# A terminology question: formally finite ??

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.

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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 (arxiv.org/abs/math/0206203) Grothendieck calls such maps quasi-finite (see pages 1 and 2 of SGA 1). – Mahdi Majidi-Zolbanin Aug 4 '11 at 15:31

## 1 Answer

In SGA 1 (arxiv.org/abs/math/0206203) Grothendieck calls such maps quasi-finite (see pages 1 and 2 of SGA 1).

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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