Deformation of F-injectivity

Determine whether F-injectivity deforms: namely, prove or disprove that a local ring R is F-injective whenever R/(z) is F-injective for a regular element z of R.

Background

The paper defines deformation of a property of local rings by requiring that the property pass from R/(z) to R whenever z is a regular element of R. It notes that F-rationality is known to deform, while F-nilpotence does not deform in general.

In contrast, the deformation behavior of F-injectivity is presented as unresolved. The question is significant because deformation properties help determine whether Frobenius-defined singularity classes behave well in families and under passage through regular hypersurface sections.

References

It is well-known, , that $F$-rationality deforms, and whether $F$-injectivity deforms remains difficult and still largely open question (,).

— $F$-depth and $F$-nilpotent rings: generalizations and applications  (2609.09558 - Maddox et al., 9 Sep 2026) in Section 2, subsection “F-nilpotent rings and their generalizations,” remark following the examples