All Questions
14 questions
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
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
8
votes
1
answer
339
views
On actions of finite groups on adic spaces
Let $K$ be an algebraically closed complete non-archimedean field and consider the unit ball $\mathbb{B}^{1}_{K}=Sp(K\langle t\rangle)$. We have an action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{B}^{1}...
- 541
5
votes
1
answer
362
views
On the noetherianess of some subalgebras of an affinoid algebra
$\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a ...
- 541
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
1
answer
147
views
Bounded torsion of quotients of affine formal models
$\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a ...
- 541
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
1
answer
381
views
Reduced complete Tate ring which is not uniform?
Recall that a topological ring $A$ is Tate if there is an open subring $A_0$ such that the induced topology on $A_0$ is t-adic for some $t \in A_0$ that becomes a unit in $A.$ One can, given a Tate ...
- 217
4
votes
1
answer
225
views
Significance of integrally closed in an affinoid algebra
A Tate affinoid k-algebra is defined as a pair $(R,R^+)$ where $R$ is a Tate algebra and $R^+$ is an open and integrally closed subring of $R$ contained in the ring of powerbounded elements.
See for ...
- 7,746
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
1
answer
765
views
Flatness of the integral closure
Let $R$ be a $p$-torsion free ring which is integrally closed in $R[1/p]$ and let $S$ be a finite etale extension of $R[1/p]$.
Is it true that an integral closure $S^+$of $R$ in $S$ is flat over $...
- 2,923
2
votes
1
answer
329
views
Representation of elements in affinoid algebra
Let $K$ be a complete, algebraically closed non-Archimedean field, and let $p \in K[x]$ be of degree $d > 0$ and norm 1. (Here the norm of a polynomial is the maximum of the norms of its ...
2
votes
1
answer
348
views
Modules with connection over $p$-adic laurent series rings
If $X$ is a smooth rigid analytic space over a $p$-adic field $K$ (of characteristic zero), then every coherent $\mathcal{O}_X$-module with integrable connection is locally free. In his paper "...
- 1,838
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