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Strongly Lech-independent ideals and Lech's conjecture

Cheng Meng

TL;DR

This work introduces strong Lech-independence as a natural strengthening of Lech-independence to study Hilbert-Samuel multiplicities under flat local extensions. The authors develop a framework based on standard sets, expansion properties, and the asymptotic Samuel function to derive precise multiplicity inequalities, including a lower bound $e(S) \ge e(R) t_1\cdots t_{r-d}$ in standard graded contexts and an upper bound $e(S) \le e(I)/(q_1\cdots q_d)$ in equal characteristic. They show that when $S$ is standard graded up to completion and $I=\mathfrak{m}S$ is strongly Lech-independent, Lech's conjecture holds in this setting, advancing the understanding of how base ring singularities compare under flat extensions. The results provide a versatile toolkit for bounding multiplicities via monomial data, liftings, and asymptotic invariants, with potential to extend to broader classes of local extensions and graded structures.

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 $e(R) \leq e(S)$.

Strongly Lech-independent ideals and Lech's conjecture

TL;DR

This work introduces strong Lech-independence as a natural strengthening of Lech-independence to study Hilbert-Samuel multiplicities under flat local extensions. The authors develop a framework based on standard sets, expansion properties, and the asymptotic Samuel function to derive precise multiplicity inequalities, including a lower bound in standard graded contexts and an upper bound in equal characteristic. They show that when is standard graded up to completion and is strongly Lech-independent, Lech's conjecture holds in this setting, advancing the understanding of how base ring singularities compare under flat extensions. The results provide a versatile toolkit for bounding multiplicities via monomial data, liftings, and asymptotic invariants, with potential to extend to broader classes of local extensions and graded structures.

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 is a flat local extension of local rings with , the completion of is the completion of a standard graded ring over a field with respect to the homogeneous maximal ideal, and the completion of is the completion of a homogeneous ideal, then .
Paper Structure (5 sections, 23 theorems, 59 equations)

This paper contains 5 sections, 23 theorems, 59 equations.

Table of Contents

  1. introduction
  2. Preliminaries, notations and assumptions
  3. Standard sets in a polynomial ring
  4. strong Lech-independence and expansion property
  5. strong Lech-independence and inequalities on multiplicities of ideals

Key Result

Lemma 2.2

Let $d$ be a positive integer, $a(z) = \sum_{i \geq 0} a_iz^i \in \mathbb{R}[z]_{(z)}$ be a rational series satisfying the following properties: (P1) $a(z)$ only has poles at roots of unity; (P2$_d$) $z = 1$ is a pole of $a(z)$ with order $d$; (P3$_d$) The orders of poles of $a(z)$ except for $1$ ar The first limit is understood by viewing $\sum_{i \geq 0} a_iz^i$ as a meromorphic function on $\ma

Theorems & Definitions (59)

  • Conjecture 1.1
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Proposition 3.5: vasconcelos2004computational, Stanley decomposition
  • Proposition 3.6: vasconcelos2004computational
  • ...and 49 more