All Questions
Tagged with galois-theory etale-cohomology
11 questions
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Galois group of shimura varieties with different level structure
Let $(G,X)$ be a shimura data, and $K$ an open compact neat subgroup of $G(\mathbb A_f)$. Suppose $K'\subset K$ is open and normal, then, I see in many references that the finite etale cover $Sh(G,X)_{...
- 775
2
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1
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367
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Groupoid of points, shape and stratified shape of $\operatorname{Sh} (X_\text{pro-ét})$
$\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\Pt{Pt}$Maybe this is well-known or even a stupid misunderstanding of something very basic. It's well-known that the groupoid of points (i.e., groupoid ...
- 2,227
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Simplification of links between idele class group and étale cohomology
I posted this question over on stack exchange and was told it would work better here.
For interest I have been looking at links between class field theory and étale cohomology. Let $k$ be a global ...
4
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1
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578
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Etale cohomology and Kummer theory
If $K$ is a field and $n \geq 1$ is such that $n \in K^{\times}$, then $H^1_{et}(\mathrm{Spec}(K),\mu_n)=K^{\times} / (K^{\times})^n$. This is easy to prove, see for instance Tamme, Etale Cohomology, ...
- 63.1k
5
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1
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406
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Does the étale topos determine the Hodge numbers?
Does the small étale topos of a smooth proper variety over a perfect field of positive characteristic determine its Hodge numbers? We consider it as a Grothendieck topos over the étale topos of the ...
user158636
1
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0
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174
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Galoisian perspective on local system tamely ramified along a smooth divisor
This question is about (1.7.8) and (1.7.11) in Deligne’s Weil II paper.
Let $X$ be a regular scheme and $D\subset X$ a smooth principal divisor cut out by the function $t$. Let $\mathcal F$ be a ...
- 1,217
3
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Conjugation action of $Gal(\bar{s}/s)$ on the tame ramification group
There is a statement in SGA 7-1 Exposé 1 (P. Deligne, Résumé des premiers exposés de A. Grothendieck, pdf of SGA7-1), (0.3.1):
$S$ is a Henselian trait (i.e. the spectrum of a henselian discrete ...
- 83
5
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1
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514
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Frobenius eigenvalues algebraic numbers
Let $X$ be a smooth projective variety over $\mathbf{F}_q$ and $\overline{X}$ its base change to $\overline{\mathbf{F}_q}$.
By Deligne’s Weil I, the eigenvalues of the geometric Frobenius acting on $...
user131191
4
votes
1
answer
283
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Interpolation of families of local fields
Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$...
user129156
11
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1
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“Algebraization" of $p$-adic fields
Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$.
Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $...
user129156
8
votes
1
answer
1k
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Relation between Galois theory and Etale Cohomology
I am a graduate student working on category theory. I am familiar with categorical Galois theory, in the way developed by Janelidze - as described for example in "Galois Theories". I am now trying to ...
