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 procategory associated to $\mathfrak{Ar}_{W(\mathbb{F})}$, as in padic Hodge Theory for RigidAnalytic 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 $\widehat{\mathfrak{Ar}}_{W(\mathbb{F})}$.
$\begingroup$
$\endgroup$
1

2$\begingroup$ Finitely generated suffices. More precisely, if the pro(finitedimensional $\mathbb{F}$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 proArtinian is just $\{(A/\mathfrak{m}_A^n\}_n$.) $\endgroup$– Jon PridhamCommented Apr 27, 2022 at 9:41
Add a comment
