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On the Equality of Test Ideals

Ian Aberbach, Craig Huneke, Thomas Polstra

Abstract

We provide a natural criterion which implies equality of the finitistic test ideal and test ideal in local rings of prime characteristic. Most notably, we show that the criterion is met by every local weakly $F$-regular ring whose anti-canonical algebra is Noetherian on the punctured spectrum.

On the Equality of Test Ideals

Abstract

We provide a natural criterion which implies equality of the finitistic test ideal and test ideal in local rings of prime characteristic. Most notably, we show that the criterion is met by every local weakly -regular ring whose anti-canonical algebra is Noetherian on the punctured spectrum.
Paper Structure (9 sections, 23 theorems, 117 equations)

This paper contains 9 sections, 23 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Annihilators of Local Cohomology
  3. Tight closure, local cohomology, and local cohomology bounds
  4. Tight closure
  5. Local Cohomology Bounds
  6. Basic properties of local cohomology bounds
  7. Equality of test ideals
  8. Local Cohomology bounds and equality of test ideals
  9. $S_2$-ification, higher Ext-modules, and local cohomology

Key Result

Theorem A

Let $(R,\mathfrak{m},k)$ be an excellent Cohen-Macaulay normal domain of prime characteristic $p>0$, of Krull dimension $d$, and $I\subseteq R$ an anti-canonical ideal.An ideal $I\subseteq R$ is an anti-canonical ideal if it represents the inverse of the canonical divisor in the class group of $R$. for every $e\in\mathbb{N}$. If $R$ is weakly $F$-regular then $R$ is strongly $F$-regular.

Theorems & Definitions (50)

  • Conjecture 1.1: The weak implies strong conjecture
  • Theorem A
  • Remark 1.2
  • Remark 1.3
  • Conjecture 1.4
  • Theorem B
  • Remark 1.5
  • Theorem 2.1: BrodmannSharpBook
  • Lemma 2.2
  • proof
  • ...and 40 more