All Questions
Tagged with rigid-analytic-geometry ac.commutative-algebra
9 questions with no upvoted or accepted answers
8
votes
0
answers
550
views
Foundational Questions on Adic Spaces
There are some foundational questions on adic spaces that I can't find in the literature. It seems that these questions are pretty natural, so I guess that an answer should be known to the experts in ...
- 2,923
7
votes
0
answers
203
views
Finite generation and finite presentation over a truncated valuation ring: is there an easier proof?
Let $K^+$ be a valuation ring which is $\pi$-adically complete for some pseudouniformizer $\pi$.
Nagata 053E proved that every finitely generated and flat (equivalently, torsion-free) $K^+$-algebra is ...
- 16.2k
5
votes
0
answers
122
views
Are affinoid algebras over nontrivially valued fields Jacobson?
It is well-known that for any field $k$ with valuation the Tate algebra $k\{T_1,\dots,T_n\}$ is Jacobson (see Bosch-Güntzer-Remmert for nontrivial valuations; for trivial valuations those are just ...
- 28.2k
4
votes
0
answers
149
views
Coherence of the I-adic completion of a local ring of a formal scheme
Let $K$ be a valued field of rank one and $K^+$ its valuation ring such that $K^+$ is $\varpi$-adically complete with respect to a pseudo-uniformizer $\varpi\in K^+$. Let $X$ be a smooth finite type $...
- 119
3
votes
0
answers
155
views
Colimits in commutative Banach algebras?
Let $K$ be a complete non-Archimedean field. It is known that the category $\mathrm{Ban}_K$ of $K$-Banach spaces with bounded linear maps does not have infinite colimits. The usual argument for $\...
- 426
3
votes
0
answers
281
views
The closed unit adic disk
I am reading the Scholze-Weinstein Berkeley lecture notes on "Perfectoid Spaces", and in particular I am stuck trying to understand the closed adic unit disk, which is the second example of ...
- 2,907
2
votes
0
answers
166
views
Theorem on formal functions when the initial data is a proper map of formal schemes
Let $\pi: X \to S:=\mathrm{Spf}\text{ } A$ be a proper morphism of $\mathbb{Z}_p$-admissible formal schemes and $\mathcal{F}$ be a coherent sheaf on $X$.
Set $S_0=\{x\}$ be a closed point of $S$ and $...
- 1,066
2
votes
0
answers
148
views
Support of Tor over affinoid algebras
Suppose $k$ is a complete nonarchimedian field, $A$ is a $k$-affinoid algebra, and $M$ is a finitely presented $A$-module. Is the set
$\tau(M)= \left\{ x \in \mathrm{Sp}(A)\,\mathrm{with}\,\mathrm{...
- 13.1k
1
vote
0
answers
81
views
Completion of $K$-algebra of finite type with respect to the residue norm
Let $K$ be a non-archimedean field. For $n \in \mathbb{N}$ let
\begin{equation*}
T_n=\{ \sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in K[\
- 473