The Stacks project

Lemma 41.3.3. Let $A$, $B$ be Noetherian local rings. Let $A \to B$ be a local homomorphism.

  1. if $A \to B$ is an unramified homomorphism of local rings, then $B^\wedge $ is a finite $A^\wedge $ module,

  2. if $A \to B$ is an unramified homomorphism of local rings and $\kappa (\mathfrak m_ A) = \kappa (\mathfrak m_ B)$, then $A^\wedge \to B^\wedge $ is surjective,

  3. if $A \to B$ is an unramified homomorphism of local rings and $\kappa (\mathfrak m_ A)$ is separably closed, then $A^\wedge \to B^\wedge $ is surjective,

  4. if $A$ and $B$ are complete discrete valuation rings, then $A \to B$ is an unramified homomorphism of local rings if and only if the uniformizer for $A$ maps to a uniformizer for $B$, and the residue field extension is finite separable (and $B$ is essentially of finite type over $A$).

Proof. Part (1) is a special case of Algebra, Lemma 10.97.7. For part (2), note that the $\kappa (\mathfrak m_ A)$-vector space $B^\wedge /\mathfrak m_{A^\wedge }B^\wedge $ is generated by $1$. Hence by Nakayama's lemma (Algebra, Lemma 10.20.1) the map $A^\wedge \to B^\wedge $ is surjective. Part (3) is a special case of part (2). Part (4) is immediate from the definitions. $\square$

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