Questions tagged [etale-covers]

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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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132 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 $G$), ...
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5 votes
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280 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 ...
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1 vote
0 answers
68 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 $...
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4 votes
1 answer
143 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. ...
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3 votes
0 answers
131 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 ...
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2 votes
1 answer
198 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 ...
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  • 1,015
2 votes
1 answer
214 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 ...
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2 votes
0 answers
181 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....
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4 votes
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221 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 ...
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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{...
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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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  • 323
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202 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 ...
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1 vote
1 answer
362 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 ...
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1 answer
270 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 ...
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6 votes
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220 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 $\...
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5 votes
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553 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}_{\...
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3 votes
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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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  • 4,283
1 vote
0 answers
131 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 ...
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2 votes
0 answers
138 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 $Y$...
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266 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 ...
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  • 1,015
8 votes
1 answer
165 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, ...
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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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1 vote
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163 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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1 vote
0 answers
113 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 ...
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  • 1,015
2 votes
0 answers
193 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 ...
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9 votes
3 answers
995 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 $\overline{K}$-...
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7 votes
1 answer
303 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 ...
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  • 597
4 votes
1 answer
410 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 $V\to X$ (...
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  • 1,614
5 votes
1 answer
433 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 ...
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1 vote
0 answers
128 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 ...
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  • 271
5 votes
1 answer
331 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 ...
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  • 271
0 votes
0 answers
236 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{...
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3 votes
0 answers
202 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{...
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14 votes
0 answers
577 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 ...
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1 vote
1 answer
198 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 ...
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1 vote
1 answer
136 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 ...
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5 votes
2 answers
341 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 ...
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3 votes
0 answers
156 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 ...
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4 votes
1 answer
638 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^...
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7 votes
1 answer
400 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 ...
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9 votes
1 answer
828 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. ...
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4 votes
0 answers
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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.
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  • 786
8 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 ...
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  • 305
3 votes
2 answers
299 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 ...
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  • 133
3 votes
0 answers
133 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 ...
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  • 1,278
4 votes
0 answers
729 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 ...
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  • 2,185
6 votes
1 answer
231 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 ...
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0 votes
0 answers
128 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_!(...
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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 $\...
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