Exercise 9, § 7

Soit $k$ a field of characteristic $p > 0$, containing a decreasing filtered family $(k_\lambda)_{\lambda \in A}$ of subfields satisfying $\bigcap_{\lambda \in A} k_\lambda = k^p$. a) Prove that the intersection of the kernels of the canonical homomorphisms of $\Omega(k)$ into $\Omega_{k_\lambda}(k)$, for $\lambda$ ranging over $A$, is reduced to $(0)$ (consider a $p$-basis of $k$). b) Let $A$ be a regular local $k$-algebra, equipped with the $m_A$-adic topology. Prove that, in order that $A$ be formally smooth over $k$, it is necessary and sufficient that $A$ be formally smooth over $k$ relative to $k_\lambda$, for every $\lambda \in A$ (use a), Prop. 8 and Exercise 7, a)).