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

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

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,

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,

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 $A$).

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