Questions tagged [etale-cohomology]
for questions about etale cohomology of schemes, including foundational material and applications.
48
questions
23
votes
2
answers
4k
views
Etale cohomology with coefficients in the integers
Here is a basic question. When does $H^1_{et}(X,\mathbb{Z})$ vanish? Using the exact sequence of constant etale sheaves $0\rightarrow\mathbb{Z}\rightarrow\mathbb{Q}\rightarrow\mathbb{Q}/\mathbb{Z}\...
- 5,022
41
votes
2
answers
9k
views
Intuition behind the Eichler-Shimura relation?
The modular curve $X_0(N)$ has good reduction at all primes $p$ not dividing $N$. At such a prime, the Eichler-Shimura relation expresses the Hecke operator $T_p$ (as an element of the ring of ...
- 116k
16
votes
0
answers
861
views
L-Functions of Varieties, Zeta Functions of Their Models
Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the ...
- 643
12
votes
1
answer
3k
views
Semisimplicity of Frobenius operation on etale cohomology?
Let $X_0$ be a variety defined over a finite field of characteristic $p \neq l$.
Is it true, that the action of the frobenius on the l-adic cohomology $H_l^*(X)$ is semisimple (say for smooth $X_0$)? ...
- 13k
45
votes
3
answers
5k
views
"Cute" applications of the étale fundamental group
When I was an undergrad student, the first application that was given to me of the construction of the fundamental group was the non-retraction lemma : there is no continuous map from the disk to the ...
- 7,250
32
votes
2
answers
2k
views
Etale cohomology can not be computed by Cech
It can be proven that if in a quasicompact scheme $X$ any finite subset is contained in an affine open subset then for any sheaf $\mathcal{F}$ on $X$ its Cech cohomology $\hat{H_{et}^{\bullet}}(X,\...
- 7,047
30
votes
4
answers
8k
views
Etale cohomology and l-adic Tate modules
$\newcommand{\bb}{\mathbb}\DeclareMathOperator{\gal}{Gal}$
Before stating my question I should remark that I know almost nothing about etale cohomology - all that I know, I've gleaned from hearing off ...
- 6,972
27
votes
2
answers
2k
views
Etale site is useful - examples of using the small fppf site?
Edit: After the answers and comments, I'm hoping for a little bit of elaboration (in the comment to the answer below.) Also, question 2 was discussed here:
Points in sites (etale, fppf, ... )
There, ...
- 3,535
25
votes
1
answer
2k
views
Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?
This question was motivated by a more general question raised by Jan Weidner here. In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}_q$ of ...
- 52.5k
22
votes
2
answers
2k
views
Comparing cohomology over ${\mathbb C}$ and over ${\mathbb F}_q$
I have the following (probably well-known) question: let $X$ be a regular scheme over
$\mathbb Z$. Let $p$ be a prime and Let us denote the reduction of $X$ mod $p$ by $X_p$.
Let also $X_{\mathbb C}$...
- 6,977
20
votes
5
answers
4k
views
Equivalent statements of the Riemann hypothesis in the Weil conjectures
In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
- 671
16
votes
2
answers
1k
views
Cohomological dimension-doubling
I'm sure this is a question which has been asked many times, if not necessarily on this site:
Why does a (smooth, projective) scheme over a field, with dimension d, behave as though it were a ...
- 7,213
15
votes
2
answers
1k
views
Motivic generalisation of Neron-Ogg-Shaferevich criterion
Given a variety $X$ over $\mathbb{Q}$ with good reduction at $p$, proper smooth base change tells us that its $l$-adic cohomology groups are unramified at $p$ (and I'd guess some $p$-adic Hodge theory ...
- 806
15
votes
1
answer
7k
views
A nice explanation of what is a smooth (l-adic) sheaf?
I would like to understand this concept. It seems to be important (for the theory of perverse sheaves), yet I don't know any nice exposition of the properties of smooth sheaves.
- 16.7k
12
votes
2
answers
2k
views
Étale cohomology of morphism whose fibers are vector spaces
Let $X\rightarrow Y$ be a morphism (may not be smooth) of varieties such that the fibres are vector spaces. Are the $l$-adic cohomologies of $X$ and $Y$ equal?
If not, under what condition (other ...
user100841