# Questions tagged [perfectoid-spaces]

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### The points of $\operatorname{Spa}\mathbb{Z}_p$

$\DeclareMathOperator\Spa{Spa}$What are the points of $\Spa\mathbb{Z}_p$? I read in Scholze-Weinstein that this adic spectrum consists of 2 points, a special point, which corresponds to the pullback ...

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### 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 ...

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### Why are there three kinds of non-archimedean geometry?

It may seem silly to ask "Why are there three types of non-Archimedean geometry?", that would be like asking why there are three (and even more) different Weil cohomologies. So I have to ...

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### Completion of infinite degree extension of perfectoid fields is perfectoid?

Is completion of infinite degree extension of perfectoid fields perfectoid ?
It is known that finite extension of perfectoid fields is also perftoid from tilting correspondence, but what about ...

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### Some questions from the paper by Scholze-Weinstein

The following is from the paper by Scholze-Weinstein on moduli of $p$ divisible groups.
My question is from a part of Lemma 4.1.7: If $R$ is a semiperfect ring, then the canonical map $W(R^{\flat}) \...

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### Relative homology in Fargues-Scholze paper

if $f:X\to Y$ is a map of small v-stacks, Scholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. They say the left adjoint exist because $f$ is a slice in $v$-site (they ...

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### Proof of Lemma 6.5 in Scholze's Perfectoid Spaces

In the proof of Lemma 6.5(approximation lemma) in Scholze's Perfectoid Spaces,
I have the following three questions about $h = f - g^\sharp_c$ and $g^\sharp_{c'}$.
(Maybe it's something you'll find ...

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### Chevalley’s affineness criterion for perfectoid space

Chevalley’s affineness criterion says that if $f: X\to Y$ is surjective and finite, $Y$ is Noetherian and $X$ is affine then $Y$ is also affine. The usual proof uses Serre's criterion and Noetherian ...

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### Does the map $\theta[1/p]: A_{\mathrm{inf}} \otimes \mathbb Q_p \to \mathbb C_p$ split?

This question might be very elementary to someone who knows p-adic hodge theory/perfectoid stuff etc.
Recall that $\mathbb C_p = \hat{\overline{\mathbb Q_p}}$ and $\mathbb C_p^\flat$ is it's tilt. We ...

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### flat resolutions of almost modules?

Recall the set-up in 2.4.10 of Almost Ring Theory by Ofer Gabber and Lorenzo Ramero:
$V$ is a commutative unital ring.
$\newcommand{\gm}{{\mathfrak m}}\gm$ is an ideal in $V$ such that $\gm^2 = \gm$.
...

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### An explicit isomorphism $K^\times/K^{\times p} \cong K^\flat/\wp K^\flat$ where $K \supset \mu_p$ is perfectoid field of mixed characteristic $(0, p)$

Let $K$ be a perfectoid field of mixed characteristic $(0, p)$, i.e. $K$ has characteristic $0$ but its residue field has characteristic $p$. Further suppose that $K$ contains all the $p$th roots of ...

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### Perfectoid approach to resolution of singularities in char $p$

Since perfectoid techniques have built a bridge between char $0$ and char $p$ worlds, it is conceivable that they can be applied to resolution of singularities in char $p$ using their successful ...

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### Proposition 5.13 (ii) in Scholze's Perfectoid Spaces

In Proposition 5.13 (ii) in Scholze's Perfectoid Spaces, we have $R \to S$ a morphism of $\Bbb F_p$-algebras and the assumption that the relative Frobenius $\Phi_{S/R}$ induces an isomorphism $R_{(\...

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### On the definition of the etale site of an adic space

I have a question related to the definition of the etale site of an adic space. As a reference, I am using Huber's book "Etale Cohomology of Rigid Analytic Varieties and Adic Spaces".
First ...

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### How many untilts?

I read the following passage in Endomorphisms of power series fields and residue fields of Fargues-Fontaine curves by Kedlaya-Temkin:
"One can construct many algebraic extensions of $\mathbb{Q}_p$...

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### Applications of the Weight Monodromy conjecture

I think of the Weight Monodromy conjecture as an analogue of the Weil conjectures in the case of bad reduction. The Weil conjectures of course have lots of applications, from point counting to ...

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### Why diamonds are only defined in characteristic $p$?

I'm trying to read Scholze's article "Etale cohomology of diamonds" (arXiv link) and both in this article and in Berkeley notes, the diamonds are defined as sheaves on the category of ...

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### Uniqueness of $\delta$-structure on a $p$-torsion ring

I was working through Bhargav's notes on $\delta$-rings and prismatic cohomology, specifically lecture 2, page 2, point 5 where he claims that the ring $\mathbb Z[x]/(px,x^p)$ has a unique $\delta$-...

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### Faithful flatness of left adjoint to almostification of algebras

I have been reading Bhatt's notes on perfectoid spaces and I have stumbled upon a fact whose proof I am unable to understand. Specifically, in Remark 4.2.8 Bhatt describes the functor $A\mapsto A_{!!}$...

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### perfectoid field of characteristic $p$

Let $L$ be perfectoid field of characteristic $p$ and $L'$ be a finite extension of $L$. Then I want to prove the trace map $\text{Tr}_{L'/L}: m_{L'}\rightarrow m_L$ is surjective. I find a proof in ...

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### Is being perfectoid a local property?

In his 2012 CDM proceedings, Peter Scholze mentions the following open question:
Let $K$ be a perfectoid field and $(A,A^+)$ a complete affinoid $K$-algebra. Suppose there exists a cover of $X = \...

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### A complete Tate Huber ring is Banachizable (maybe not)?

I have questions of technical nature.
A complete Tate Huber ring is a complete topological (commutative) ring $A$ admitting an open subring $A_0$ whose topology is the $\varpi A_0$-adic topology, for ...

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### Looking for basic example in the theory of adic spaces

I have heard that there exist examples of Huber pairs $(A,A^{\circ})$ and $(B,B^+)$ such that $\operatorname{Spa}(B,B^+)$ is a rational open of $\operatorname{Spa}(A,A^{\circ})$ and such that $B^+ \...

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### Can Scholze's perfectoid spaces bridge the gap for twin prime conjecture? [closed]

It seems than an analogue of the twin prime conjecture for polynomials in finite fields has been solved: see https://www.quantamagazine.org/big-question-about-primes-proved-in-small-number-systems-...

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### Iwasawa theory and perfectoid spaces

Have there been any applications of perfectoid theory to Iwasawa theory? At a first glance, this seems like a natural choice. For instance, the field $\mathbb Q_p(\mu_p^{1/p^\infty})$ is studied in ...