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.
Categories: math.AC math.AG