Exercise 23, § 5

Let $k$ be a field of characteristic exponent $p$, $A$ a finitely generated $k$-algebra, $p$ a prime ideal of $A$. Now prove that the following conditions are equivalent: (i) the ring $A_p \otimes_k k'$ is regular for every extension $k'$ of $k$; (ii) the ring $A_p \otimes_k {k'}^{-\infty}$ is regular; (iii) the ring $A_p \otimes_k k'$ is regular for every finite and purely inseparable extension $k'$ of $k$. (To exercise prove (ii) $\Rightarrow$ (iii), use Exercise 14, a), p. 96. To exercise prove (iii) $\Rightarrow$ (i), it is enough to show that $A_p \otimes_k k'$ is regular when $k'$ is a finitely generated extension of $k$ (Exercise 21), purely inseparable extension of a purely transcendental extension (Exercises 22 and 14, a), and domination of the extension $k'$ by a finite quasi-Galois extension of a purely transcendental extension). Now dominate $k'$ by an extension $k''(X_1, ..., X_n)$ where $k''$ is a finite and purely inseparable extension of $k$ and reduce to the case $k' = k''(X_1, ..., X_n)$ by Exercise 14, a). On putting $B = A_p \otimes_k k''$, a regular algebra by (iii), we have $A_p \otimes_k k' = B \otimes_{k''} k''(X_1, ..., X_n)$, a regular algebra by Exercise 6, e), p. 94.) We say that $p \in \mathrm{Spec}(A)$ is smooth if it has the equivalent properties above.