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The vanishing of the first tight Hilbert coefficient for Buchsbaum rings

Duong Thi Huong, Nguyen Tuan Long, Pham Hung Quy

TL;DR

The paper investigates when the vanishing of the first tight Hilbert coefficient $e_1^*(Q)$ forces $F$-rationality in local rings of positive characteristic. It proves that if $R$ is excellent, reduced, Buchsbaum and satisfies $(S_2)$, then $e_1^*(Q)=0$ for some parameter ideal $Q$ implies that $R$ is $F$-rational, with the converse following from known results. The argument hinges on the inequality $e_1^*(Q)\ge e_0(Q)-\ell(R/Q^*)+e_1(Q)$ and a detailed analysis of limit closures $Q^{\lim}$ together with local cohomology data; the equality case further yields Cohen–Macaulayness and thus $F$-rationality. The authors also show that the $(S_2)$ condition cannot be dropped and discuss extensions to mixed characteristic via big Cohen–Macaulay approaches.

Abstract

We prove that if the first tight Hilbert coefficient vanishes then ring is $F$-rational provided it is a Buchsbaum local ring satisfying the $(S_2)$ condition.

The vanishing of the first tight Hilbert coefficient for Buchsbaum rings

TL;DR

The paper investigates when the vanishing of the first tight Hilbert coefficient forces -rationality in local rings of positive characteristic. It proves that if is excellent, reduced, Buchsbaum and satisfies , then for some parameter ideal implies that is -rational, with the converse following from known results. The argument hinges on the inequality and a detailed analysis of limit closures together with local cohomology data; the equality case further yields Cohen–Macaulayness and thus -rationality. The authors also show that the condition cannot be dropped and discuss extensions to mixed characteristic via big Cohen–Macaulay approaches.

Abstract

We prove that if the first tight Hilbert coefficient vanishes then ring is -rational provided it is a Buchsbaum local ring satisfying the condition.
Paper Structure (6 sections, 12 theorems, 50 equations)

This paper contains 6 sections, 12 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hilbert coefficients
  4. Buchsbaum and tight Buchsbaum rings
  5. Limit closure of a parameter ideal
  6. The main results

Key Result

Theorem 1.2

Conjecture 1 holds if we further assume that R is reduced and Buchsbaum.

Theorems & Definitions (27)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 2.1: VMSVGGH10
  • Theorem 2.2: GHMMQ24
  • Theorem 2.3: MQ24GMV
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 17 more