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Results tagged with galois-cohomology
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user 5101
6
votes
Accepted
$H^1$ and fractional ideals group
This vanishing is very different from Hilbert's theorem 90. I will only sketch a proof as it is rather standard and it will be good for you to fill in the details.
Your module is a permutation module …
- 20.6k
11
votes
Accepted
"Forms" of quadrics
It is not too difficult to see that any automorphism of a smooth quadric hypersurface
$$X : Q(x) = 0,$$ over a field $k$ must be a projective automorphism (see for instance the argument I give in Auto …
- 20.6k
1
vote
Accepted
Twists of projective automorphisms
Thanks to the hint from Ulrich, I think I am now able to answer the question.
The set $\mathrm{H}^1(k, \mathrm{Aut}(X_{\bar k},L_{\bar k}))$ classifes the following objects:
$k$-Isomorphism classes …
- 20.6k
11
votes
Accepted
Embedding torsors of elliptic curves into projective space
Suppose that $C \subset X$ is a smooth projective curve of genus $1$ embedded in a Brauer-Severi surface over a field $k$. We have $C^2 = 9$ since this holds after passing to the algebraic closure, wh …
- 20.6k
16
votes
Accepted
Third Galois cohomology group
The group $H^3(K,\bar{K}^\times)$ naturally arises when trying to calculate the Brauer group of a variety. Explicitly, the Hochschild-Serre sequence yields the exact sequence
$$0 \to \mathrm{Br}_1(X)/ …
- 20.6k
4
votes
Accepted
Computing $H^1$ with coefficients in a torsion-free abelian group
I will focus attention on smooth projective varieties $X$ over $k$ with $\mathrm{Pic}(X_{\bar{k}})$ a free finitely generated abelian group, as they illustrate all the essential behaviour relevant to …
- 20.6k
4
votes
Applications of the Galois embedding problem
Shafarevich made heavy use of embedding problems in his resolution of the inverse Galois problem for solvable groups. The (naive) viewpoint is that solving the relevant embedding problems allowed him …
- 20.6k
9
votes
Accepted
Elements of arbitrary large order in the first Galois cohomology of an elliptic curve
Here is the kind of method I had in mind.
We have the elliptic curve Kummer sequence
$$0 \to E[n] \to E \to E \to 0,$$
Here I denote by $E[n]$ the $n$-torsion group scheme of $E$. Applying Galois coh …
- 20.6k