Questions tagged [etale-covers]
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99 questions
33
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An etale version of the van Kampen theorem
Let $V$ be a smooth connected algebraic variety over an algebraically closed field $k$. Let $W_1, W_2$ be closed subvarieties of $V$ of positive codimension whose intersection $W_1 \cap W_2$ has ...
- 114k
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votes
3
answers
2k
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Explicit computations of the étale homotopy type?
Hi,
I'm currently trying to learn about etale homotopy for schemes as introduced by Artin-Mazur. I know that by the Artin-Mazur comparision theorem, it is possible to compute the etale homotopy type ...
- 1,071
17
votes
2
answers
1k
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A short proof for simple connectedness of the projective line
The Riemann-Hurwitz formula implies that the projective line $\mathbb{P}^1_K$ over any algebraically closed field $K$ is simply connected (i.e., $\pi_1^{et}(\mathbb{P}^1_K) = 1$; equivalently, if $\...
- 3,362
15
votes
0
answers
777
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Fundamental group of formal punctured disc and punctured affine line
On a course that ended some time ago, I was handed the following problem:
Problem: Compute $\pi_1^{ét} (\mathbb{A}^1_{\mathbb C} \setminus \{ 0 \}, \overline x)$.
Hint: Find all finite ...
- 421
15
votes
0
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517
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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 ...
- 16.1k
14
votes
2
answers
951
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Relationship between étale and topological $K(\pi,1)$s
I was trying to find a proof, or a counterexample to the claim that if $X/\mathbb{C}$ is connected smooth projective, then $X$ is a $K(\pi^{\mathrm{\acute{e}t}},1)$ if and only if $X^\mathrm{an}$ is a ...
- 787
14
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2
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1k
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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
14
votes
1
answer
897
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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
11
votes
1
answer
2k
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Galois theory for products of fields (aka finite etale extensions)
Let $F$ be a field. By a Galois algebra over $F$ I mean a finite etale extension, that is, a product $K = K_1 \times \cdots \times K_r$ of finite (separable) field extensions, of total degree $[K : F]...
- 980
11
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1
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413
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Why can we take the colimit over the category of elements?
I'm trying to understand J. P. Murre's Tata notes on Grothendieck's theory of the fundamental group. For a Galois category $\mathcal C$ (which I'm taking to be locally small) with fundamental functor $...
10
votes
1
answer
1k
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Are higher etale homotopy groups topological groups in a natural way?
Since etale fundamental group of a scheme $X$ is the group of natural automorphisms of the fibre functor of the category of finite etale covers of $X$, it comes with structure of a topological group. ...
- 723
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
9
votes
1
answer
2k
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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
9
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1
answer
1k
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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=...
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9
votes
2
answers
1k
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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
305
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Formally etale algebras over fields of characteristic 0
I was wondering if anyone might have a non-trivial example of a formally etale algebra over a field of characteristic 0 which is not ind-etale (i.e. a union of etale extensions).
For some motivation, ...
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votes
1
answer
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Why only finite morphisms in etale fundamental group?
Can one define a version of etale fundamental group which takes into account infinite etale covers? What properties of the usual etale fundamental group would fail for it?
P.S.: here one can find ...
- 305
7
votes
1
answer
386
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On a quasi-separated assumption in a lemma for the homotopy exact sequence of the etale fundamental group
Background:
I've seen two versions of the homotopy exact sequence for etale fundamental groups. One from Stacks:
Stacks 0BTX: Let $k$ be a field with algebraic closure $\overline{k}$. Let $X$ be a ...
- 701
7
votes
1
answer
695
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Constructible étale sheaves on X are étale algebraic spaces over X
I saw the following statement in a paper of Bhatt-Mathew:
Let $X$ be a quasicompact quasiseparated scheme. Then there is an equivalence of categories between constructible étale sheaves (of sets) on ...
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7
votes
1
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538
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Is the quotient presheaf $\mathbb{G}_m/\mu_p$ an étale sheaf?
I apologize if the question is bit trivial for mathoverflow, but I asked on stack exchange a while ago and haven't got any answer.
Let $k$ be a field of characteristic $p > 0$. Consider the ...
- 536
7
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0
answers
330
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Künneth formula for $\pi_1$-proper morphisms
Context:
Let $X$ and $Y$ be connected qcqs schemes over an algebraically closed field $k$. Denote by $\pi_1(X)$, $\pi_1(Y)$ their étale fundamental groups (base points omitted). Grothendieck proved ...
- 71
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0
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312
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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
6
votes
1
answer
471
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Étale fundamental group of multiplicative group over an algebraically/separably closed field
This is a repost of my question here.
Do we know the structure of the étale fundamental group $\pi^\text{et}_1(\mathbb{G}_{m,K^\text{sep}})$ of the multiplicative group, for a given field $K$? For ...
- 311
6
votes
1
answer
292
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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
6
votes
0
answers
377
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Fundamental group of a product in characteristic 0
It is proven in SGA1 that if $k$ is an algebraically closed field, if $X$ is a proper $k$-scheme and if $Y$ is a locally noetherian $k$-scheme (say, $X$ and $Y$ are non-empty and connected) then $\...
- 1,584
6
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0
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301
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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
5
votes
2
answers
456
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Finite etale covers of products of curves
Probably this question can be phrased in a much greater generality, but I will just state it in the generality I require. I work over $\mathbb{C}$.
Let $C_1, C_2 \subset \mathbb{P}^1$ be non-empty ...
- 21.4k
5
votes
1
answer
1k
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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 ...
- 16.1k
5
votes
1
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434
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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
5
votes
1
answer
739
views
Surjective étale morphisms étale locally split
The actual question is slightly more general than that in the title:
Let $p: U\to Y$ be a surjective étale morphism and $Y\to X$ be a finite morphism of schemes. Is there an étale cover $V\to X$ (...
- 1,906
5
votes
1
answer
337
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$\mathbb{A}^1$-invariance of categories of Finite Etale Covers
Let $k$ be algebraically closed with characteristic $0$. For a scheme $X$, let $FEt(X)$ be the category of finite etale covers of $X$. What can be said about $FEt(X \times \mathbb{A}^1)$ and the ...
- 2,327
5
votes
1
answer
261
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Two curves of genus $g \geq 2$ in characteristic $p >0 $ with isomorphic abelianizations
Let $k$ be an algebraically closed field of characteristic $p>0.$ How can I construct two projective curves $C_1,C_2$ of genus $ g \geq 2$ so that the abelianizations $\pi_1(C_i)^{ab},i=1,2$ are ...
5
votes
1
answer
378
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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
5
votes
1
answer
371
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How the automorphism group of an elliptic curve acts at the localization of the stack $\mathcal{M}_{1, 1, k}$ at the corresponding point
I am studying the enlightening article "The Picard Group of $\mathcal{M}_{1, 1, S}$", written by Fulton and Olsson, but I have some problems with a proof.
Setting
Let $\mathcal{M}_{1, 1, k}$ ...
- 53
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
0
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316
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To what extent are geometric methods being used to attack the inverse Galois problem?
My limited knowledge so far is that some groups have been constructed geometrically using the theory of covering spaces, then applying Hilbert irreducibility.
Is there a deeper way in which inverse ...
- 131
5
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0
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628
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Étale homotopy type of $\text{Spec}(\mathbb{Z}) \cup \{ \text{place}_\infty \}$
Has anyone formally calculated the étale homotopy type of $\text{Spec}(\mathbb{Z}) \cup \{ \text{place}_{\infty} \}$?
According to arithmetic topology, $\text{Spec}(\mathbb{Z}) \cup \{ \text{place}_{\...
user30211
5
votes
0
answers
349
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Algebraic spaces as quotients of schemes (Definition from wikipedia)
I think that wikipedia article on Algebraic spaces contains a serious content error in the part on the definition of Algebraic spaces as quotients of schemes and I would like to discuss if it is ...
- 6,038
5
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0
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1k
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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
4
votes
1
answer
704
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Do higher etale homotopy groups of spectrum of a field always vanish?
Let $k$ be a field. In what generality is it true that higher etale homotopy groups
of $\mathrm{Spec}\,k$ vanish?
If the absolute Galois group is finite, we have a universal cover $\mathrm{Spec}\,k^...
- 231
4
votes
2
answers
1k
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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
4
votes
1
answer
625
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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
190
views
Regarding a 'global' version of Chase-Harrison-Rosenberg exact sequence for rings
If $R$ is a commutative ring with identity with a 'nice' action of a finite group $G$, the subring $R^G\subset R$ gives a Galois extension of rings. In this case, S.U. Chase, D.K. Harrison, A. ...
- 737
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1
answer
369
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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
4
votes
0
answers
97
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Is there a concept of a map of Grothendieck sites having dense image?
Someone recently asked if one can talk about a map being etale dense just like one can talk about it being Zariski dense. My main question is: has anyone discussed such a notion?
On a simple ...
- 15.4k
4
votes
0
answers
343
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Building intuition for the étale topology
My Honours supervisors have charged me with building intuition for étale morphisms and the étale topology. Their suggestions were to "compute the étale topology in a few simple cases", such ...
- 721
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
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0
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395
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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
3
votes
1
answer
2k
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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 ...
