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