The etale-covers tag has no wiki summary.
2
votes
1answer
121 views
Which algebras can be presented as filtered colimits of f.g. regular ones with smooth connecting morphisms?
Let $R$ be a regular (commutative associative unitial) algebra over a prime field $F$ (i.e. $F=F_p$ or $F=\mathbb{Q}$); assume that it is noetherian excellent (and even of Krull dimension $1$). What ...
0
votes
1answer
110 views
A question regarding etale descent
We will always work with finite-type, smooth schemes over a field $k$. Let $\pi: Y \to X$ be an etale map of $k$-schemes. Let $Z$ be another $k$-scheme admitting a morphism $f: Y \to Z$. Suppose ...
9
votes
1answer
214 views
Example: Principal G bundle that is not Zariski locally trivial, G not finite and G simply connected
Let $G$ be an affine algebraic group over $\mathbb{C}$. It is well known that when working with principal $G$ bundles it is too restrictive to require bundles to be locally trivial in the Zariski ...
3
votes
2answers
251 views
homotopy exact sequence for the étale fundamental group
I have been trying to understand the homotopy exact sequence for the étale fundamental group which says
$ 1 \rightarrow \pi_1 \bar{X},\bar{x_0})\rightarrow \pi_1 (X,x_0)\rightarrow Gal(k)\rightarrow ...
9
votes
2answers
200 views
Henselian couples and finite etale morphisms
Let $S$ be a scheme and $S_0 \subset S$ a closed subscheme. Then $(S, S_0)$ is said to be a Henselian couple if for every finite $X \rightarrow S$, setting $X_0 := X\times_S S_0$, the natural map from ...
2
votes
0answers
127 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 ...
4
votes
1answer
164 views
Étale covers and birationality of varieties
All varieties are assumed to be projective over $\mathbb{C}$. Let $f_1: Y \to X$ and $f_2: Y' \to X$ be étale morphisms with same finite Galois groups (to be honest, I don't know what does Galois ...
3
votes
1answer
169 views
Structure of fundamental groups arising from smooth projective morphisms
Let $f:X\to B$ be a smooth projective morphism of complex algebraic varieties.
If $f$ is of relative dimension zero, i.e., $f$ is a finite etale cover, then the image of the topological fundamental ...
5
votes
1answer
235 views
étale covers and torsion line bundles
Let $n \geq 2$ be an integer, $X$ a smooth variety over a field $k$ containing $\mu_n$ and $G$ a cyclic group of order $n$ acting on it. Assume that the action is free. Then the morphism $\pi: X \to ...
2
votes
1answer
234 views
The étale fundamental group in the non-normal case
It is known, that the étale fundamental group of a normal connected scheme equals the galois group of the maximal unramified extension of its function field.
This is not true for integral schemes in ...
2
votes
2answers
276 views
About “de-Rham” and “l-adic” local systems - comparison
Hello,
Suppose that $k$ is an algebraically closed field of char. 0.
Let $X$ be a smooth connected variety over $k$.
Then I have the category $A$ of Regular Singular smooth $D$-modules on $X$ (i.e. ...
3
votes
2answers
255 views
purity for finite flat group schemes
Let $X$ be a "nice" scheme and $Z \hookrightarrow X$ closed of codimension $\geq 2$. Let $Y$ over $X \setminus Z$ be a torsor for a finite flat group scheme $G/X$.
Does $Y$ spread out to a $G$-torsor ...
4
votes
0answers
160 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 ...
17
votes
2answers
532 views
Explicit computations of the étale homotopy type?
Hi,
I'm currently trying to learn about etale homotopy for schemes as introduced by Artin-Mazur. I know that by the Artin-Mazur comparision theorem, it is possible to compute the etale homotopy type ...
22
votes
1answer
2k views
An etale version of the van Kampen theorem
Let $V$ be a smooth connected algebraic variety over an algebraically closed field $k$. Let $W_1, W_2$ be closed subvarieties of $V$ of positive codimension whose intersection $W_1 \cap W_2$ has ...
1
vote
1answer
288 views
Does a curve over a number field have a finite etale cover of given degree
Let $X$ be a (smooth projective geometrically connected) curve over a number field $K$ of genus $g\geq 2$. Let $d\geq 2$ be an integer.
Does there exist a curve $Y$ over $K$ with a finite etale ...
5
votes
1answer
254 views
Picard groups of abelian étale covers
Let $X$ be a scheme (you can assume that $X$ is proper and smooth over an algebraically closed field) and $T$ is a finite subgroup of $\text{Pic } X$ (of order prime to the characteristic). Does there ...
