# Questions tagged [etale-covers]

The etale-covers tag has no usage guidance.

91
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### Extending étale covers from the regular locus to a resolution of singularities

Let $X$ be a normal proper variety with rational singularities (or terminal if that is necessary) and $X_{\text{reg}} \to X$ the regular locus. Let $\pi : \tilde{X} \to X$ be a resolution of ...

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### What are étale coverings of the spectrum of a discrete valuation ring?

This question comes when I try the valuative criterion on properness of the moduli space of stable sheaves. Let $X$ be a projective scheme over $\Bbbk$ with an ample line bundle $\mathcal L$. Let $P(t)...

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### Behaviour of cycles modulo algebraic equivalence on an etale covering

I found a neat result in Beauville's paper "VARIÉTÉS DE PRYM ET JACOBIENNES INTERMÉDIAIRES" : if $U \subset \mathbb{P}^n$ is an open and $V \to U$ is a conic bundle whose fibres are all ...

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### Beauville Exercise VII.7 (3)-A proof that $\kappa(X)\geq \kappa(Y)$ for $f\colon X\to Y$ surjective morphism of smooth projective varieties

Here $\kappa(X)$ denotes the Kodaira dimension of a smooth projective variety $X$.
Question 1:
I would like to solve Exercise VII.7 (3) from the Beauville book "Complex Algebraic Surfaces":
...

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1
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236
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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 $\mathcal{M}_{1, 1, k}$ ...

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250
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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 ...

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### Is the theta characteristic attached to an etale double cover of a plane quintic arising from a cubic threefold even in characteristic two?

We work over an algebraically closed field $k$ of characteristic $2$. Let $X$ be a cubic threefold realized as a conic bundle via $f: X \to \mathbb{P}^2$ (after blowing up some line). Let $C \to \...

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175
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### Reference request: Kummer étale topology and tame topology

In Theorem 7.6 of Illusie - An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology it is stated that if $X$ is a log regular fs log scheme and $U$ is the open ...

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160
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### Question regarding étale sheaf under finite étale surjective morphism

Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$, and suppose we have a surjective finite étale morphism $f:X\rightarrow Y$ (actually $Y=X/G$ for a free action of a finite group $G$), ...

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293
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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 ...

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136
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### Pushforward of sheaves along finite etale map

Suppose $\pi : Y \to X$ is a finite 'etale map of degree d. I want a formula for $\pi_* \mathcal O_Y$. I'm happy with a formula in $K$ theory.
There is a $S_d$-torsor $P \to X$ of local isomorphisms $...

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158
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### 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. ...

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141
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### Lifting isogeny over étale cover

I am in the situation where I need to lift a particular isogeny over an étale cover, and I am not sure how to justify the existence of such a lift. I am trying to fill in the details of the proof of ...

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242
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### Some facts about sheafification functor on étale site

I'm studying the book Etale cohomology and the Weil conjecture by Freitag, Kiehl and I have some questions on the subchapter introducing the machinery associating to an étale presheaf
a sheaf (that is ...

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255
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### 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 ...

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### 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....

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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 ...

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123
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### 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{...

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174
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### 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 $\mathbf{P}^2_{\mathbf{F}_q}$?

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### 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 ...

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416
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### 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 ...

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326
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### 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 ...

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294
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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 $\...

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586
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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}_{\...

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### 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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### 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 ...

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147
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### 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 $Y$...

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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 ...

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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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### 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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### 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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### 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 ...

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### 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 ...

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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 $\overline{K}$-...

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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 ...

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### 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 $V\to X$ (...

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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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### 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 ...

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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 ...

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### 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{...

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### 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{...

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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 ...

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### 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 ...

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159
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### 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 ...

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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 ...

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### 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 ...

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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^...

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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 ...

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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. ...

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### 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.