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Suppose that $R \subseteq S$ is a generically separable extension of 1-dimensional normal domains (you can assume that $R$ is local if you'd like) of equal-characteristic $p > 0$ (for simplicity, as you'll see, it's not really needed).

Suppose that for each point $x \in \text{Spec} R$, there exists a point $y \in \text{Spec} S$ such that $R_x \subseteq S_y$ has tame ramification (ie, the residue field extension is separable and if we write, $r = us^n$ where $r$ and $s$ are uniformizers for $R_x$ and $S_y$ respectively, then $p \not | n$).

Question: Does this condition have a name? (ie, weakly tame, domesticated :-), etc)

I think this condition is also the same as requiring that the trace map $\text{Tr}: S \to R$ is surjective. In the case that $R \subseteq S$ is generically Galois, Kerz and Schmidt have called this surjective trace condition cohomologically tamely ramified. Has anyone else seen a name for this surjective trace condition especially in the non-1-dimensional case?

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+1 for domesticated :-) If there is no standard terminology, then you have a good candidate there. –  Alex B. Dec 7 '10 at 1:36
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Not that I have any idea if this has a name or that I should mingle in the naming business, but I would say that domesticated sounds "tamer" to me, than tame. In other words, I could easily imagine something that's tame, yet not domesticated, but less so for something that's domesticated but not tame. At the same time, I don't know what would be a good word that's sort of tame, but a bit less so. Maybe docile? Then again, one should always keep in mind Miles Reid's comment about non-native speakers' ideas about naming things.... –  Sándor Kovács Dec 7 '10 at 6:33
    
Sandor, I agree, domesticated sounds tamer than tame. I'm still hoping this has been named something else. Actually, I think the suggestion for domesticated was Aaron Bertram's when I asked some people at tea yesterday if they knew if this had a name.. –  Karl Schwede Dec 7 '10 at 15:04

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