Proposition 6

Let $k$ be a field of characteristic $p \neq 0$, $k'$ a purely inseparable extension of $k$, of finite degree and of height $\leqslant 1$, and $A$ a regular local $k$-algebra. Then $A_{(k')}$ is a local ring and the following conditions are equivalent:

(i) the ring $A_{(k')}$ is regular; (ii) the $\kappa_A$-linear mapping

deduced from the canonical injection ${k'}^p \to A$ is injective.

Let indeed $(x_i)_{i \in I}$ be a finite $p$-base of $k'$ over $k$ (A, V, p. 98); for each $i \in I$, put $f_i = x_i^p \in k$. The $k$-algebra $k'$ is identified with the quotient of $k[(T_i)_{i \in I}]$ by the ideal generated by the polynomials $T_i^p - f_i$, hence the $A$-algebra $A_{(k')}$ with the quotient of $A[(T_i)_{i \in I}]$ by the ideal generated by the polynomials $T_i^p - f_i 1_A$.

Moreover, $(f_i)_{i \in I}$ is a $p$-base of ${k'}^p$ over $k^p$, and the ${k'}^p$-vector space $\Omega_{k^p}({k'}^p)$ has as basis the family of the $df_i$ (A, V, p. 97, th. 1). Proposition 6 then follows from lemma 2.