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Consider a profinite group $S$ acting trivially on $\mathbb{F}_p$. Choose $\chi \neq 0\in H^1(S, \mathbb{F}_p)$ and set $T = \ker(\chi)$. Let $X$ be the $S$-Module of all functions $S/T \rightarrow \mathbb{F}_p$.

Show that there exists a canonical isomorphism $$ H^q(S,X) \cong H^q(T,\mathbb{F}_p). $$

The case $q=0$ is obvious, since $X^S$ consists of the constant functions and $\mathbb{F}_p^T = \mathbb{F}_p$ may be embedded into $X$ as the constant functions.

But how do I proceed from there? Dimension shifting doesn't seem to work here or am I missing something?

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You don't need to write \text{ker}; you can just write \ker. That automatically provides proper spacing before an after "ker" in expressions like $5\ker f$. (I editing the question accordingly.) – Michael Hardy Oct 3 '12 at 0:35
Thanks. I did wonder if this is supported, but didn't try it – p.kelly Oct 3 '12 at 21:21
That looks like Shapiro's Lemma – i. m. soloveichik Oct 3 '12 at 21:30
Yes it does. Thank you for the hint – p.kelly Oct 5 '12 at 9:39
up vote 1 down vote accepted

Thanks to i. m. soloveichik I was able to solve my question.

There exists an isomorphism

$$ \text{Ind}^T_S(\mathbb{F}_p) \cong \text{Map}(S/T, \mathbb{F}_p) = X $$ given by $$ x(\sigma) \mapsto y(\sigma T) = \sigma x (\sigma^{-1}) $$

By Shapiro's Lemma we get the isomorphism

$$ H^q(S, \: \text{Ind}^T_S(\mathbb{F}_p)) \cong H^q(T,\mathbb{F}_p). $$

A proof is found in Neukirch/Schmidt/Wingberg: 'Cohomology of Number Fields' p. 60.

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