Questions tagged [etale-covers]
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103 questions
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Etale descent for K theory of algebraic stacks
We know that etale descent fails for algebraic stacks as one can see for the finite etale map $Spec k \to BG$ for a finite algebraic group scheme $G $ over a field of characteristic 0. It seems the ...
- 1,545
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What is the right notion of étale fundamental groupoid?
Given a connected and locally Noetherian scheme $S$ and a geometric point $p \colon s \to S$, the étale fundamental group $\pi_1(S,s)$ satisfies
Theorem A: For each finite étale cover $Y \to S$, the ...
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Compactifying varieties with the same Etale fundamental group
Let $X$ be a non-singular, affine variety (of finite type) over $\mathbb{C}$. Does there exist a non-singular projective variety $\bar{X}$ containing $X$ as an open subset such that the Etale ...
- 2,639
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Étale morphisms, a priori
I don't know what the etiquette is for asking old questions from MSE, but I would very much like to re-ask this one here at MO:
Background: The notion of an étale morphism has proved itself to be ...
- 1,315
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Normal subgroup of the geometrical fundamental group is the normal subgroup of the arithmetic fundamental group
I asked some questions on a descending lemma in Lawrence-Venkatesh 4 days ago, but it has not received any answer. I understood (2) now but I'm still confused on (1).
I want to ask a new question here....
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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 ...
- 16.1k
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Characterize descents of geometric finite étale cover by means of homotopy exact sequence
Let $X/k$ be a geometrically connected $k$-variety (=separated of finite type, esp quasi-compact; the base field $k$ assumed to be separable, so $\overline{k}=k^{\text{sep}}$), $\overline{X} := X \...
- 4,142
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Galois action on étale path torsors
TLDR: How is the Galois action on étale path torsors defined?
Let $X$ be a smooth proper scheme, over a field $k$, and let $x,y\in X(k)$ be a pair of points. Let $\pi_1^{\text{ét}}(\overline{X},\...
- 3,416
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Why can we take the colimit over the category of elements?
I'm trying to understand J. P. Murre's Tata notes on Grothendieck's theory of the fundamental group. For a Galois category $\mathcal C$ (which I'm taking to be locally small) with fundamental functor $...
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Fundamental group of a quotient by a group action
Suppose I have a quotient $X \to S$ by a finite abelian group $G$ action (I have several cases, but in all of them the group $G$ and the action could be written explicitly), where $X,S$ are surfaces (...
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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 \...
- 4,480
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Base change for fundamental group prime to p in mixed characteristic?
I found the answer to this question while typing it up, but since I've already written it, it is probably worthwhile to post-and-answer in case someone finds it useful.
Let $S=\operatorname{Spec}\...
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Two curves of genus $g \geq 2$ in characteristic $p >0 $ with isomorphic abelianizations
Let $k$ be an algebraically closed field of characteristic $p>0.$ How can I construct two projective curves $C_1,C_2$ of genus $ g \geq 2$ so that the abelianizations $\pi_1(C_i)^{ab},i=1,2$ are ...
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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 ...
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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)...
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