# When does completion preserve injectivity?

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.

• Trivia: if $f$ is an homeomorphism onto its image (e.g., $A$ and $B$ are DVR), so is $\hat{f}$. Jun 2 '18 at 15:01
• Uhm, yes I'm interested in the DVR case. But why? Jun 2 '18 at 15:21
• Assume that $A$ and $B$ are DVRs with uniformizing elements $\pi_A \in A$ and $\pi_B \in B$. Write $f(\pi_A) = u \pi_B^{\nu}$ with $u$ a unit of $B$ and $\nu$ a positive integer. Then $f(A) \cap \pi_B^{n \nu} B \subseteq f(\pi_A^{n}A)$ for every $n \ge 0$, hence $f$ is open. Jun 2 '18 at 18:46
• The natural condition is that the topology on $A$ induced by powers of its maximal ideal and the topology induced by the powers of the maximal ideal of $B$ intersected with $A$ are the same. Jun 3 '18 at 20:41
• Trivia continued: If $\hat{f}$ is injective and if $\hat{A}$ is compact (i.e, the residue field of $A$ is finite), then $f$ is an homeomorphism onto its image. So the "natural condition" is necessary under the compactness assumption. Jun 4 '18 at 20:19

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.

• Wait, but what about the following example: $A=\mathbb Z[t]_{(p,t)}$ and $B=\mathbb Z[t]_{(t)}$. Clearly $A\subset B$, but $\hat A= \mathbb Z_p[[t]]$ and $\hat B=\mathbb Q[[t]]$ Jul 18 '18 at 19:54
• Moreover I'm looking at EGA I (chapter 0 of Springer edition), and I cannot find any 3.9.8. Jul 18 '18 at 20:04
• @manifold: your example is not a local homomorphism (the map $A_{\mathfrak p} \to A_{\mathfrak q}$ for $\mathfrak p \supsetneq \mathfrak q$ never is). Jul 18 '18 at 21:41
• @manifold: it's 3.9.8 of chapter 1 (not chapter 0), on page 255. Jul 19 '18 at 7:25