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Let $A$ be a commutative ring and $\mathfrak{p}$ be a prime ideal of $A$. Under which assumptions for $A$ and $\mathfrak{p}$ does localization by $\mathfrak{p}$ and completion with respect to $\mathfrak{p}$ commute? To be more precise,

when is $\widehat{A}_\widehat{\mathfrak{p}}$ (completion w.r.t. $\mathfrak{p}$) isomorphic to $\widehat{A_\mathfrak{p}}$ (completion w.r.t. $\mathfrak{p} A_\mathfrak{p}$)?

For example, is it true under the assumptions $A$ noetherian and $\mathfrak{p}$ a maximal ideal?

It seems to me that one important ingredient for a possible proof (under the right assumptions) is that localisation and building factor rings commutes. So a side question: Is it always true that $A_\mathfrak{p}/(\mathfrak{p}A_\mathfrak{p})^k \cong (A/\mathfrak{p}^k)_{\overline{\mathfrak{p}}}$?

Thanks

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    Didn't you just ask this on MO and get an answer.2011-05-10
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    ([MO question](http://mathoverflow.net/questions/64399) BBischof mentioned)2011-05-10
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    It doesn't look like it was asked by the same person to me. The notation is not the same and usually when questions are posted on both sites they are written in exactly the same way. In my opinion it is a different question.2011-05-10
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    Incidentally, it is not true that localization (in general) commutes with completion. For instance, if $A$ is a complete ring, $A_f$ may not be for $f \in A$.2011-05-11

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