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$F$-Volumes

Wágner Badilla-Céspedes, Luis Núñez-Betancourt, Sandra Rodríguez-Villalobos

Abstract

In this work we define a numerical invariant called $F$-volume. This number extends the definition of $F$-threshold of a pair of ideals $I$ and $J$, $c^J(I)$ to a sequence of ideals $J$, $I_1, \ldots, I_t$. We obtain several properties that emulate those of the $F$-threshold. In particular, the $F$-volume detects $F$-pure complete intersections. In addition, we relate this invariant to the Hilbert-Kunz multiplicity.

$F$-Volumes

Abstract

In this work we define a numerical invariant called -volume. This number extends the definition of -threshold of a pair of ideals and , to a sequence of ideals , . We obtain several properties that emulate those of the -threshold. In particular, the -volume detects -pure complete intersections. In addition, we relate this invariant to the Hilbert-Kunz multiplicity.

Paper Structure

This paper contains 5 sections, 21 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Existence and definition
  3. First properties
  4. Properties for $F$-pure rings
  5. Relations with Hilbert-Kunz multiplicities

Key Result

Theorem 1

Let $\underline{I}=I_1,\ldots, I_t\subseteq R$ be a sequence of ideals, and $J\subseteq R$ be an ideal such that $I_1,\ldots, I_t\subseteq \sqrt{J}$. Let Then, the limit converges, and it is called the $F$-volume of $\underline{I}$ with respect to $J$.

Theorems & Definitions (55)

  • Theorem 1: see Theorem \ref{['ThmLimitExistsSec']} and Definition \ref{['def1']}
  • Theorem 2: see Theorem \ref{['ThmFpureCI']}
  • Theorem 3: see Theorem \ref{['ThmHK']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Definition 2.6
  • Example 2.8
  • Lemma 2.9
  • ...and 45 more