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On Partial Trace Ideals

Souvik Dey, Shinya Kumashiro

Abstract

We investigate the notion of partial trace ideals, recently introduced by Maitra. We first establish several properties of partial trace ideals and give affirmative answers to questions posed by Maitra. We then study the invariant defined by the partial trace ideal of the canonical module, and obtain an upper bound that recovers one direction of a result of Kobayashi. Moreover, in the case of numerical semigroup rings generated by three elements, we provide an explicit formula for this invariant.

On Partial Trace Ideals

Abstract

We investigate the notion of partial trace ideals, recently introduced by Maitra. We first establish several properties of partial trace ideals and give affirmative answers to questions posed by Maitra. We then study the invariant defined by the partial trace ideal of the canonical module, and obtain an upper bound that recovers one direction of a result of Kobayashi. Moreover, in the case of numerical semigroup rings generated by three elements, we provide an explicit formula for this invariant.
Paper Structure (6 sections, 20 theorems, 41 equations)

This paper contains 6 sections, 20 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Trace ideals and partial trace ideals
  3. Trace ideals
  4. Partial trace ideals
  5. Discussions on the h-invariant of the canonical module
  6. three-generated numerical semigroup rings

Key Result

Theorem 1.2

(Proposition p21 and Theorem 1dim) Let $S=R\setminus \bigcup_{\mathfrak{p}\in \operatorname{Spec} (R)\setminus \operatorname{Max} (R)}\mathfrak{p}$ be a multiplicatively closed subset of $R$. Consider the following conditions. Then, (1)$\Leftrightarrow$(2)$\Rightarrow$(3)$\Leftrightarrow$(4) hold. (3)$\Rightarrow$(2) holds if $R$ is local and $\dim R=1$.

Theorems & Definitions (59)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.4
  • Remark 2.5
  • proof
  • Proposition 2.6
  • proof
  • Lemma 2.7
  • ...and 49 more