an interesting class of $F$-coherent rings.

These observations also raise the following question. Let $(R,{{\mathfrak m}})$ be a local Noetherian ring of positive characteristic. Then under which condition of $R$ is a system of parameters of $R$ a regular sequence on the perfect closure $R^{\infty}$, or is $R^{\infty}$ a big Cohen-Macaulay $R$-algebra?

We prove by using a valuation, that if $R$ is $F$-coherent, then every system of parameters of $R$ is a regular sequence on $R^{\infty}$. This result is seen to be related to the recent work by Roberts, Singh, and Srinivas. Quite generally, if $R$ is any complete local domain of positive characteristic, then $R^{\infty}$ is almost Cohen-Macaulay ([@RSS] for this terminology). Note, however, that even if $R^{\infty}$ is almost Cohen-Macaulay, it does not necessarily follow that $R$ is Cohen-Macaulay. Finally, we mention that homological aspects of perfect closures, as well as absolute integral closures of Noetherian domains are studied by Asgharzadeh [@Asg]

Preliminaries on tight closure
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Let $R$ be a ring of characteristic $p>0$. Let $F^{e}_{R}:R \to R^{(e)}$ be the $e$-th iterated Frobenius map in which $R^{(e)}=R$ as a left $R$-module, while the right $R$-module structure of $R^{(e)}$ is provided by $a \cdot r=a F^{e}(r)$. The $\textit{Frobenius}$ 