Symbolic power containments in singular rings in positive characteristic

TL;DR

This paper investigates the containment problem between symbolic and ordinary powers of ideals in singular rings of positive characteristic, extending known results and developing new criteria for $F$-purity and strong $F$-regularity.

Contribution

It introduces Fedder and Glassbrenner type criteria for $F$-purity and strong $F$-regularity in singular rings and extends prime characteristic containment results to these settings.

Findings

Developed criteria for $F$-purity and strong $F$-regularity in singular rings.

Extended containment results to ideals of finite and infinite projective dimension.

Proved a variation of the containment still holds in singular rings.

Abstract

The containment problem for symbolic and ordinary powers of ideals asks for what values of $a$ and $b$ we have $I^{(a)} \subseteq I^b$. Over a regular ring, a result by Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede partially answers this question, but the containments it provides are not always best possible. In particular, a tighter containment conjectured by Harbourne has been shown to hold for interesting classes of ideals - although it does not hold in general. In this paper, we develop a Fedder (respectively, Glassbrenner) type criterion for $F$-purity (respectively, strong $F$-regularity) for ideals of finite projective dimension over $F$-finite Gorenstein rings and use our criteria to extend the prime characteristic results of Grifo-Huneke to singular ambient rings. For ideals of infinite projective dimension, we prove that a variation of the containment still holds, in the spirit of work by Hochster-Huneke and Takagi.