All Questions
Tagged with fundamental-group etale-cohomology
10 questions
47
votes
3
answers
5k
views
"Cute" applications of the étale fundamental group
When I was an undergrad student, the first application that was given to me of the construction of the fundamental group was the non-retraction lemma : there is no continuous map from the disk to the ...
- 7,320
11
votes
2
answers
2k
views
locally constant constructible sheaves and finite etale coverings
Maybe it is well known to experts or maybe it is just a stupid idea, but I will ask any way.
We know that if $X$ is a topological space, then there is an equivalence of categories between the ...
- 314
9
votes
1
answer
2k
views
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 ...
- 2,974
8
votes
1
answer
403
views
Is $H_{et}^1(X,F) = H^1(\pi_1^{et}(X), F(\bar{k}))$ true?
Let $X$ be a smooth geometrically connected scheme over a field $k$ of characteristic 0 (but not necessarily algebraically closed, I can take it to be a number field). Let $F$ be a finite algebraic ...
- 83
6
votes
1
answer
471
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 ...
- 311
4
votes
1
answer
1k
views
Fundamental Group and Etale Cohomology
I encountered the following statement without a reference many times. For a smooth variety $X$ over a perfect field $k$.
$Hom(H^1_{et}(X, \mathbb{Z}/n), \mathbb{Z}/n) \cong \pi^{ab}_1(X)/n$
Is there ...
- 235
2
votes
0
answers
93
views
Unramified section associated to a rational point
This is a question for those familiar with the section conjecture, so I'll do away with the definition of a ramification map in this case. Here is the definition of a ramification map from an etale ...
- 895
2
votes
0
answers
881
views
Question about the specialization map for Etale Fundamental Groups
Let $A$ be a complete, discrete valuation ring, and let $s$ (resp $\eta$) be the special (resp. generic) point of $S=Spec(A)$. Let $\phi:X \rightarrow S$ be a proper morphism and fix geometric base ...
- 133
1
vote
0
answers
238
views
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 $\mathbf{P}^2_{\mathbf{F}_q}$?
- 333
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