the identity map on $R$.

Let $R \to S$ be a ring extension of reduced Noetherian rings of characteristic $p>0$ and suppose that $R$ is an algebra retract of $S$ and $S$ is $F$-coherent. Then $R$ is $F$-coherent.

Let $\phi:S \to R$ be a retraction map. Then we can extend $\phi$ to a ring homomorphism $\phi_{\infty}:S^{\infty} \to R^{\infty}$ such that the restriction $\phi_{\infty} \vert_{R^{\infty}}$ is the identity map as follows. The composition of ring homomorphisms: $$\begin{CD}
S^{1/q} @>\simeq >> S @>\phi>> R @>\simeq>> R^{1/q}, \\
\end{CD}$$ where the first and the third maps are given by the Frobenius maps, yields a compatible sequence of retraction maps $\phi_{e}$ with $\phi_{0}=\phi$ for every $q=p^e$ and taking its direct limit, we find that $\varinjlim_{e} \phi_{e}$ is the desired map. Now we apply ([@Glaz], Theorem 4.1.5) to conclude that $R^{\infty}$ is coherent, as desired.

I do not know if the above theorem holds, only assuming that $R$ is a direct summand of $S$ as an $R$-module. The situation in the above theorem is to be seen in the case of affine semigroup rings. Let $k[C]$ be an affine semigroup ring and let $F$ be a face of the cone $\mathbb{R}_{+}C$. Then $k[F \cap C]$ 