Questions tagged [etale-covers]

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votes
1answer
149 views

Equivalence of categories between finite étale covers of connected scheme and finite continuous permutation representations of étale fundamental group

The title and tags say it all: I am looking for a clean statement and proof of the equivalence of categories between finite étale covers of a connected $k$-scheme and finite continuous permutation ...
2
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0answers
165 views

Missing Detail in Construction of Étale Fundamental Group

$\DeclareMathOperator\Aut{Aut}$I am currently trying to consolidate my understanding of the étale fundamental group, and there is a small detail in the construction that I do not understand in general....
4
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0answers
179 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 ...
1
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0answers
118 views

Rational varieties over finite fields admit an open set isomorphic to an affine space

This paper roughly claims that given a projective variety $X$ over a finite field, there is a finite map $f:X\rightarrow \mathbb{P}^n$ such that if $H$ is a hyperplane in $\mathbb{P}^n$ and $U=\mathbb{...
1
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0answers
104 views

What is the étale fundamental group of projective spaces over finite fields?

Is there any convenient way to understand the étale fundamental group of projective spaces over finite fields, in particular, the étale fundamental group of $\mathbf{P}^2_{\mathbf{F}_q}$?
2
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0answers
175 views

Killing cohomology of structure sheaf by pullback along Frobenius and finite etale covers

On a smooth projective variety $X$ over a finite field, you can pullback any element in $H^1(X,\mathcal{O}_X)$ by a combination of Frobenius and finite etale cover so it gets killed. In order to prove ...
1
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1answer
313 views

Is every etale cover a principal bundle?

Let $f: X\rightarrow Y$ be proper etale morphism between varieties over the field of complex numbers. Does there exists a finite group $G$ such that $Y$ is the categorical quotient of $X$ under the ...
2
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1answer
195 views

Vector bundles that are fixed under pull-back by the absolute Frobenius

Are there algebraic projective curves over finite fields other than $\mathbb{P^1}$ that if a vector bundle on it, is stable under Frobenius i.e. $F^*E\cong E$ implies that $E$ is a trivial bundle? If ...
6
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0answers
171 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 $\...
5
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0answers
523 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}_{\...
3
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0answers
95 views

Weakly contractible cover in étale homotopy theory

It is easy enough to construct the separable closure $k^{sep}$ of a field $k$, which then has $\pi_0(k^{sep}) = 0$ (profinite set of connected components), $\pi_1(k^{sep}) = 0$ (there are no ...
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0answers
109 views

Pro-étale locally simply connected schemes

In topology, topological manifolds are locally simply connected. However, in the étale topology of schemes, the analogue statement is not true: If $k$ is a field then finite separable field extensions ...
2
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0answers
125 views

Finite étale morphism from a scheme to an algebraic space

Let $f : X \to Y$ be a finite, surjective étale morphism of algebraic spaces (say, of finite type over some noetherian scheme). Assume that $X$ is a scheme. Does this imply that $Y$ is a scheme? Is $Y$...
5
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0answers
220 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 ...
8
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1answer
138 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, ...
1
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0answers
95 views

Essential Image of the Étale Homotopy type

For any scheme $X$ we can associate the étale homotopy type $Et(X)$, which is a pro-object in the homotopy category of CW-complexes. My question is, do we have a good understanding of the essential ...
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0answers
137 views

automorphisms of an étale cover of a curve

The base field is algebraically closed and of chatacteristic zero. If $X$ is a smooth projective curve and $Y\to X$ is an étale covering of $X$ of degree $d$, then what can we say about the ...
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0answers
100 views

Projection from closure of locally closed subscheme is Etale

Let $S$ an arbitrary scheme and denote by $\Delta: S \to S \times S$ the diagonal immersion and $p_i: S \times S \to S$ the both projections to first resp second factor. (in following we will wlog ...
2
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0answers
163 views

Extending etale morphisms

Let $Y$ be an affine, integral, Gorenstein surface. Let $y \in Y$ be a closed point such that there exists a finite, etale morphism $f: X \to Y\backslash \{y\}$ from an integral variety $X$ to the ...
9
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3answers
937 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 $\overline{K}$-...
7
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1answer
265 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 ...
4
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1answer
263 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 $V\to X$ (...
5
votes
1answer
275 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 ...
1
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0answers
110 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 ...
5
votes
1answer
276 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 ...
0
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0answers
189 views

Galois cover corresponding to finite quotient of the étale fundamental group

Let $X$ be a connected scheme,$\pi_1(X,\bar{x})$ its étale fundamental group for some geometric point $\bar{x} : Spec(K) \rightarrow X$ and $E = \pi_1(X,\bar{x})/N$ a finite quotient of $\pi_1(X,\bar{...
3
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0answers
198 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{...
14
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0answers
470 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 ...
1
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1answer
178 views

Étale morphism over unirational/uniruled variety

Suppose we have an étale morphism between smooth quasi-projective (complex) varieties $X \rightarrow Y$ and assume that $Y$ is unirational. I am wondering whether we can somehow deduce that $X$ is ...
1
vote
1answer
129 views

Base change for prime-to-$p$ fundamental group

Let $k$ be an algebraic closure of $\mathbb{F}_p$. Let $X$ be a connected smooth quasi-projective $k$-scheme. If $K$ is an algebraically closed field containing $k$, is the prime-to-$p$ etale ...
5
votes
2answers
306 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 ...
3
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0answers
152 views

Recover an etale fundamental group from local fundamental groups?

Suppose we have a scheme $S$ and an etale covering $\{ U_i \to S \}$. How much information about $\pi_1^{et} (S)$ can we recover from all the $\pi_1^{et} (U_i)$ ? Are there special conditions we can ...
4
votes
1answer
576 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^...
7
votes
1answer
361 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 ...
9
votes
1answer
661 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. ...
4
votes
0answers
195 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.
7
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1answer
660 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 ...
3
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2answers
270 views

English literature close to “Algébre et Théories Galoisiennes” by Régine and Adrien Douady

I'm currently working on my undergraduate dissertation. I'm working on covering sapces of Riemann surfaces so my supervisor asked me to read the book I mention in the title: "Algébre et Théories ...
3
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0answers
104 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 ...
4
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0answers
540 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 ...
6
votes
1answer
222 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 ...
0
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0answers
124 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_!(...
17
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2answers
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 $\...
2
votes
0answers
247 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 ...
5
votes
1answer
378 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 ...
2
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0answers
422 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. ...
8
votes
1answer
902 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]...
12
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0answers
352 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 ...
9
votes
1answer
817 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\...
14
votes
1answer
689 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 ...