Search Results

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 $1 \in \widehat{\mathbf{Z}}$). Then one comp …

The Galois cohomology of finite fields is pretty straightforward: a $G_k$-module $M$ is entirely determined by the action of Frobenius $\varphi$, and we have $H^0(k, M) = M^{\varphi = 1}$, $H^1(k, M) …

In the paper of Bloch and Kato in the Grothendieck Festschrift, and some other papers relating to the Bloch-Kato conjecture and the ETNC, the cohomology groups $H^i_{\mathrm{et}}(\operatorname{Spec} …

Cohomology of $\widehat{\mathbb{Z}}$-modules is something you can just write down. If $V$ is a (finite) module with a (continuous) action of $\widehat{\mathbb{Z}}$, so the generator acts by some endom …

You should read the Bloch--Kato paper in the Grothendieck Festschrift. This was, I believe, the first paper to consider Selmer groups of Galois representations defined by local conditions coming from …

No, life is not so easy I'm afraid. For instance, the group $\mathrm{Ext}^1_{G_{\mathbf{Q}}}(\mathbf{Z}_\ell, \mathbf{Z}_\ell(n)) = H^1(\mathbf{Q}, \mathbf{Z}_\ell(n))$ has positive rank for all odd i …

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Br{Br}\DeclareMathOperator\U{U}\DeclareMathOperator\disc{disc}\DeclareMathOperator\Nm{Nm}\DeclareMathOperator\diag{diag}\D …

There are three approaches I know of to studying $H^1_{\mathrm{f}}(K, V)$, where $V = Sym^k(h^1(E))(n))$. All rely on $E$ being modular, so let me assume this henceforth (of course, this is no assumpt …