All Questions
Tagged with ag.algebraic-geometry etale-covers
84 questions
10
votes
1
answer
1k
views
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
1
vote
0
answers
175
views
Canonical étale path between a point and its ''nearby'' point
Consider the punctored line $X=\Bbb{A}^1_k\setminus \{s_1,\ldots,s_n\}$ over some field $k$. A(n étale) path in $X$ between two geometric points $x$ and $y$ is, by definition, an isomorphism between ...
- 311
3
votes
0
answers
211
views
Étale endomorphism of $\operatorname{GL}_n$ surjective over an algebraic closure
I am currently reading chapter 1, exposé XXII of SGA7 and I am stuck at the following argument, left without explanation. It can be formulated like this:
Let $k$ be a separably closed field and $\bar{...
- 1,479
9
votes
1
answer
2k
views
Under what conditions is the induced map of etale fundamental groups surjective?
Let $f:X \to Y$ be a morphism of schemes. I am interested in sufficient conditions on $f$ which would ensure that the induced map $\pi_1^{et}(X) \to \pi_1^{et}(Y)$ of etale fundamental groups is ...
- 2,964
5
votes
0
answers
1k
views
Is it true that any étale morphism is quasi-affine?
Let $\phi:X\to Y$ be an étale morphism of Noetherian schemes. Does $\phi$ have to be quasi-affine? In other words, if $Y$ is affine does it mean that $X$ is quasi-affine?
It will follow from the ...
- 2,649
6
votes
1
answer
292
views
Finite étale covers of concentrated schemes and extension of base field
Let $k'/k$ be an extension of algebraically closed fields of characteristic $0$, and $X$ a concentrated (i.e. quasi-compact and quasi-separated) scheme over $k$.
Question: is the pullback functor ...
- 1,000
5
votes
1
answer
434
views
A weak version of high dimensional Abhyankar's conjecture
I am encountering the following situation which is similar to the Abhyankar's
higher dimensional conjecture on étale fundamental groups, but with much
stronger assumptions:
Let $S$ be a finitely ...
- 53
14
votes
2
answers
1k
views
Example: Principal G bundle that is not Zariski locally trivial, G not finite and G simply connected
Let $G$ be an affine algebraic group over $\mathbb{C}$. It is well known that when working with principal $G$ bundles it is too restrictive to require bundles to be locally trivial in the Zariski ...
- 3,968
4
votes
0
answers
279
views
Is there a Seifert–van Kampen theorem for etale fondemental group?
Is there a Seifert–van Kampen theorem for etale fondemental group? (for example for varieties over a non-algebraically closed field) Any example is welcome.
- 806
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
3
votes
0
answers
526
views
An Explicit Example of Galois Theory for Schemes
I'm currently attempting to understand Galois theory for schemes, largely following the books Galois Theory for Schemes by Henrik Lenstra and Galois Groups and Fundamental Groups by Tamas Szamuely. ...
user106566
3
votes
0
answers
152
views
exact sequence of fundamental groups associated to "almost" smooth families of curves
Suppose I have a proper, flat family of curves $X \to S$ that has a section. Fix a basepoint $s \in S$ and let $X_s$ denote the corresponding fiber. Let $\mathbb{L}$ be a set of primes which does not ...
- 1,298
2
votes
0
answers
256
views
Proof of this ‘lemme connu’
In the proof of Corollary 10.12 of Exposé I of SGA 1 something like the following is asserted as a ‘known lemma’:
Let $k$ be an infinite field and $B$ a finite $k$-algebra. If $B$ is not a product ...
- 1,217
3
votes
2
answers
2k
views
homotopy exact sequence for the étale fundamental group
I have been trying to understand the homotopy exact sequence for the étale fundamental group which says
$$ 1 \rightarrow \pi_1 (\bar{X},\bar{x_0})\rightarrow \pi_1 (X,x_0)\rightarrow Gal(k)\...
- 319
0
votes
0
answers
133
views
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_!(...
user120812
2
votes
3
answers
651
views
question about the induced homomorphism of etale fundamental groups
Background/Setup
For any connected scheme $S$, let $\text{FEt}_S$ denote the category of finite etale $S$-schemes. Let $f : X\rightarrow Y$ be a morphism of connected schemes, then for any finite ...
- 10.7k
15
votes
0
answers
517
views
Zariski vs etale torsors over abelian varieties
Question. Let $A$ be an abelian variety (say, over the complex numbers), $G$ an algebraic group, $c$ a class in $H^1_{\rm et}(A, G)$. Denote the multiplication by $N$ map on A by $m_N:A\to A$. Does ...
- 16.2k
3
votes
1
answer
2k
views
The étale fundamental group in the non-normal case
It is known, that the étale fundamental group of a normal connected scheme equals the galois group of the maximal unramified extension of its function field.
This is not true for integral schemes in ...
0
votes
1
answer
528
views
Artin approximation of a diagram
Let consider $f:(X,x)\to (Z,z)$ and $g:(Y,y)\to (Z,z)$ morphisms of pointed $k$-schemes of finite type ($k$ is a field). Suppose that there exists a map on the level of formal neighborhoods $\phi:X_{x}...
- 3,472
9
votes
2
answers
1k
views
Henselian couples and finite etale morphisms
Let $S$ be a scheme and $S_0 \subset S$ a closed subscheme. Then $(S, S_0)$ is said to be a Henselian couple if for every finite $X \rightarrow S$, setting $X_0 := X\times_S S_0$, the natural map from ...
- 5,019
9
votes
1
answer
1k
views
étale covers and torsion line bundles
Let $n \geq 2$ be an integer, $X$ a smooth variety over a field $k$ containing $\mu_n$ and $G$ a cyclic group of order $n$ acting on it. Assume that the action is free. Then the morphism $\pi: X \to Y=...
- 91
4
votes
2
answers
1k
views
About "de-Rham" and "l-adic" local systems - comparison
Hello,
Suppose that $k$ is an algebraically closed field of char. 0.
Let $X$ be a smooth connected variety over $k$.
Then I have the category $A$ of Regular Singular smooth $D$-modules on $X$ (i.e. ...
- 5,562
5
votes
1
answer
398
views
Covering of schemes and flatness
Let $f:X \to Y$ be a finite surjective morphism of quasi-projective schemes over $\mathbb{C}$, $X$ is reduced and $Y$ is integral. Suppose that there exists an integer $n$ such that for every closed ...
- 833
5
votes
1
answer
1k
views
Picard groups of abelian étale covers
Let $X$ be a scheme (you can assume that $X$ is proper and smooth over an algebraically closed field) and $T$ is a finite subgroup of $\text{Pic } X$ (of order prime to the characteristic). Does there ...
- 16.2k
6
votes
0
answers
301
views
Overview and/or reference of theory of pro-universal covers?
This question will contain very little in the way of concrete information, because I don't have much to go on. I've heard whispers of something called a "pro-universal cover," which is the inverse ...
- 693
3
votes
2
answers
471
views
purity for finite flat group schemes
Let $X$ be a "nice" scheme and $Z \hookrightarrow X$ closed of codimension $\geq 2$. Let $Y$ over $X \setminus Z$ be a torsor for a finite flat group scheme $G/X$.
Does $Y$ spread out to a $G$-torsor ...
user19475
1
vote
1
answer
609
views
Descend of etale morphism
I am not sure whether the title is appropriate for this question or not. I am sorry if there is anyone who is confused with the title and the contents.
What I want to ask is the following: let $k$ be ...
- 601
5
votes
1
answer
378
views
Étale covers and birationality of varieties
All varieties are assumed to be projective over $\mathbb{C}$. Let $f_1: Y \to X$ and $f_2: Y' \to X$ be étale morphisms with same finite Galois groups (to be honest, I don't know what does Galois ...
- 3,472
7
votes
0
answers
312
views
Etale local isomorphism to the tangent cone
Let $X$ be a scheme and $p\in X$ a closed point. We say that $(X,p)$ is etale locally isomorphic to $(Y,q)$ if there exists an etale neighborhood of $p$ in $X$, and etale neighborhood of $q$ in $Y$, ...
- 2,537
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
1
answer
369
views
Structure of fundamental groups arising from smooth projective morphisms
Let $f:X\to B$ be a smooth projective morphism of complex algebraic varieties.
If $f$ is of relative dimension zero, i.e., $f$ is a finite etale cover, then the image of the topological fundamental ...
- 41
2
votes
0
answers
385
views
branch locus of the discriminant map $\overline{\mathcal{H}}_{g',r} \to \overline{\mathcal{M}}_{g,n}$
Let $\overline{\mathcal{M}}_{g,n}$ be the moduli space of pointed, stable, genus $g$ curves. Let $\overline{\mathcal{H}}_{g',r}$ be the hurwitz space of cyclic covers of degree $r$ of genus $g$ curves ...
- 3,779
0
votes
1
answer
203
views
A question regarding etale descent
We will always work with finite-type, smooth schemes over a field $k$. Let $\pi: Y \to X$ be an etale map of $k$-schemes. Let $Z$ be another $k$-scheme admitting a morphism $f: Y \to Z$. Suppose ...
- 179
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