ring $(\mathbb{Z}/p\mathbb{Z})[t^{2},t^{3}]$ is $F$-coherent, due to Corollary \[coherent-3\]. Hence the generic fiber of ${\operatorname{Spec}}\mathbb{Z}[x,y]/(x^{3}-y^{2}) \to {\operatorname{Spec}}\mathbb{Z}$ is supposed to be of $F$-coherent type. On the other hand, let us take $\mathbb{Z}[x,y]/(y^{2}-x^{3}-x^{2})$, an ordinary double point. Then $\mathbb{Z}[x,y]/(y^{2}-x^{3}-x^{2}) \simeq \mathbb{Z}[t,t\sqrt{t+1}]$, so we have that $(\mathbb{Z}/p\mathbb{Z})[t,t\sqrt{t+1}] \to (\mathbb{Z}/p\mathbb{Z})[t,\sqrt{t+1}]$ is the normalization map and purely inseparable only when $p=2$. In view of Corollary \[coherent-5\], the property of fibers being $F$-coherent can be isolated in the arithmetic deformation.

Tight closure in perfect rings and coherence
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In this section, we show that Brenner and Monsky’s recent example [@BM] that tight closure does not commute with localization is related to the coherence of certain perfect rings. While the failure of coherence for the absolute integral closure was already found by Aberbach and Hochster [@AH], the main idea discussed in this section has the advantage that it can throw some light on the nature of general perfect rings.

We first establish some basic properties of tight closure of finitely generated ideals on certain perfect rings. Let $A$ be a perfect ring, that is, the Frobenius map is bijective on $A$. Then the Frobenius functor $\mathbf{F}_{A}$ is obviously flat.

\[perfect\] Let $A$ be a perfect coherent ring 