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This is a "technical" question that I came across in my research.

Let $A = \textbf{Z}_{p}[\![t_1, \cdots, t_a ]\!]<z_1, \cdots, z_b>$ be the $(p, t_1, \cdots, t_a)$-adic completion of the polynomial ring $\textbf{Z}_{p}[t_1, \cdots, t_a, z_1, \cdots, z_b]$. Let $B$ be a subring of $A$ such that $B$ is complete and separated with the linear topology given by decreasing chain of ideals $I_1 \supset I_2 \supset I_3 \supset \cdots$, making the inclusion $B \hookrightarrow A$ continuous (where $A$ is given the $(p, t_1, \cdots, t_a)$-adic topology).

Suppose that $B$ is local and $B/I_i$ for each $i$ is finite (so in particular, $B$ is a profinite local ring). Then is $B$ necessarily Noetherian?

After trying to find a counterexample in all possible ways I know, I'm guessing that such $B$ should be Noetherian, but do not know how to prove it. Any suggestion would be greatly appreciated.

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  • $\begingroup$ Do I miss something, or is the question equivalent to whether any compact subring of $\mathbf{Z}_p[[t_1,\dots,t_a]]$ is noetherian? (indeed it seems that a compact subring of $A$ is necessarily contained therein, and the inclusion being continuous and injective, it's a homeomorphism onto its image by Hausdorff compactness) $\endgroup$
    – YCor
    Apr 19, 2014 at 21:34
  • $\begingroup$ @Yves: could you explain why you think that a compact subring of '$A$' is necessarily contained in '$\mathbf{Z}_p [\![t_1, \cdots, t_a ]\!]$'? For example, '$B = \mathbf{Z}_p [\![ pz_1]\!]$' satisfies above conditions but not contained in '$\mathbf{Z}_p [\![t_1, \cdots, t_a]\!]$'. $\endgroup$
    – david
    Apr 20, 2014 at 4:36
  • $\begingroup$ ok, I see, you're right. $\endgroup$
    – YCor
    Apr 20, 2014 at 7:50

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