Rings whose Frobenius is flat

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

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?

MathOverflow Asked by anon1432 on December 30, 2020

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