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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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  • $\begingroup$ 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 $\endgroup$
    – David White
    Commented Jul 30, 2011 at 14:19
  • $\begingroup$ 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. $\endgroup$
    – the L
    Commented Jul 30, 2011 at 14:35
  • $\begingroup$ May be analytically finite? Google does not show any thing like that though. $\endgroup$
    – Hailong Dao
    Commented Aug 1, 2011 at 0:50
  • $\begingroup$ I'll be happy to use analytically finite, if there is no name for this. $\endgroup$
    – Mahdi Majidi-Zolbanin
    Commented Aug 1, 2011 at 19:03
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    $\begingroup$ 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). $\endgroup$
    – Mahdi Majidi-Zolbanin
    Commented Aug 4, 2011 at 15:31

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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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    $\begingroup$ 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. $\endgroup$
    – Laurent Moret-Bailly
    Commented Sep 21, 2011 at 6:29

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