We explore the singularity classes F-nilpotent, weakly F-nilpotent, and generalized weakly F-nilpotent under faithfully flat local ring maps. As an application, we show that the loci of primes in a Noetherian ring of prime characteristic which define either weakly F-nilpotent or F-nilpotent local rings are open with respect to the Zariski topology whenever R is F-finite or essentially of finite type over an excellent local ring.
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Kenkel et al. (2023) studied this question.