# Tagged Questions

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### 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 ...
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### 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 ...
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### Relationship between étale and topological $K(\pi,1)$s

I was trying to find a proof, or a counterexample to the claim that if $X/\mathbb{C}$ is connected smooth projective, then $X$ is a $K(\pi^{\mathrm{\acute{e}t}},1)$ if and only if $X^\mathrm{an}$ is a ...
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### Examples of étale covers of arithmetic surfaces

Define an arithmetic scheme $X$ to be a separated, integral scheme, flat and finite type over $\mathbb{Z}$. I am interested in obtaining examples of finite étale covers of arithmetic schemes. I am ...
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### 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 ...
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### Under what conditions is the induced map of etale fundamental groups surjective?

Let $f:X \to Y$ be a morphism of schemes. I am interested in sufficient conditions on $f$ which would ensure that the induced map $\pi_1^{et}(X) \to \pi_1^{et}(Y)$ of etale fundamental groups is ...
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### 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 ...
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### $\mathbb{A}^1$-invariance of categories of Finite Etale Covers

Let $k$ be algebraically closed with characteristic $0$. For a scheme $X$, let $FEt(X)$ be the category of finite etale covers of $X$. What can be said about $FEt(X \times \mathbb{A}^1)$ and the ...
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### 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)\...
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### 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 ...
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### Covering of schemes and flatness

Let $f:X \to Y$ be a finite surjective morphism of quasi-projective schemes over $\mathbb{C}$, $X$ is reduced and $Y$ is integral. Suppose that there exists an integer $n$ such that for every closed ...
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### 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 ...
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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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $K$-...
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### 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 ...
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### 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 ...
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### question about the induced homomorphism of etale fundamental groups

Background/Setup For any connected scheme $S$, let $\text{FEt}_S$ denote the category of finite etale $S$-schemes. Let $f : X\rightarrow Y$ be a morphism of connected schemes, then for any finite ...
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### Stalks of higher direct image under open embedding

Let $U$ be an open subset of $\mathbb P^1$ without two points (say $t=0$ and $t=\infty$) and $j: U\to \mathbb P^1$ be an open immersion. Ground field $k$ is algeraically closed. Let $G$ be the group ...
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### Descend of etale morphism

I am not sure whether the title is appropriate for this question or not. I am sorry if there is anyone who is confused with the title and the contents. What I want to ask is the following: let $k$ be ...
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 ...
Let consider $f:(X,x)\to (Z,z)$ and $g:(Y,y)\to (Z,z)$ morphisms of pointed $k$-schemes of finite type ($k$ is a field). Suppose that there exists a map on the level of formal neighborhoods \$\phi:X_{x}...