$s_{1},\ldots,s_{n}$. We may then apply this fact together with ([@Mat], Theorem 7.4) to conclude that $(0:_{R}a)$ and $I \cap J$ are both finitely generated for finitely generated ideals $I$, $J$ and $a \in R$. So if $S$ is $F$-coherent, so is $R$.

It is not clear as to what to expect on the topology of the $F$-coherent locus of ${\operatorname{Spec}}R$ in general. For example, if $R$ is an excellent domain, then the $F$-coherent locus contains an open subset, since the regular locus is open.

As a natural extension of the argument used to show that a regular ring is $F$-coherent, it can be shown that the flat colimit of $F$-coherent rings is $F$-coherent, if the colimit is Noetherian. In fact, this follows easily from the fact that the perfect closure of a ring $R$ of characteristic $p>0$ is obtained as the colimit: $$\begin{CD}
\varinjlim \{R @>F>> R @>F>> R @>F>> \cdots \},
\end{CD}$$ where the map is the Frobenius map, together with the fact that if the map from one direct system to another system is flat, its colimit is also flat.

Although the next corollary is simple, it serves as a useful way for producing sufficiently many $F$-coherent rings.

