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 ...
- 286
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 ...
- 21.3k
25
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 $\...
- 14.1k
24
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 ...
- 148k
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 ...
- 50.6k
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
18
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{...
- 21.3k
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 $\...
- 21.3k
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 ...
- 13.6k
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 $R$.
- 31.2k
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
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 ...
- 166
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 ...
- 16.1k
15
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 ...
- 47.4k
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 $X$ ...
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 ...
- 7,736
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}\...
- 28.7k
13
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:...
- 21.3k
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 ...
- 35.2k
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 ...
- 21.3k
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 ...
- 148k
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 ...
- 66.3k
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 ...
- 21.3k
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....
- 148k
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 ...
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 \...
- 52.6k
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 ...
- 28.7k
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 ...
- 12.3k
11
votes
Accepted
Is it true that $ H^{2r} ( X , \, \mathbb{Q}_{ \ell } (r) ) \simeq H^{2r} ( \overline{X} , \, \mathbb{Q}_{ \ell } (r) )^G $?
This is false for a general field $k$. It is true for some special fields, like finite fields.
Counterexample: Take $k = \mathbb C((t))$, $E$ an elliptic curve over $\mathbb C$ base-changed to $\...
- 148k
10
votes
Accepted
Is $H_{et}^1(X,F) = H^1(\pi_1^{et}(X), F(\bar{k}))$ true?
This is true, yes. More generally, if $X$ is a scheme and $F$ is a locally constant étale sheaf of finite abelian groups on $X$, then
$$
H^1_{et}(X,F) = H^1(\Pi_1^{et}(X), \tilde F),
$$
where $\Pi_1^{...
- 8,972
Only top scored, non community-wiki answers of a minimum length are eligible