is an algebra retract of $k[C]$. This follows from the fact that $C \backslash F$ is a (prime) ideal. Let $P_{F}$ be an ideal generated by the monomials $X^{c}$ for $c \in C \backslash F$. Then $k[C]=k[C \cap F]+P_{F}$ and clearly $k[C \cap F] \cap P_{F}=0$. Hence $k[C \cap F]$ is an algebra retract of $k[C]$.

The next proposition shows that we can always reduce to the henselian local rings to study the $F$-coherent property of local rings.

Let $(R,{{\mathfrak m}})$ be a reduced local ring of characteristic $p>0$. Then $R$ is $F$-coherent if and only if the henselization $R^{\mathrm{h}}$ is so.

Since $R \to R^{\mathrm{h}}$ is faithfully flat, $R$ is $F$-coherent, if $R^{\mathrm{h}}$ is so. Conversely, assume that $R$ is $F$-coherent. Then we show that $F$-coherence passes to any finite étale extension $R \to S$. Since $S$ is étale over $R$, the natural ring homomorphism $R^{1/q}\otimes_{R} S  \to S^{1/q}$ is an isomorphism for all $q > 0$. Hence we have $R^{\infty} \otimes_{R} S \simeq S^{\infty}$. Since $R^{\infty} \otimes_{R} S$ is a module-finite free extension of $R^{\infty}$, it is coherent as well. We also proved that the localization of an $F$-coherent ring is $F$-coherent. Therefore, any standard étale extension of 