Stability and deformation of F-singularities

TL;DR

This paper investigates the stability of various F-singularities under small perturbations in local rings, establishing stability for some types and revealing instability for others, with connections to deformation theory.

Contribution

It proves $rak{m}$-adic stability for F-rationality universally, and for F-injectivity, F-purity, and strong F-regularity under specific conditions, clarifying their deformation behavior.

Findings

$rak{m}$-adic stability holds for F-rationality in all cases.

F-injectivity, F-purity, and strong F-regularity are not always stable.

Connections between stability and deformation phenomena are established.

Abstract

We study the problem of $\mathfrak{m}$-adic stability of F-singularities, that is, whether the property that a quotient of a local ring $(R,\mathfrak{m})$ by a non-zero divisor $x \in \mathfrak{m}$ has good F-singularities is preserved in a sufficiently small $\mathfrak{m}$-adic neighborhood of $x$. We show that $\mathfrak{m}$-adic stability holds for F-rationality in full generality, and for F-injectivity, F-purity and strong F-regularity under certain assumptions. We show that strong F-regularity and F-purity are not stable in general. Moreover, we exhibit strong connections between stability and deformation phenomena, which hold in great generality.