Regular rings with F-finite field of fractions

Question

Let $S$ be a regular domain of characteristic $p>0$ with fraction field $K$. Assume that $K$ is $F$-finite, meaning that $K$ is a finite module over $K^p$. Does it follow that $S$ is also $F$-finite?

Diego

Answer 1

I believe this is false. [Datta–Smith, Ex. 4.5.1] give an example of a DVR that is not $F$-finite, whose fraction field is $\mathbf{F}_p(x,y)$.

A way to produce more examples is the following:

Proposition [Datta–Smith, Prop. 2.6.1]. A noetherian domain of positive characteristic is $F$-finite if and only if it is excellent and its fraction field is $F$-finite.

Thus, any non-excellent regular domain of positive characteristic with $F$-finite fraction field is a counterexample to your claim.

There are many examples of non-excellent regular domains in the literature: see [Nagata, (E3.3)], [Matsumura, (34.B)], and [Raynaud, Exp. I, §11] for more examples. It was not clear to me, however, which of those existing examples have $F$-finite function field. On the other hand, combining Prop. 11.6 in Raynaud with Datta–Smith's proposition above gives a systematic way to construct non-excellent DVR's with $F$-finite fraction field.

Edit. Datta–Smith recent posted another preprint, which discusses these questions in more detail. See §1.2 and §3 in particular.

Answer 2

I believe the result must be true though I don't have the answer right now, but maybe this can help (or not):

Denote $\phi: A\rightarrow A$, $x\mapsto x^p$ the Frobenius morphism, and still denote $\phi: K\rightarrow K$ its extension to $K$.

Let $n\ge 1$ be an integer and assume $\mathcal{B}=\{f_1, ..., f_n\}$ is a $\phi(K)$-basis of $K$. We can always assume that $f_1$, ..., $f_n$ belong to $A$ (if $f_i=\frac{a_i}{b_i}$, $a_i$, $b_i\in A$, then $\{b_1^{p-1}a_1, ..., b_n^{p-1}a_n\}$ is still a $\phi(K)$-basis of $K$).

Identify $K$ to $\phi(K)^n$ through $\mathcal{B}$, and denote $M\subset A$ the $\phi(A)$-module (free) of rank $n$ generated by $\mathcal{B}$. Then $A/M$ is a torsion module over $\phi(A)$, by assumption. Now I think we have to look for properties about torsion modules over regular domains (maybe first assume that $R$ is local ?), which I don't know much (but I'll look at it).

Hope this helped a bit.