Definition 19.1.1.
Recall that for \(R\) an \(\FF_p\)-algebra, we have defined the coperfection of \(R\) as the image \(R_{\perf}\) of \(R\) under the left adjoint of the forgetful functor from perfect \(\FF_p\)-algebras to arbitrary \(\FF_p\)-algebras. Concretely,
\begin{equation*}
R_{\perf} = \colim(R \stackrel{\phi}{\to} R \stackrel{\phi}{\to} R \to \cdots).
\end{equation*}
Now suppose that \(R\) is an algebra over a perfect field \(k\) of characteristic \(p\text{.}\) Let \(R^{(1)} \to R\) be the relative Frobenius map (Definition 14.1.2); then the induced map \((R^{(1)})_{\perf} \to R_{\perf}\) is an isomorphism. See also Exercise 19.6.2.