Example 15.40.2. Let $k$ be a field of characteristic $p > 0$. Suppose that $a \in k$ is an element which is not a $p$th power. A standard example of a geometrically regular local $k$-algebra whose residue field is purely inseparable over $k$ is the ring
Namely, $A$ is a localization of a smooth algebra over $k$ hence $k \to A$ is formally smooth, hence $k \to A$ is formally smooth for the $\mathfrak m$-adic topology. A closely related example is the following. Let $k = \mathbf{F}_ p(s)$ and $K = \mathbf{F}_ p(t)^{perf}$. We claim the ring map
is formally smooth for the $(x)$-adic topology on $A$. Namely, $\Omega _{k/\mathbf{F}_ p}$ is $1$-dimensional with basis $\text{d}s$. It maps to the element $\text{d}x + \text{d}t = \text{d}x$ in $\Omega _{A/\mathbf{F}_ p}$. We leave it to the reader to show that $\Omega _{A/\mathbf{F}_ p}$ is free on $\text{d}x$ as an $A$-module. Hence we see that condition (5) of Theorem 15.40.1 holds and we conclude that $k \to A$ is formally smooth in the $(x)$-adic topology.
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