All Questions
Tagged with perfectoid-spaces ag.algebraic-geometry
15 questions
2
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0
answers
152
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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 ...
9
votes
0
answers
963
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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 ...
- 15.4k
1
vote
0
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227
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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 $\...
- 282
3
votes
0
answers
264
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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 ...
- 2,907
3
votes
0
answers
280
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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
8
votes
1
answer
2k
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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}) \...
- 81
2
votes
0
answers
504
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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 ...
- 1,083
6
votes
0
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449
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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 ...
- 61
3
votes
0
answers
287
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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 ...
- 1,083
19
votes
1
answer
2k
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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 ...
- 199
6
votes
1
answer
1k
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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_{(\...
- 435
9
votes
1
answer
1k
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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 ...
- 228
15
votes
0
answers
439
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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 ...
- 7,736
10
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0
answers
873
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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
4
votes
0
answers
244
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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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