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If $R$ is a quasi-excellent ring, then is $R_{\mathfrak p}$ also quasi-excellent for every prime ideal $\mathfrak p$ of $R$ ?

I think Matsumura's commutative ring theory book says that localization of Excellent rings are again Excellent, but I do not know about quasi-excellence ...

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  • $\begingroup$ Not very helpful: it is claimed without proof in [Tag 07QU]. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jul 2 at 15:45
  • $\begingroup$ @R.vanDobbendeBruyn: I see ... do you happen to know what happens to localization of $J$-$2$ rings? They still remain $J$-$2$ then? $\endgroup$
    – Alex
    Commented Jul 2 at 16:17

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Yes, localizations of quasi-excellent rings are quasi-excellent by (34.A) in Matsumura's Commutative algebra.

As you noted in the comments, the G-ring condition is already local, so the key point is that being J-2 localizes. This follows from characterization (3) of the J-2 property in [Matsumura, Theorem 73] (see also [EGAIV2, Théorème 6.12.4] and [Stacks, Tag 07PC].) The original reference is [Nagata 1959, Theorem 2(i)].

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  • $\begingroup$ Thanks Takumi ... I'm guessing that what Nagata calls "formation of rings of quotients " is just arbitrary localization in our modern terminology? $\endgroup$
    – Alex
    Commented Jul 2 at 21:30
  • $\begingroup$ @Alex Yes, that is correct. For example, this is the terminology Nagata uses in Chapter I, §6 of Local rings. $\endgroup$
    – Takumi Murayama
    Commented Jul 2 at 23:02

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