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Let $(A,m)$ be a noetherian local ring, and let $p \subseteq A$ be a prime ideal.

From this data, we can construct two rings: 1. We may localize $A$ at $p$, and then complete, obtaining the $pA_p$-adic completion of $A_p$.

  1. We may complete $A$ at $m$, then localize $\hat{A}$ at $p\hat{A}$, and then take the adic completion of $\hat{A}_{p\hat{A}}$ at $p\hat{A} \cdot \hat{A}_{p\hat{A}}$.

My question: are these two rings isomorphic?

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    $\begingroup$ Welcome new contributor. No, those are not the same. Consider the case when $A$ is the localization of $k[t]$ at the maximal ideal generated by $t$ and where $p$ is the zero ideal. Then $A_p$ is the fraction field $k(t)$, which also equals the $pA_p$-adic completion of $A_p$. However, the completion of $\widehat{A}$ at the image ideal equals the Laurent series field $k((t))$. $\endgroup$ – Jason Starr Jan 15 at 18:00

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