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 ...
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, ...
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 $\...
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 := ...
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 ...
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 $R$.
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 ...
Community wiki
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 $\...
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 $\...
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 ...
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 ...
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 ...
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. ...
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 ...
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 $X$ ...
Community wiki
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 ...
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
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 ...
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 ...
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 ...
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 ...
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 ...
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
$$H^i(X,\mathbb Z_\ell) = \left\{\begin{array}{ll}\...
12
votes
Accepted
Complete intersections in toric varieties
Any smooth projective toric variety is rational, in particular simply connected.
Then, by the Lefschetz hyperplane theorem for global complete intersections, if $\dim X \geq 3$ is a smooth complete ...
11
votes
Accepted
Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?
It follows from results of Ribet in "On l-adic representations attached to modular forms" (Invent. Math. 28 (1975), 245–275) that the Mumford-Tate conjecture holds for the motives attached to modular ...
Community wiki
11
votes
Accepted
Cohomology of a constant etale sheaf
The etale cohomology groups $H^i(X_{et},V)$ vanish for $i > 0$ if $X$ is a quasi-compact, quasi-separated, normal scheme, and $V$ is a $\mathbb{Q}$-vector space. (In fact, one may replace "normal" ...
11
votes
Accepted
Under what conditions is the induced map of etale fundamental groups surjective?
There is a very general criterion for a map on $\pi_1$ to be surjective. Recall that for $X$ connected, the category of finite étale covers of $X$ is equivalent to the category $\pi_1(X)\text{ -}\...
11
votes
Accepted
Why define étale cohomological dimension as it is defined?
I think it's a combination of two things:
First, we do etale cohomology with torsion sheaves only, because of pathologies with non-torsion sheaves, and so we care about the cohomological dimension ...
11
votes
Accepted
Derived completion of complexes
As I mentioned in a comment, $K \to L$ must also be an isomorphism after rationalizing. For example, if $0 \to R \to F \to \Bbb Q \to 0$ is a free resolution of $\Bbb Q$, then let
$$
K = \dots \to 0 \...
Only top scored, non community-wiki answers of a minimum length are eligible
Related Tags
etale-cohomology × 625ag.algebraic-geometry × 512
arithmetic-geometry × 141
nt.number-theory × 69
reference-request × 59
galois-representations × 39
cohomology × 33
motives × 31
sheaf-theory × 30
ac.commutative-algebra × 26
hodge-theory × 24
sheaf-cohomology × 23
galois-cohomology × 23
at.algebraic-topology × 22
algebraic-groups × 20
schemes × 19
algebraic-number-theory × 18
perverse-sheaves × 16
homotopy-theory × 14
motivic-cohomology × 14
homological-algebra × 13
abelian-varieties × 13
derived-categories × 13
grothendieck-topology × 13
etale-covers × 13