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



When does completion preserve injectivity?
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.


