A sufficient condition for F-purity

TL;DR

This paper establishes a new sufficient condition for F-purity in local rings based on properties of the canonical ideal and the quotient ring, advancing understanding in the area of F-singularities.

Contribution

It proves that an equidimensional S_2 local ring with a canonical ideal whose quotient is F-pure is itself F-pure, providing a new criterion for F-purity.

Findings

R/I being F-pure implies R is F-pure under certain conditions

Examples show not all Cohen-Macaulay F-pure rings satisfy this property

The result links canonical modules to F-purity in local rings

Abstract

It is well known that nice conditions on the canonical module of a local ring have a strong impact in the study of strong F-regularity and F-purity. In this note, we prove that if (R,m) is an equidimensional and S_2 local ring that admits a canonical ideal I such that R/I is F-pure, then R is F-pure. We also provide examples to show that not all Cohen-Macaulay F-pure local rings satisfy this property.