All Questions
Tagged with etale-covers arithmetic-geometry
8 questions
14
votes
1
answer
897
views
Examples of étale covers of arithmetic surfaces
Define an arithmetic scheme $X$ to be a separated, integral scheme, flat and finite type over $\mathbb{Z}$. I am interested in obtaining examples of finite étale covers of arithmetic schemes. I am ...
- 1,338
11
votes
3
answers
1k
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Are "large enough" finite etale covers arithmetic?
Let $X$ be a variety over a number field $K$. Then it is known that for any topological covering $X' \to X(\mathbb{C})$, the topological space $X'$ can be given the structure of a $\overline{K}$-...
- 437
10
votes
1
answer
1k
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Which of these 4 definitions of Galois coverings of integral schemes are equivalent?
Here are four possible definitions for an etale, finite, surjective map $X\rightarrow Y$ between integral schemes to be considered Galois:
There exists a finite group $G$, and an action $\varphi: G\...
- 460
4
votes
1
answer
625
views
Does a curve over a number field have a finite etale cover of given degree
Let $X$ be a (smooth projective geometrically connected) curve over a number field $K$ of genus $g\geq 2$. Let $d\geq 2$ be an integer.
Does there exist a curve $Y$ over $K$ with a finite etale $K$-...
- 1,213
4
votes
0
answers
395
views
Do regular noetherian schemes of dimension one only have finitely many etale covers of bounded degree
Let $X$ be a regular noetherian scheme of dimension one. Let $d$ be an integer.
Question. Are there only finitely many finite etale morphisms $Y\to X$ of degree $d$?
I want to exclude finite etale ...
- 381
2
votes
1
answer
94
views
Base change for fundamental group prime to p in mixed characteristic?
I found the answer to this question while typing it up, but since I've already written it, it is probably worthwhile to post-and-answer in case someone finds it useful.
Let $S=\operatorname{Spec}\...
- 371
2
votes
1
answer
148
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Galois action on étale path torsors
TLDR: How is the Galois action on étale path torsors defined?
Let $X$ be a smooth proper scheme, over a field $k$, and let $x,y\in X(k)$ be a pair of points. Let $\pi_1^{\text{ét}}(\overline{X},\...
- 2,907
0
votes
0
answers
133
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Operations on étale sheaves
Which of the following operations on étale sheaves $A$ commute with tensor powers? (eg. for instance $i^*(A^{\otimes n})=(i^*(A))^{\otimes n}$?)
$i^*(A)$, $i$ closed immersion.
$i_*(A)$
$i^!(A)$
$i_!(...
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