# Completion of localization of completion

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?

• 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 $k((t))$. – Jason Starr Jan 15 at 18:00