Questions tagged [etale-covers]

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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 ...
Ben C's user avatar
  • 2,549
2 votes
0 answers
67 views

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)...
Display Name's user avatar
1 vote
0 answers
70 views

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 ...
TCiur's user avatar
  • 381
1 vote
1 answer
209 views

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": ...
Federico Fallucca's user avatar
5 votes
1 answer
236 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
7 votes
0 answers
250 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
3 votes
1 answer
244 views

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 \...
TCiur's user avatar
  • 381
3 votes
1 answer
175 views

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 ...
user197402's user avatar
0 votes
0 answers
160 views

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 $G$), ...
Hajime_Saito's user avatar
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293 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
1 vote
0 answers
136 views

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 $...
Leo Herr's user avatar
  • 1,004
4 votes
1 answer
158 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
3 votes
0 answers
141 views

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 ...
Martin Skilleter's user avatar
2 votes
1 answer
242 views

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 ...
user267839's user avatar
  • 4,956
2 votes
1 answer
255 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 ...
Martin Skilleter's user avatar
2 votes
0 answers
183 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....
Martin Skilleter's user avatar
4 votes
0 answers
259 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
1 vote
0 answers
123 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{...
user127776's user avatar
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1 vote
0 answers
174 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 $\mathbf{P}^2_{\mathbf{F}_q}$?
hennlu's user avatar
  • 323
2 votes
0 answers
228 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 ...
user127776's user avatar
  • 5,607
1 vote
1 answer
416 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 ...
S.D.'s user avatar
  • 482
2 votes
1 answer
326 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 ...
user127776's user avatar
  • 5,607
6 votes
0 answers
294 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
5 votes
0 answers
586 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}_{\...
Cayley-Hamilton's user avatar
3 votes
0 answers
120 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 ...
Cayley-Hamilton's user avatar
1 vote
0 answers
145 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 ...
curious math guy's user avatar
2 votes
0 answers
147 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 $Y$...
Mellon's user avatar
  • 197
5 votes
0 answers
312 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
  • 4,956
9 votes
1 answer
238 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
1 vote
0 answers
108 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 ...
curious math guy's user avatar
1 vote
0 answers
195 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 ...
user190964's user avatar
1 vote
0 answers
121 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 ...
user267839's user avatar
  • 4,956
2 votes
0 answers
248 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 ...
user45397's user avatar
  • 2,013
9 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
7 votes
1 answer
338 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
  • 617
5 votes
1 answer
534 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
  • 1,724
5 votes
1 answer
516 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
1 vote
0 answers
148 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 ...
Pippo's user avatar
  • 271
5 votes
1 answer
370 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
  • 271
0 votes
0 answers
275 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{...
Moutand Mohammed's user avatar
3 votes
0 answers
205 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{...
Stabilo's user avatar
  • 1,345
14 votes
0 answers
644 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
1 vote
1 answer
214 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 ...
pirignao's user avatar
1 vote
1 answer
159 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 ...
user avatar
5 votes
2 answers
361 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
3 votes
0 answers
159 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 ...
JJJ's user avatar
  • 167
4 votes
1 answer
677 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
7 votes
1 answer
466 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
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
4 votes
0 answers
243 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
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