All Questions
Tagged with etale-cohomology galois-cohomology
24
questions
2
votes
0
answers
66
views
Finite dimensionality of Galois cohomology
Let $K_S$ denote the maximal extension of $\mathbb{Q}$, unramified outside a finite set of primes $S$, and let $G_S$ denote the Galois group of $K_S/\mathbb{Q}$.
It is known that for any finitely ...
- 2,113
1
vote
1
answer
637
views
Cohomology with coefficients in $\mu_\infty$
I'm encountering a lot of problems when dealing with the root of unity sheaf $\mu_\infty := \mathrm{colim}_n\mu_n$.
Let $X$ be a smooth geometrically integral variety over a number field $k$. Although ...
- 885
3
votes
0
answers
151
views
Obtaining an exact sequence of Galois modules via derived functors
This question has two parts, the first part will be to obtain the desired exact sequence while the second will be to study it in the corresponding derived category and try to obtain it from there.
Let ...
- 885
3
votes
2
answers
384
views
Is it true that $ H^{2r} ( X , \, \mathbb{Q}_{ \ell } (r) ) \simeq H^{2r} ( \overline{X} , \, \mathbb{Q}_{ \ell } (r) )^G $?
Let $ k $ be a field and let $ X $ be a smooth projective variety over $ k $ of dimension $ d $.
We denote by $ \overline{X} = X \times_k \overline{k} \ $ the base change of $ X $ to the algebraic ...
- 595
8
votes
2
answers
2k
views
The Mumford-Tate conjecture
The Mumford-Tate conjecture asserts that, via the Betti-étale comparison isomorphism, and for any smooth projective variety $ X $, over a number field $ K $, the $ \mathbb{Q}_{ \ell } $-linear ...
- 595
10
votes
1
answer
2k
views
What is known about the cohomological dimension of algebraic number fields?
What is the cohomological dimension of algebraic number fields like $\Bbb{Q}$, $\Bbb{Q}[i]$, $\Bbb{Q}[\sqrt{3}]$ or similar? I'm interested in computing the cohomological dimension of $\Bbb{A}^1_k$ ...
- 301
2
votes
0
answers
156
views
Fundamental Group of small Zariski open set
Let $Y$ be an integral affine variety over $\mathbb{C}$ and $K$ be its function field. How to find a sufficiently small Zariski open set of $Y$ such that it is isomorphic to $K(\pi,1)$? Here $\pi$ is ...
- 677
3
votes
1
answer
273
views
Semi-simple Galois actions on étale cohomology
Assume that semi-simplicity of the Galois action on $\ell$-adic cohomology of all smooth projective varieties over finite fields, were known.
Can one deduce that the Galois action on $\ell$-adic ...
user113393
6
votes
0
answers
359
views
Galois invariants in étale cohomology
Suppose $X$ is a smooth projective variety over a field $k$, with separable closure $\overline{k}$, Galois group $G$, and let $\overline{X}$ be $X_{\overline{k}}$.
Do we have
$$(H^j(\overline{X},\...
user120812
2
votes
0
answers
256
views
etale cohomology of tori
Let $k$ be an algebraically closed field.
Let $A$ be a strictly henselian local ring which is a $k$-algebra.
Let $T$ a torus over $k((t))$.
Can we compute $H^{1}(A((t)),T)$?
- 3,452
9
votes
3
answers
2k
views
Etale cohomology with coefficients in $\mathbb{Q}$
Let $X$ be a smooth variety of a field $k$. Then is
$$H_{et}^i(X, \mathbb{Q}) = 0$$
for all $i > 0$?
The result is true for $i=1$. This follows from the same argument given for $\mathbb{Z}$-...
- 20.9k
3
votes
0
answers
243
views
Relation between Galois and etale cohomologies
Let $D$ be the ring of integers of a number field $F$.
Let $X=\mathrm{Spec} ~D$, and let $\pi$ be the etale fundamental group of $X$.
There are natural maps from $H^i(\pi, \mathbf{Z}/n)$ to $H^i_{...
- 367
8
votes
0
answers
155
views
defining Selmer groups using étale cohomology
Concerning http://swc.math.arizona.edu/aws/1999/99RubinES.pdf, especially section I.5:
Can one define the Selmer groups and the unramified cohomology groups as étale
cohomology groups of certain ...
user19475
3
votes
1
answer
610
views
Galois cohomology of a non-abelian group over a function field
Let $k$ be an algebraically closed field, and $X$ a connected smooth projective curve over $X$. Let $F$ be the function field of $k$. Let $G$ be an algebraic group over $k$ (assume that it is smooth, ...
- 5,522
8
votes
1
answer
421
views
When does the continuous Galois(=etale) cohomology of fields coincide with the naive one? Often true by the Bloch-Kato conjecture?
For a field $F$ I am interested in its $l$-adic (Galois=\'etale) cohomology; here $l$ is a prime distinct from the characteristic of $F$ (for simplicity one may assume that the latter is $0$).
For $...
- 16.7k