TITLE: When does completion preserve injectivity?
QUESTION [9 upvotes]: Let $ f:A\to B$ be an injective, local homorphism between two Noetherian local rings. Consider the completions $\hat A$ and $\hat B$  with respect the maximal ideals. We have an induced homomorphism $\hat f: \hat A \to\hat B$. What assumptions do we need in order to ensure that also $\hat f$ is injective?
My main interest is geometric, so for example the local map induced by a surjective morphism between Noetherian schemes.
Many thanks in advance

REPLY [4 votes]: This may be useful:

Proposition (Zariski) (see EGA I, (3.9.8) in Springer edition)
Let $f: (A,\mathfrak{m})\to (B,\mathfrak{n})$ be a local homomorphism of noetherian local rings. Assume that:

$f$ is injective.
$\hat{A}$ is a domain.
$f$ is essentially of finite type.

Then the $\mathfrak{m}$-adic topology on $A$ is induced by the $\mathfrak{n}$-adic topology on $B$.

Of course this implies that $\hat{f}$ is injective. If we don't assume that $\hat{A}$ is a domain (but $A$ is) it is easy to construct counterexamples.