Proposition 8

Let $k$ be a field and $A$ a noetherian local $k$-algebra. The following conditions are equivalent:

(i) A is formally smooth over $k$ for the $\mathfrak{m}_A$-adic topology ; (ii) A is regular and the $\kappa_A$-linear mapping

deduced from the canonical injection $k \to A$ is injective ; (iii) A is absolutely regular ; (iv) for every purely inseparable extension $k'$ of $k$, of finite degree and of height $\leqslant 1$, the local ring $A_{(k')}$ is regular.

(ii) $\Leftrightarrow$ (i): it is enough to apply Prop. 7 and Remark 1 above, taking for $k_0$ the prime subfield of $k$; in fact, $k$ and $\kappa_A$ are formally smooth over $k_0$ (No. 3, Theorem 1). (i) $\Rightarrow$ (iii): this follows from Corollary 2 of Theorem 2 (No. 5).

If k is of characteristic 0, it follows from Corollary 1 of Theorem 2 (No. 5) that (iv) implies (i), whence the proposition in this case. Suppose k of characteristic $p \neq 0$ and prove (iv) ⇒ (ii). Let $k'$ be a purely inseparable extension of k, of finite degree and of height $\leq 1$. If A and $A_{(k')}$ are regular, the canonical mapping $\kappa_A \otimes_{k'} \Omega_{k^p}({k'}^p) \longrightarrow \kappa_A \otimes_A \Omega(A)$ is injective (No. 6, Prop. 6). By Theorem 1, b) of A, V, p. 97, applied to the extension k of $k^p$, the k-vector space $\Omega(k)$, which coincides with $\Omega_{k^p}(k)$, is the increasing filtered union of the subspaces $k \otimes_{{k'}^p} \Omega_{k^p}({k'}^p)$, where $k'$ runs through the set of finite purely inseparable extensions of k of height $\leq 1$ contained in a fixed algebraic closure of k. Assertion (ii) follows.