals, deriving new inequalities on multiplicities and providing partial results related to Lech's conjecture in commutative algebra.

Contribution

It defines strongly Lech-independent ideals and uses this concept to establish inequalities on multiplicities in flat local extensions, advancing understanding of Lech's conjecture.

Findings

Established inequality e(R) ≤ e(S) under specific conditions

Generalized Lech-independent ideals to strongly Lech-independent ideals

Provided new tools for analyzing multiplicities in local ring extensions

Abstract

We introduce the notion of strongly Lech-independent ideals as a generalization of Lech-independent ideals defined by Lech and Hanes, and use this notion to derive inequalities on multiplicities of ideals. In particular we prove that if $(R,\mathfrak{m}) \to (S,\mathfrak{n})$ is a flat local extension of local rings with $\dim R = \dim S$, the completion of $S$ is the completion of a standard graded ring over a field $k$ with respect to the homogeneous maximal ideal, and the completion of $\mathfrak{m}S$ is the completion of a homogeneous ideal, then $e(R) \leq e(S)$.