MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is there a name for a local homomorphism $\varphi:A\longrightarrow B$ of local rings $A$ and $B$, whose completion $\hat{\varphi}:\hat{A}\longrightarrow\hat{B}$ is a finite homomorphism? (that is, $\hat{B}$ is a module finite extension of $\hat{A}$ via $\hat{\varphi}$). Perhaps formally finite?

EDIT: Assume $A$ and $B$ are Noetherian.

share|cite|improve this question
I figured I'd tag as algebraic geometry since there is probably a nice geometric way to understand such maps and they may have a term for them. I've never come across anything with this description personally, but I'm not an algebraic geometer – David White Jul 30 '11 at 14:19
Actually, there is a notion of a formally finite map: A map between two adic rings $(A,\mathfrak{a}) \to (B,\mathfrak{b})$ is called formally finite if $B/\mathfrak{b}$ is a finite $A$-module. – the L Jul 30 '11 at 14:35
May be analytically finite? Google does not show any thing like that though. – Hailong Dao Aug 1 '11 at 0:50
I'll be happy to use analytically finite, if there is no name for this. – Mahdi Majidi-Zolbanin Aug 1 '11 at 19:03
I just found out that in SGA 1 ( Grothendieck calls such maps quasi-finite (see pages 1 and 2 of SGA 1). – Mahdi Majidi-Zolbanin Aug 4 '11 at 15:31

In SGA 1 ( Grothendieck calls such maps quasi-finite (see pages 1 and 2 of SGA 1).

share|cite|improve this answer
Right, but now quasifinite morphisms are assumed of finite type (see footnote at the same place in SGA1). For instance, $A\to\hat{A}$ is not quasifinite in general. – Laurent Moret-Bailly Sep 21 '11 at 6:29

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.