Stability and deformation of F-singularities

We study the problem of m-adic stability of F-singularities, that is, whether the property that a quotient of a local ring ( \(R,\mathfrak{m}\) $$R,\mathfrak{m}$$ ) by a non-zero divisor \(x\in\mathfrak{m}\) $$x\in \mathfrak{m}$$ has good F-singularities is preserved in a sufficiently small \(\mathfrak{m}\) $$\mathfrak{m}$$ -adic neighborhood of x. We show that \(\mathfrak{m}\) $$\mathfrak{m}$$ -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.