Questions tagged [etale-covers]

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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 ...
Terry Tao's user avatar
  • 108k
24 votes
3 answers
2k views

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 ...
Dedalus's user avatar
  • 1,071
17 votes
2 answers
1k views

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 $\...
Lior Bary-Soroker's user avatar
15 votes
0 answers
486 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 ...
Piotr Achinger's user avatar
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 ...
solbap's user avatar
  • 3,938
14 votes
1 answer
837 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 ...
PrimeRibeyeDeal's user avatar
14 votes
0 answers
718 views

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 ...
Jędrzej Garnek's user avatar
13 votes
2 answers
903 views

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 ...
Alex Youcis's user avatar
11 votes
3 answers
1k views

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 $\overline{K}$-...
David Urbanik's user avatar
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\...
Quinlan Aktaş's user avatar
10 votes
1 answer
1k views

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]...
Evan O'Dorney's user avatar
10 votes
1 answer
367 views

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 $...
themathandlanguagetutor's user avatar
9 votes
1 answer
1k views

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. ...
geometer's user avatar
  • 713
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 ...
Yellow Pig's user avatar
  • 2,490
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=...
user36795's user avatar
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 ...
Kestutis Cesnavicius's user avatar
9 votes
1 answer
262 views

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, ...
SAIKYO's user avatar
  • 113
8 votes
1 answer
1k views

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 ...
man's user avatar
  • 305
7 votes
1 answer
363 views

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 ...
KReiser's user avatar
  • 659
7 votes
1 answer
495 views

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 ...
Taisong Jing's user avatar
7 votes
0 answers
291 views

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 ...
Benedikt's user avatar
6 votes
1 answer
429 views

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 ...
Pippo's user avatar
  • 291
6 votes
1 answer
278 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 ...
Giulio Bresciani's user avatar
6 votes
0 answers
339 views

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 $\...
Antoine Ducros's user avatar
6 votes
0 answers
276 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 ...
peterx's user avatar
  • 693
6 votes
0 answers
293 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 $Y$, ...
Nicholas Proudfoot's user avatar
5 votes
2 answers
402 views

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 ...
Daniel Loughran's user avatar
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 ...
Piotr Achinger's user avatar
5 votes
1 answer
413 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 ...
John Z.'s user avatar
  • 53
5 votes
1 answer
643 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 $V\to X$ (...
Lao-tzu's user avatar
  • 1,856
5 votes
1 answer
298 views

$\mathbb{A}^1$-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 ...
Elden Elmanto's user avatar
5 votes
1 answer
254 views

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 ...
Crystallineperiodic's user avatar
5 votes
1 answer
597 views

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 ...
Steve's user avatar
  • 453
5 votes
1 answer
366 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 ...
Li Yutong's user avatar
  • 3,362
5 votes
1 answer
301 views

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 $\mathcal{M}_{1, 1, k}$ ...
PIELEO13's user avatar
5 votes
1 answer
383 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 ...
user46578's user avatar
  • 823
5 votes
0 answers
309 views

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 ...
Nicolas Banks's user avatar
5 votes
0 answers
613 views

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}_{\...
Ronald J. Zallman's user avatar
5 votes
0 answers
341 views

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 ...
user267839's user avatar
  • 5,948
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 ...
Rami's user avatar
  • 2,571
4 votes
1 answer
697 views

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^...
rori's user avatar
  • 231
4 votes
2 answers
1k 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)\...
ozheidi's user avatar
  • 229
4 votes
2 answers
933 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. ...
Sasha's user avatar
  • 5,492
4 votes
1 answer
610 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 $K$-...
Harry's user avatar
  • 1,203
4 votes
1 answer
176 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. ...
Hajime_Saito's user avatar
4 votes
1 answer
355 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 ...
123's user avatar
  • 41
4 votes
0 answers
309 views

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 ...
Martin Skilleter's user avatar
4 votes
0 answers
255 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.
Bonbon's user avatar
  • 806
4 votes
0 answers
393 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 ...
Masse's user avatar
  • 381
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 ...
Andreas Mihatsch's user avatar