18
votes
Accepted
Are higher etale homotopy groups topological groups in a natural way?
TL;DR The higher étale homotopy groups are the homotopy groups of the profinite completion of the shape of the étale topos. As such they are profinite groups. If you choose to see profinite groups as ...
- 16.5k
18
votes
Accepted
A short proof for simple connectedness of the projective line
You can deduce this from the classification of vector bundles on $\mathbf{P}^1$. Say $f:C \to \mathbf{P}^1$ is a connected finite etale Galois cover of degree $n$. We must show $n=1$.
The sheaf $E := ...
- 411
14
votes
Accepted
Do higher etale homotopy groups of spectrum of a field always vanish?
The étale topos of a field $k$ is just the topos of sets with a continuous $\mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-...
- 16.5k
12
votes
Accepted
Are "large enough" finite etale covers arithmetic?
Let's assume that $X$ admits a $K$-point $x$ and use the corresponding geometric point as the base point. The existence of a rational point is in fact necessary for a positive answer, as explained by ...
- 7,377
10
votes
Galois theory for products of fields (aka finite etale extensions)
Let $K$ be a separable algebraic closure of $k$. Then the functor sending an etale $k$-algebra $A$ to $Hom(A,K)$ is an equivalence to the category of finite sets with a continuous action of the Galois ...
- 101
10
votes
Are "large enough" finite etale covers arithmetic?
Adding on Will's and Sasha's answers, the condition of having a rational point, or at least a "1-truncated homotopy fixed point" for the action is necessary. For example, let $C_2$ act on ...
- 4,189
10
votes
Are "large enough" finite etale covers arithmetic?
Here's a simple argument assuming $X$ admits a $K$-rational point, and that $X$ has a finitely generated geometric fundamental group. In fact the "further" covering $X''$ can be chosen to be ...
- 10.7k
9
votes
Accepted
On a quasi-separated assumption in a lemma for the homotopy exact sequence of the etale fundamental group
This is more a comment than an answer: a few years back, in 2011, while working with some friends on SGA1, we also found out that we could not prove this statement without the hypothesis that $X$ is ...
- 3,998
9
votes
Accepted
Why only finite morphisms in etale fundamental group?
As mentioned in other comments, there is a "pro-étale fundamental group" considered by Bhatt and Scholze. It is introduced in Chapter 7. of their article "The pro-étale topology for schemes". It is a ...
- 381
8
votes
A short proof for simple connectedness of the projective line
You can apply the following statement to $X = \mathbb{P}^1_K$ and $L = O(1)$ when $K$ is a separably closed field.
Let $L$ be a line bundle on a reduced connected scheme $X$ such that $H^{0}(X,...
- 7,239
7
votes
Accepted
A weak version of high dimensional Abhyankar's conjecture
Here's a sketch of a possible way: Since $p$ is large enough, we can assume that $G$ is of order prime to $p$. If $X$ was projective, then the prime-to-$p$ completions of $\pi_1(X_{\mathbb{C}})$ and $...
- 16.1k
7
votes
Constructible étale sheaves on X are étale algebraic spaces over X
In case anyone wants to know a reference, I found it now:
SGA4, Exp. IX, Prop. 2.7
Statement (Translated):
Proposition 2.7 Let $X$ be a quasicompact and quasiseparated scheme, and let $F$ be a ...
- 493
7
votes
Étale fundamental group of multiplicative group over an algebraically/separably closed field
I am not sure if much is known about the structure of $\pi_1(\mathbf{G}_m)$ itself, but at least there is a complete characterization of its finite quotients. This is the content of the so-called ...
- 2,923
7
votes
Accepted
Is every etale cover a principal bundle?
Not every etale covering is a principal bundle under a group $G$. This is easiest to see using the Galois correspondence for etale coverings: the category of finite etale coverings of $X$ is ...
- 1,790
6
votes
Accepted
Surjective étale morphisms étale locally split
We can work locally on $X$ and even (by standard limit arguments) assume that $X=\mathrm{Spec}(R)$ where $R$ is local and strictly henselian. Then $Y=\coprod_{i=1}^{n}Y_i$ where each $Y_i$ is local ...
- 19.6k
6
votes
Finite etale covers of products of curves
The question already has a beautiful answer, but here's a different point of view which you may find helpful.
Let $F_i = \pi_1(C_i, x_i)$, which is a free group on $\#(\mathbf{P}^1\setminus C_i) - 1$ ...
- 16.1k
6
votes
Accepted
Finite etale covers of products of curves
The answer is no, at least in general, as shown by the following counterexample.
Take a double cover $\bar{f} \colon \bar{X} \to \mathbb{P}^1 \times \mathbb{P}^1$, branched over a reducible
curve of ...
- 66.3k
5
votes
Accepted
Finite étale covers of concentrated schemes and extension of base field
Yes, you can reduce to the finite type case by noetherian approximation (Appendix C in Thomason-Trobaugh). Namely, you can write $X=lim_\alpha X_\alpha$ where $X_\alpha$ is of finite type over $k$. ...
- 8,972
5
votes
Accepted
Equivalence of categories between finite étale covers of connected scheme and finite continuous permutation representations of étale fundamental group
I like the notes from the 2016-2017 edition of the Stanford number theory learning seminar. The result that you want is Theorem 3.4 here.
5
votes
Accepted
Two curves of genus $g \geq 2$ in characteristic $p >0 $ with isomorphic abelianizations
See
Nakajima, Shoichi, On generalized Hasse-Witt invariants of an algebraic curve, Galois groups and their representations, Proc. Symp., Nagoya/Jap. 1981, Adv. Stud. Pure Math. 2, 69-88 (1983). ...
- 156k
5
votes
Accepted
Why can we take the colimit over the category of elements?
If I'm not mistaken, a cosequence of the arguments in Murre's proof is that the inclusion $J^{\text{op}}\colon\mathcal I^{\text{op}}\hookrightarrow \mathcal E ^{\text{op}}$ is a final functor: It's ...
- 499
4
votes
Accepted
Vector bundles that are fixed under pull-back by the absolute Frobenius
For a finite flat cover $\pi:Y\to X$ the pushforward $E:=\pi_*\mathcal{O}_Y$ comes with a morphism $F^*E\to E$ induced by the Frobenius on $Y$. If $\pi$ is etale this morphism is an isomorphism: over ...
- 7,377
4
votes
Étale morphism over unirational/uniruled variety
In the case where $Y$ is unirational and projective, there exists no non-trivial étale cover of $Y$. In fact, the fundamental group of a complex, projective, smooth unirational variety is trivial, see
...
- 66.3k
4
votes
Accepted
Example: Principal G bundle that is not Zariski locally trivial, G not finite and G simply connected
I think a construction like the one for ${\rm PGL}_2$ in the question should work. The bundle in the question corresponds to the quaternion algebra over $\mathbb{C}[s^{\pm},t^{\pm}]$ whose norm form ...
- 17.4k
3
votes
Accepted
English literature close to "Algébre et Théories Galoisiennes" by Régine and Adrien Douady
O. Forster "Lectures on Riemann Surfaces" (Springer) is a good starting point before taking on T. Szamuely "Galois Groups and Fundamental Groups". After all, as Szamuely writes on page 65 at the ...
- 1,561
3
votes
Reference request: Kummer étale topology and tame topology
He did not say that the sites are equal as this claim is wrong in general, but that the categories of covers, i.e. finite morphisms which cover the log scheme (See definition 3.1. of the same paper), ...
3
votes
Accepted
Regarding a 'global' version of Chase-Harrison-Rosenberg exact sequence for rings
I'm converting my comment to an answer. Let $\pi:X\to Y$ be a Galois étale cover, with Galois group $G$. One has a Hochschild-Serre spectral sequence
$$E_2 = H^p(G, H^q(X_{et},\mathbb{G}_m))\...
3
votes
Accepted
Beauville Exercise VII.7 (3)-A proof that $\kappa(X)\geq \kappa(Y)$ for $f\colon X\to Y$ surjective morphism of smooth projective varieties
It is necessary to assume that $k$ has characteristic zero (the book is probably only considering $k = \mathbb{C}$) since otherwise there are surjective inseparable maps $\mathbb{P}^2 \to X$ where $X$ ...
- 3,625
3
votes
Normal subgroup of the geometrical fundamental group is the normal subgroup of the arithmetic fundamental group
For $H$ a normal subgroup of $G$, $G/H$ acts on the set of normal subgroups of $H$. Then $H$ is a fixed point of this action if and only if it is a normal subgroup of $G$.
The quotient in this case is ...
- 148k
2
votes
Accepted
Is the theta characteristic attached to an etale double cover of a plane quintic arising from a cubic threefold even in characteristic two?
I'll show the rank is at most 4 in characteristic 2, modulo some claims that I think can be justified.
Your exact sequence
$$0 \to \mathcal{O}_C \to \pi_*\mathcal{O}_{\widetilde{C}} \to \mathcal{O}_C
...
- 148k
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