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Let $\mathbb{F}$ be a finite field, and $W(\mathbb{F})$ its associated ring of Witt's vectors. On page 6 of the following lecture notes Deformations of Galois Representations, the category $\mathfrak{Ar}_{W(\mathbb{F})}$ is defined to be the category of local finite artinian $W(\mathbb{F})$-algebras with residue field $\mathbb{F}$. On the aforementioned notes, it also defines the category $\widehat{\mathfrak{Ar}}_{W(\mathbb{F})}$ as the category of noetherian, complete, local algebras with residue field $\mathbb{F}$. Let now $P\mathfrak{Ar}_{W(\mathbb{F})}$ be the pro-category associated to $\mathfrak{Ar}_{W(\mathbb{F})}$, as in p-adic Hodge Theory for Rigid-Analytic Varieties proposition 3.2. It seems clear that $\widehat{\mathfrak{Ar}}_{W(\mathbb{F})}$ should be a full subcategory of $P\mathfrak{Ar}_{W(\mathbb{F})}$. I am interested in understanding the relationship between these two categories. For example, it would be nice to know when an element in $P\mathfrak{Ar}_{W(\mathbb{F})}$ is in $\widehat{\mathfrak{Ar}}_{W(\mathbb{F})}$.

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    $\begingroup$ Finitely generated suffices. More precisely, if the pro-(finite-dimensional $\mathbb{F}$-vector space) $\mathfrak{m}_A/\mathfrak{m}_A^2$ is in fact a f.d. vector space, then $A$ is Noetherian. (For a f.g. complete local ring $A$, the assoc pro-Artinian is just $\{(A/\mathfrak{m}_A^n\}_n$.) $\endgroup$ Commented Apr 27, 2022 at 9:41

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