Questions tagged [perfectoid-spaces]

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2
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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 $\gm^2 = \gm$. ...
9
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1answer
440 views

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 ...
18
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1answer
1k views

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 ...
5
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1answer
1k views

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_{(\...
8
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1answer
615 views

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 ...
6
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1answer
673 views

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 $\mathbb{Q}_p$...
15
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222 views

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 ...
10
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344 views

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 ...
3
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0answers
142 views

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 $\delta$-...
3
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1answer
150 views

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 $A\mapsto A_{!!}$...
2
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1answer
206 views

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 ...
5
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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 = \...
2
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183 views

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 ...
8
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158 views

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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1answer
618 views

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-...
10
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1answer
650 views

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 ...