\[coherent-1\] Let $R$ be a Noetherian ring of characteristic $p>0$.

1.  Any regular ring is $F$-coherent.

2.  Let $S$ be a multiplicative subset of an $F$-coherent ring $R$. Then the localization $S^{-1} R$ is $F$-coherent as well.

3.  If $U=\{{{\mathfrak p}}\in {\operatorname{Spec}}R~|~R_{{{\mathfrak p}}}~\mbox{is $F$-coherent}\}$ is constructible, then $U$ is a non-empty Zariski open subset.

4.  Let $R \subseteq S$ be a faithfully flat extension. Then if $S$ is $F$-coherent, so is $R$.

For $(1)$, we note that $R \to R^{1/p} \to \cdots \to R^{1/p^e} \to \cdots$ is a direct system with flat transition maps by using a theorem of Kunz [@Ku], so that $R^{\infty}$ is coherent since the flat colimit of coherent rings is coherent.

For $(2)$, let $S$ be a multiplicative subset of $R$. We show that there is an isomorphism: $(S^{-1} R)^{\infty} \simeq S^{-1}(R^{\infty})$. Since the problem depends only on $R_{\mathrm{red}}$, we may assume that $R$ is already reduced. Then the natural map $R \to R^{\infty}$ extends to an injection $S^{-1}R \to S^{-1}(R^{\infty})$. Since this map is purely inseparable and the localization of a perfect ring is again perfect, the required isomorphism follows. On the other hand, the property of coherence is stable under localization, so the isomorphism 