Questions tagged [perfectoid-spaces]
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18 questions with no upvoted or accepted answers
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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 ...
- 1,083
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Geometrization of the global Langlands correspondence?
Fargues-Scholze famously describe arithmetic local Langlands via global geometric Langlands on the Fargues-Fontaine (FF) curve.
The FF curve acts like an algebraic curve over $\mathbb{C}_p$ (its ...
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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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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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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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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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3
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Algebraic properties of Witt vectors $W(K^{\flat\circ})$, $K$ a characteristic 0 perfectoid field
Let $K$ be as in the title with tilt $K^\flat$. $W = W(K^{\circ\flat})$ satisfies a universal property: it is the unique $p$-adically complete $p$-torsion free $\mathbb{Z}_p$-algebra $A$ with $A / pA \...
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3
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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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3
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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 ...
- 2,907
3
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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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3
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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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2
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Uniqueness and existence of maps
I am currently reading the Berkeley lectures on Perfectoid Spaces by Scholze and Weinstein. In the section "The adic open unit disk over $\mathbb{Z}_p$" we encounter from Proposition 4.2.6 ...
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Vector bundles on pro-etale topology over a field
Suppose $K$ is a finite extension of $\mathbb Q_p$. Consider the one-point adic space $X=\operatorname{Spa}K$, and let $C=\hat {\bar K}$, $G=\operatorname{Gal}(\bar K/K)$. I heard that the category of ...
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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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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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Deformations over $A_{\inf}$
Setup:
Let $K$ be a perfectoid field of characteristic $0$ with tilt $K^{\flat}$.
Let $A_{\inf}=W(\mathcal{O}_{K^{\flat}})$ be the infinitesimal period ring.
Let $\mathcal{X}$ be a flat, projective $\...
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