11
votes
Accepted
For which subgroups the transfer map kills a given element of a group?
The answer to Q1 is yes, the order of $a$ might be smaller than the gcd: Let $G=\langle x,y\mid x^8=y^2=1,x^y=x^3\rangle$ be the semidihedral group of order $16$. Let $a=[x]\in G_{\text{ab}}=C_2\times ...
- 1,604
9
votes
Accepted
Example of continuous cohomology vs cohomology
Consider the circle $G=\mathbb{R}/\mathbb{Z}$ and its trivial representation on $M=\mathbb{R}$.
Since $G$ is abelian, $H^1(G,M)\cong \text{Hom}(G,M)$ and $H^1_c(G,M)\cong \text{Hom}_c(G,M)$ (these are ...
- 11.4k
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, ...
- 131k
7
votes
Accepted
Ker of corestriction of Galois cohomology
(Not sure why this question comes up naturally. The more interesting question, and the one analogue to the kernel of restriction, is to ask what is the cokernel of corestriction. That turns up a lot. ...
- 7,986
6
votes
Accepted
Classification of algebraic groups of the types $^1\! A_{n-1}$ and $^2\! A_{n-1}$
Have you tried Chapter 17, section 17.1 from Springer's book on algebraic groups? I believe that is as down-to-earth as it can get, and it is certainly rather detailed.
- 6,454
6
votes
Twisted forms with real points of a real Grassmannian
Let $X$ be a smooth projective variety over a field $K$ of characteristic zero such that
$$
X_{\bar{K}} \cong \mathrm{Gr}(k,n)_{\bar{K}}
$$
and $X(K) \ne \varnothing$.
First, consider the case $n \ne ...
- 36.4k
6
votes
Accepted
The second Tate-Shafarevich group of a permutation module is trivial
We write $G_w={\rm Gal}(L_w/K_v)$.
Definition. For $n\ge 1$, we denote
$$Ш_\omega^n(G,M)=\ker\Big(H^n(G,M)\to\prod_C H^n(C,M)\Big)$$
where $C$ runs over the cyclic subgroups of $G$.
Remark. $Ш^2(L/...
- 13.4k
5
votes
Accepted
Equivalence between twists of a curve and torsors of its automorphism group
Actually, this has little to do with varieties and is just some general phenomenon in topos theory. For simplicity, let $\mathscr C$ be a (small) category with finite limits, endowed with a ...
- 26.5k
5
votes
Accepted
Galois cohomology of Tate modules
Let $S$ be a finite set containing all places where an elliptic curve $E$ has bad reduction as well as $p$ and $\infty$. Write $T$ for $T_pE$ and $G_S$ for the Galois group of the maximal extension ...
- 7,986
4
votes
Biquadratic extension of global function fields with cyclic decomposition groups
For a different type of answer: over any field $K$, take two double coverings $C\rightarrow \mathbb{P}^1$, $D\rightarrow \mathbb{P}^1$ with disjoint branch loci; let $\sigma $ and $\tau $ be the ...
- 36.9k
4
votes
Example of continuous cohomology vs cohomology
Free pro-p group of rank at least $2$ is an example of such behaviour.
In the article "On discrete homology of a free pro-p-group", S. O. Ivanov, R. Mikhailov publication arXiv it's proven ...
- 4,299
4
votes
Accepted
Deformations of Galois cohomology
The answer to your example question is "No".
Let $G$ be isomorphic to $\widehat{\mathbf{Z}}$, and $\phi$ the topological generator of $G$ (corresponding to $1 \in \widehat{\mathbf{Z}}$). ...
- 35.8k
4
votes
Accepted
Decomposition groups for the Galois module $\mu_8$
If I understand this, $E = \mathbf Q(\sqrt{7},\zeta_8) = F(\zeta_8)$. Since $\mathbf Q(\zeta_8)/\mathbf Q$ ramifies only at $2$ (I am ignoring infinite places), $E/F$ can ramify only at a place over $...
- 49.1k
4
votes
Accepted
Twisted forms with real points of a real Grassmannian
Updated
Let $G:={\rm Aut}(X)^0$. Then it is well-known that
$$G_{\mathbb C}= ({\rm Aut}(X)^0)_{\mathbb C}= ({\rm Aut}(X)_{\mathbb C})^0= {\rm Aut}(X_{\mathbb C})^0\cong {\rm PGL}(n,\mathbb C).$$ So $G$...
- 15k
4
votes
Accepted
Group homology for a metacyclic group
The name metacyclic is normally used for a group which is cyclic-by-cyclic (ie. a group $G$ with a cyclic normal subgroup $N$ such that $G/N$ is also cyclic). I will therefore refer to a finite group $...
- 1,604
3
votes
Group homology for a metacyclic group
Everything follows from the p-primary decomposition theorem, which is nicely explained in Ken Brown's group cohomology bible (Kasper's answer is the restriction-corestriction argument written circa ...
- 16.9k
3
votes
Accepted
Pontryagin dual of cokernel, $(\operatorname{coker} F)^* \cong \hat{H}^0(\operatorname{Gal}(L/K),E(L)), $
$\newcommand{\coker}{{\rm coker}}
$
I construct the required homomorphism.
Let $A$ be an abelian variety over a number field $K$, and let $L/K$ be a finite Galois extension.
We denote by $A'$ the dual ...
- 13.4k
3
votes
Accepted
Torus gerbes over curves
I am just posting my comment as one answer. Let $K$ be a field, and let $T$ be a $K$-group scheme such that there exists a field extension $K'/K$ that is finite and separable (i.e., étale), and there ...
2
votes
Accepted
Local Tate duality for F-vector space
EDIT: I treat the general case.
Write ${\Bbb F}={\Bbb F}_q$ where $q=p^l$ for some natural $l$.
Then ${{\Bbb F}}_q\supseteq {{\Bbb F}}_p={\Bbb Z}/p{\Bbb Z}$.
Using a comment of @DavidLoeffler,
we ...
- 13.4k
1
vote
$\mathbb{Q}$-forms of $\operatorname{SL}_4(\mathbb{R})$ inside $\operatorname{SL}_8(\mathbb{R})$
$\newcommand{\nN}{{\mathcal N}}
\newcommand{\SL}{{\rm SL}}
\newcommand{\G}{{\bf G}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\R}{{\Bbb R}}
\newcommand{\C}{{\Bbb C}}
$The answer is No.
Write $\nN$ for the ...
- 13.4k
Only top scored, non community-wiki answers of a minimum length are eligible