37
votes
Accepted
"Cute" applications of the étale fundamental group
Using the étale fundamental group one can construct an injective group homomorphism
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \hookrightarrow \operatorname{Out}(\widehat{F_2})$
which is ...
- 276
26
votes
Accepted
Steenrod operations in etale cohomology?
You maybe want to have a look at
P. Brosnan and R. Joshua. Comparison of motivic and simplicial operations in mod-$\ell$ motivic and étale cohomology. In: Feynman amplitudes, periods and motives, ...
- 16.5k
22
votes
What is known about the cohomological dimension of algebraic number fields?
By definition, an algebraic number field is a finite extension
of the field of rational numbers $\Bbb Q$.
An algebraic number field $K$ is called totally imaginary
if it has no embeddings into $\...
- 11.5k
19
votes
Clarifying the connection between 'etale locally' and 'formally locally'
For local noetherian rings, the henselization has much more direct algebro-geometric meaning than the completion since it is built from local-etale algebras. Hence, your statement that the completion ...
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 := ...
- 381
16
votes
Steenrod operations in etale cohomology?
Your first map fits in an action of the Steenrod algebra.
In fact $H^*_{ét}(X;\mathbb{F}_2)$ is the homology of $C^*_{ét}(X;\mathbb{F}_2)=R\Gamma(X;\mathbb{F}_2)$, an element of the derived category ...
- 15.6k
16
votes
Accepted
Commutative algebra counterexample
Let $R=\mathbb{Z}$ and let $M=\mathbb{Z}$ with $x$ acting by $2$. Then $M\otimes_{R[x]}R[x,x^{-1}]\cong \mathbb{Z}[1/2]$ is not finitely generated over $R$.
- 29.9k
16
votes
Accepted
Étale cohomology of morphism whose fibers are vector spaces
The answer below is slightly reorganized to incorporate the edits. I thank @PiotrAchinger and @S.D. for their comments correcting and clarifying this answer.
Definition 1. An affine space morphism ...
15
votes
Is there a "universal" cohomology theory for varieties over p-adic fields?
I'm seeing this old question just now, and simply wanted to remark that the situation may be slightly better.
Namely, enlarging the category of $\mathbb Q$-vector spaces into the larger semisimple $\...
- 14k
14
votes
Accepted
Etale cohomology of $\mathrm{Spec}(k\{X,Y\})\backslash\langle0,0\rangle$
I'm not sure what "easy" means in the context of etale cohomology but there is a way of passing from Artin's result to the stated one.
Let $j$ from $\mathbb A^2 \backslash \langle 0,0\rangle$ to to $\...
- 115k
14
votes
Accepted
Hodge standard conjecture for étale cohomology
I'm very interested myself on a better answer to this question, but let me point out the obvious: the main problem is that there is no Hodge theory on positive characteristic.
The proof in ...
- 17k
14
votes
"Cute" applications of the étale fundamental group
I think a fantastic application of the étale fundamental group is in extensions of the Chabauty-Coleman(-Kim) method. The original idea was that we could bound the rational points on a curve $C$ by ...
- 6,834
13
votes
Idea of using etale site
I can tell you how they are related.
Before Riemann people would say, for example, the complex square root function (for $z\neq 0$) is two valued, but for any small region of (non-zero) complex ...
- 10.2k
13
votes
The Weil numbers and modulus of an elliptic curve
If $E$ does not have CM, then $\tau$ will be transcendental over $\mathbb Q$, so it's hard to imagine a relation with the eigenvalues of Frobenius, which are integers in an imaginary quadratic field. ...
- 42.1k
13
votes
Accepted
Etale cohomology can not be computed by Cech
Let $k$ be an algebraically closed field. Glue two copies of $\text{Spec}(k[[x]])$ along $\text{Spec}(k((x)))$. This gives a scheme $X = U \cup V$ such that any etale covering of $X$ can be refined by ...
- 146
13
votes
Accepted
Etale cohomology with coefficients in $\mathbb{Q}$
The following is surely expressing whatever is in the core non-formal aspect of Joe Berner's answer (which is above my pay grade); it is offered as an alternative version of the same ideas.
Let $X$ ...
13
votes
Accepted
Tate twists and cohomology of $\mathbf{P}^1$
The Tate twist is what we need to express Poincaré duality without making any choice. Such a choice appears in the choice of an orientation of the affine line minus the origin, and have shadows in the ...
- 11.6k
13
votes
Accepted
Etale cohomology of localizations of henselian rings
TL;DR: Your expectation is right. In fact, there is a third object to compare with $R[1/p]$ and $\hat R[1/p]$, the affinoid rigid space ${\rm Spf}(\hat R)^{\rm rig}$. The cohomology comparison is ...
- 13.5k
13
votes
Accepted
Cohomology of resolution of singularity
In general, the answer is no. It already fails for a nodal curve. In fancier terms, you can understand the obstruction as follows: if $X$ has a resolution $\tilde X$, and the cohomology of $X$ injects ...
- 31.7k
13
votes
Accepted
On the definition of the etale site of an adic space
Great question!
The short answer is that Huber simply wanted to be in a setting where everything is (stably) sheafy, and so put some assumptions ensuring this. Note that Huber's work remained somewhat ...
- 14k
13
votes
Accepted
Does $0\to I\to\mathrm{Gal}_K\to\mathrm{Gal}_k\to 0$ always split?
Good question! Let me try to guess what Gabber had in mind there. (Note that he only says "known" (to him), not "well-known"...)
The claim is that the extension splits. Note that ...
- 14k
12
votes
Accepted
Example of a variety over a number field with non-semisimple Galois representation on $\ell$-adic cohomology
Here's an example, if I'm not mistaken. Let $E / K$ be an elliptic curve and $x \in E$ a non-torsion $K$-point. Then the image of the divisor $\{x\} - \{\infty\}$ under the etale cycle class map is a ...
- 31.1k
12
votes
Flat versus etale cohomology
Here is a good example of some pathological behaviour that will shatter all your hopes and dreams.
Claim. Let $k$ be a perfect field, and let $X$ be a smooth, proper, integral $k$-scheme. Then the ...
- 20.2k
12
votes
Accepted
Computing the etale cohomology of spheres
This is true over any algebraically closed field $k$ of characteristic different from $2$. More generally, if $\ell$ is invertible in $k$, then
$$H^i(X,\mathbb Z_\ell) = \left\{\begin{array}{ll}\...
- 20.2k
12
votes
Accepted
Complete intersections in toric varieties
Any smooth projective toric variety is rational, in particular simply connected.
Then, by the Lefschetz hyperplane theorem for global complete intersections, if $\dim X \geq 3$ is a smooth complete ...
- 61.9k
11
votes
Accepted
Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?
It follows from results of Ribet in "On l-adic representations attached to modular forms" (Invent. Math. 28 (1975), 245–275) that the Mumford-Tate conjecture holds for the motives attached to modular ...
11
votes
Accepted
Cohomology of a constant etale sheaf
The etale cohomology groups $H^i(X_{et},V)$ vanish for $i > 0$ if $X$ is a quasi-compact, quasi-separated, normal scheme, and $V$ is a $\mathbb{Q}$-vector space. (In fact, one may replace "normal" ...
- 126
11
votes
Accepted
Under what conditions is the induced map of etale fundamental groups surjective?
There is a very general criterion for a map on $\pi_1$ to be surjective. Recall that for $X$ connected, the category of finite étale covers of $X$ is equivalent to the category $\pi_1(X)\text{ -}\...
- 20.2k
11
votes
Accepted
Why define étale cohomological dimension as it is defined?
I think it's a combination of two things:
First, we do etale cohomology with torsion sheaves only, because of pathologies with non-torsion sheaves, and so we care about the cohomological dimension ...
- 115k
11
votes
Accepted
Derived completion of complexes
As I mentioned in a comment, $K \to L$ must also be an isomorphism after rationalizing. For example, if $0 \to R \to F \to \Bbb Q \to 0$ is a free resolution of $\Bbb Q$, then let
$$
K = \dots \to 0 \...
- 47.4k
Only top scored, non community-wiki answers of a minimum length are eligible