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Consider a commutative unital ring $A$ and its ideal $I$. Let $\hat{A}$ be a completion of $A$ with $I$-adic topology. I want to know whether $\hat{I^n}=\hat{I}^n$ holds or not in general. (Note that $I$-adic topology of $I^n$ concides with its relative topology, even without noetherian hypothesis.)

It is true when $A$ is noetherian, since $\hat{I}=I\hat{A}$ holds. (Atiyah-MacDonald Prop10.15)

My attempt: Write down $\hat{I^n}$ as a subset of $\prod_l A/I^l$. Then $\hat{I}^n\subseteq \hat{I^n}$doesn't seem to be true. So I want to make a counterexample.

nessy
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  • In the power series ring $A = \mathbb{Q}\left[\left[x_1, x_2, x_3, \ldots\right]\right]$, take the maximal ideal $I = \left(\text{all power series with constant coefficient } 0\right)$. Then, $x_2^2 + x_3^3 + x_4^4 + x_5^5 + \cdots \in \widehat{I^2}$, but I would be surprised if this belongs to $\widehat{I}^2$. Can you prove this? – darij grinberg Aug 28 '20 at 10:41
  • Let $f = x_2^2+x_3^3+\cdots \in A$. For the first statement, I think it is enough to show that "$\overline{f} \in I^2/(I^2 \cap I^n)$ for arbitrary $n$". But I can't even prove that $\overline{f}\in I^2$. I'm missing something? I have no idea for the second statement because I can't determine what actually $\hat{I}^2$ is. – nessy Aug 28 '20 at 12:25
  • Oh, I see -- you're probably right; I thought that $f = \lim_{n \to \infty} \left(x_2^2 + x_3^3 + \cdots + x_n^n\right)$ would be obvious, but it is not so clear-cut. – darij grinberg Aug 28 '20 at 14:56

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