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,m) by a non-zero divisor x is an element of m has good F-singularities is preserved in a sufficiently small m-adic neighborhood of x. We show that 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.