# 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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