38
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 ...

29
votes

Accepted

### When (or why) is a six-functor formalism enough?

When defining a homotopy-coherent structure, you have to strike the correct balance between supplying enough data (so that all isomorphisms (between isomorphisms, ...) that you need later are actually ...

24
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 $\...

23
votes

Accepted

### Why do we care about the eigenvalues of the Frobenius map?

The Riemann hypothesis is very important for the relationship between the cohomology and combinatorics of the variety.
First, the Riemann hypothesis lets us read off the Betti numbers from the point ...

20
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 ...

20
votes

### Why do we care about the eigenvalues of the Frobenius map?

Here are a few different uses of knowing how large the eigenvalues are as complex numbers.
Application 1: Bounding exponential sums. Many classical exponential sums can be interpreted essentially as a ...

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 := ...

17
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 $\...

17
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 ...

17
votes

Accepted

### Is there a ring stacky approach to $\ell$-adic or rigid cohomology?

This is an interesting question.
First, I think the [PS] reference does not give the "correct" Betti stack. In my notes on 6 functors, I define a different stack $X_B$ such that $D_{\mathrm{...

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

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 ...

15
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 ...

14
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

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 ...

14
votes

### Why do we care about the eigenvalues of the Frobenius map?

All of the other answers give good reasons why the eigenvalues and their absolute values are important, but it should be noted that the eigenvalues can be used to give an exact point count via the ...

13
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}\...

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 ...

12
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 ...

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 ...

12
votes

Accepted

### Homology of the étale homotopy type

I'm sure there are easier and better ways to think about this, but here's how I like to think about it.
Work on the big pro-etale site on all schemes, which maps to the pro-etale site of a point, $\pi:...

12
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

### The Mumford-Tate conjecture

Yes.
Under the Hodge conjecture, the Hodge cycles are the algebraic cycles, so the $\mathbb Q_\ell$-linear combinations of Hodge cycles are the $\mathbb Q_\ell$-linear combinations of algebraic cycles....

12
votes

Accepted

### Cohomology of Grothendieck topology

Artin, M. Grothendieck topologies. (English) Zbl 0208.48701 Cambridge, Mass.: Harvard University. 133 p. (1962). (pdf copy)
These notes seem to fit your description precisely. They are concise, start ...

Community wiki

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 \...

11
votes

Accepted

### Liftable rational varieties

Two such examples were given by Achinger and Zdanowicz [AZ17], both of which satisfy a whole bunch of other good properties (e.g. their classes in the Grothendieck ring of varieties are polynomials in ...

11
votes

Accepted

### Why care about Grothendieck topology?

Etale topology, required to define etale cohomology, is not a topology in the usual sense. It is Grothendieck topology only.
In the category of topological manifolds, an etale cover of $X$ is a ...

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