Suppose that $A$ is an algebra of finite type over a field $k$ and let $\mathfrak{p}$ be a prime ideal of $A$. Let $\hat{A}$ denote the completion of $A$ along $\mathfrak{p}$. Is is true that $$ \hat{A} \otimes_k \hat{A} $$ is Noetherian?
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don't think this holds in general see here – Simonsays Oct 04 '22 at 14:08
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I wouldn’t be so quick in judgement. For instance, the tensor product $A_{\mathfrak{p}} \otimes_k A_{\mathfrak{p}}$ is Noetherian despite not satisfying the hypothesis of the question you pointed me to. – ofiz Oct 04 '22 at 21:40