The Stacks project

Lemma 15.49.1. Let $A \to B$ be a local homomorphism of Noetherian local rings. Let $D : A \to A$ be a derivation. Assume that $B$ is complete and $A \to B$ is formally smooth in the $\mathfrak m_ B$-adic topology. Then there exists an extension $D' : B \to B$ of $D$.

Proof. Denote $B[\epsilon ] = B[x]/(x^2)$ the ring of dual numbers over $B$. Consider the ring map $\psi : A \to B[\epsilon ]$, $a \mapsto a + \epsilon D(a)$. Consider the commutative diagram

\[ \xymatrix{ B \ar[r]_1 & B \\ A \ar[u] \ar[r]^\psi & B[\epsilon ] \ar[u] } \]

By Lemma 15.37.5 and the assumption of formal smoothness of $B/A$ we find a map $\varphi : B \to B[\epsilon ]$ fitting into the diagram. Write $\varphi (b) = b + \epsilon D'(b)$. Then $D' : B \to B$ is the desired extension. $\square$

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