On the Equality of Test Ideals

TL;DR

This paper establishes a criterion ensuring the equality of finitistic and test ideals in local rings of prime characteristic, particularly for weakly F-regular rings with Noetherian anti-canonical algebras.

Contribution

It introduces a natural criterion for the equality of test ideals and demonstrates its applicability to a broad class of weakly F-regular rings.

Findings

Criterion implies equality of test ideals in certain local rings

All local weakly F-regular rings with Noetherian anti-canonical algebra meet the criterion

Enhances understanding of test ideal behavior in prime characteristic rings

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.