Let $\mathfrak{p}$ be a prime ideal of a ring $A$. The completion $\hat{A}$ of $A$ with respect to its adic-topology is used to simplify $A$ beyond the localization $A_{\mathfrak{p}}$. For a multiplicative subset $S \subset A$, we have the localization $S^{-1} A$. Is there an analogous operation for completion? I would like some analogous ring, whose intuition would then be that we are "simplifying $A$ beyond $S^{-1} A$".
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What do you mean by ''simplifying'' a ring? – ΑΘΩ Mar 15 '19 at 06:04
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You mean completion of the localization at a maximal ideal to obtain a complete DVR where Hensel lemma holds, a considerable simplification ? In noetherian ring with an ideal you can always look at $R \hookrightarrow \varprojlim R/I^n$, but there are also things like $k[x,y] \hookrightarrow \varprojlim_{n,m} k[x,y]/(x^n,y^m) = k[[x,y]]$ – reuns Mar 15 '19 at 06:05
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What is the situation in which one gets a DVR? I mean the inverse limit of rings of the form $R / \mathfrak{m}^n$, so the adic completion. What you have said is useful, but I was hoping there would be something that takes in $S$ and $A$ as an input. Perhaps it doesn't exist. – Ronald J. Zallman Mar 15 '19 at 06:43