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Let $R$ be a ring of characteristic $p>0$. The (absolute) Frobenius is the map of rings $F_R:R\rightarrow R$ defined by $x\mapsto x^p$.

I am interested in rings for which $F_R$ is flat (hence faithfully flat). Here are some families of examples of rings $R$ with this property.

-Regular rings (Kunz)

-Perfect rings (rings for which $F_R$ is an isomorphism)

-Valuation rings (see Theorem 3.1 of "Frobenius and valuation rings" -Datta, Smith)

In fact, Kunz famously showed that if $R$ is noetherian, then $F_R$ is flat if and only if $R$ is regular. Note that rings of the latter two types are rarely noetherian. Furthermore, it is not hard to see that the above list is by no means exhaustive.

I would like to know what is known about the class of rings with $F_R$ flat in general (without noetherian hypothesis). Specifically, is there is a non-tautological characterization of the class of (not necessarily noetherian) rings such that $F_R$ is flat (ala Kunz)? Or perhaps some characterization among a large class of rings properly containing the class of noetherian rings?

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    $\begingroup$ This is probably a question better suited for MO. Any questions involving removal of Noetherian hypotheses are pretty niche, and are much more likely to receive traction at MO. $\endgroup$
    – Alex Youcis
    Commented Aug 25, 2020 at 18:55
  • $\begingroup$ I agree with Alex Youcis; Datta-Smith at the very least is maaaybe 5 years old. This lies firmly in the realm of research. $\endgroup$
    – Tabes Bridges
    Commented Aug 25, 2020 at 19:00

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