All Questions
Tagged with ag.algebraic-geometry etale-covers
35 questions with no upvoted or accepted answers
15
votes
0
answers
777
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 ...
- 421
15
votes
0
answers
517
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 ...
- 16.2k
7
votes
0
answers
330
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 ...
- 71
7
votes
0
answers
312
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 $Y$, ...
- 2,537
6
votes
0
answers
377
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 $\...
- 1,584
6
votes
0
answers
301
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 ...
- 693
5
votes
0
answers
316
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 ...
- 131
5
votes
0
answers
628
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}_{\...
user30211
5
votes
0
answers
349
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 ...
- 6,038
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 ...
- 2,649
4
votes
0
answers
97
views
Is there a concept of a map of Grothendieck sites having dense image?
Someone recently asked if one can talk about a map being etale dense just like one can talk about it being Zariski dense. My main question is: has anyone discussed such a notion?
On a simple ...
- 15.4k
4
votes
0
answers
343
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 ...
- 721
4
votes
0
answers
279
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.
- 806
4
votes
0
answers
395
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 ...
- 381
3
votes
0
answers
140
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 ...
user30211
3
votes
0
answers
211
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{...
- 1,479
3
votes
0
answers
152
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 ...
- 1,298
3
votes
0
answers
526
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. ...
user106566
2
votes
0
answers
111
views
Is the connecting map $\pi_2(B) \to \pi_1(F)$ ever nonzero in smooth proper families?
Suppose that $X, B$ are smooth irreducible varieties over $\mathbb{C}$ and $f : X \to B$ is a smooth proper morphism. Then we can consider the homotopy exact sequence:
$$ \pi_2(B) \to \pi_1(F) \to \...
- 3,720
2
votes
0
answers
189
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....
- 721
2
votes
0
answers
332
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 ...
- 5,901
2
votes
0
answers
174
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 $Y$...
- 197
2
votes
0
answers
325
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 ...
- 2,323
2
votes
0
answers
256
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 ...
- 1,217
2
votes
0
answers
385
views
branch locus of the discriminant map $\overline{\mathcal{H}}_{g',r} \to \overline{\mathcal{M}}_{g,n}$
Let $\overline{\mathcal{M}}_{g,n}$ be the moduli space of pointed, stable, genus $g$ curves. Let $\overline{\mathcal{H}}_{g',r}$ be the hurwitz space of cyclic covers of degree $r$ of genus $g$ curves ...
- 3,779
1
vote
0
answers
81
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 ...
- 679
1
vote
0
answers
127
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{...
- 5,901
1
vote
0
answers
161
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 ...
- 4,418
1
vote
0
answers
118
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 ...
- 4,418
1
vote
0
answers
274
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 ...
- 317
1
vote
0
answers
135
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 ...
- 6,038
1
vote
0
answers
175
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 ...
- 311
0
votes
0
answers
196
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 $G$), ...
- 737
0
votes
0
answers
392
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{...
- 181
0
votes
0
answers
133
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_!(...
user120812