Stability and deformation of F-singularities

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.

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References in corpus (3)

• F-signature exists
• Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras
• An analog of adjoint ideals and PLT singularities in mixed characteristic

Cited by in corpus (3)

• Arithmetic and geometric deformations of $F$-pure and $F$-regular singularities
• $F$-purity deforms in $\mathbb{Q}$-Gorenstein rings
• $\mathfrak{m}$-adic Perturbations in Noetherian Local Rings