My question, in the easiest case, is the following: given a complete local Noetherian $k$-algebra $\hat{A}$ with residue field $k$, when is $\text{Spf}(\hat{A})$ algebraizable? Or in other words, when does there exist a $k$-algebra $R$ of finite type and an ideal $I\subset R$ such that the $I$-adic completion of $R$ is isomorphic to $\hat{A}$? If this does not always hold, then what would be the easiest $\hat{A}$ which gives a counterexample? Furthermore, how is the situation if we do not assume anything on the residue field of $\hat{A}$?
Some more context: by a result of Cohen, we know that $\hat{A}\cong k[[x_1,\ldots,x_n]]/J$, so it seems to me that my question (in the case of $k=\mathbb{C}$) is equivalent to asking when a germ of an analytic subset is (analytically) isomorphic to a germ of an algebraic one (using arguments of Artin). From what I've gathered so far, this is not an easy question in general. But maybe I am missing something here.
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