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Results tagged with galois-cohomology
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user 18060
3
votes
Accepted
action of automorphisms on the Galois cohomology of the function field of a variety
since $\left(k(C)^\times\right)^n)$ certainly includes $k$, $\sigma(h)/h$ lies in it if and only if its divisor class is an $n$-fold multiple of a principal divisor.
One can easily compute the diviso …
- 131k
5
votes
Accepted
Cohomology with coefficients in $\mu_\infty$
Question 2. Does $H^2(\bar{X},\mu_\infty)$ have trivial Galois action? If so, is it true for all $H^i(\bar{X}, \mu_\infty)$?
The answers are "often not" and "almost never".
The exact sequence $1 \to …
- 131k
11
votes
Accepted
Is it true that $ H^{2r} ( X , \, \mathbb{Q}_{ \ell } (r) ) \simeq H^{2r} ( \overline{X} , \...
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 $\mathb …
- 131k
3
votes
Accepted
Local triviality of Galois cohomology classes over $\mathbb{Q}$
I think it follows again from Chebotarev. Represent the cohomogy class as an extension of the trivial representation by $A$. Such an extension is itself a Galois action on a finitely-generated $\mathb …
- 131k
5
votes
Accepted
Taking quotient of a variety by the additive group
For 1, yes. In fact, any smooth morphism of varieties admits a section locally in the etale topology everywhere.
Proof: A generic hypersurface section is smooth of dimension one lower over any particu …
- 131k
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 …
- 131k
8
votes
Accepted
Biquadratic extension of global function fields with cyclic decomposition groups
$E= \mathbb F_q ( \sqrt{t}, \sqrt{t^2-1} ) $ over $F =\mathbb F_q(t)$ does the trick if $q$ is congruent to $1$ mod $4$. It suffices to check that at each place where one of the extensions ramifies, t …
- 131k