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Let us consider $R_n = \mathbb Z_\ell[\theta_n]/(\theta^{\ell^n}-1)$, an auxiliary prime power $q\equiv 1 \pmod \ell$ with an action of $\mathbb Z = \langle \sigma\rangle$ by $\sigma(\theta_n) = \theta_n^q$. It acts through a quotient $\mathbb Z/\ell^{n-n_0}$ for some $n_0$.

Is there a nice description of the cohomology groups $H^1(\mathbb Z, GL_k(R_n))$? At least for $k = 1$? I am having a hard time finding references.

(The question has been edited after I realized that I wanted to solve a slightly different problem.)

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  • $\begingroup$ I deleted my comment. Perhaps you should edit the title of the question? $\endgroup$
    – David Loeffler
    Commented Aug 26, 2020 at 17:01
  • $\begingroup$ I edited my title but I am not sure it's the best for the question. $\endgroup$
    – Asvin
    Commented Aug 26, 2020 at 17:03
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    $\begingroup$ Personally, I think "What is $\operatorname H^1(\mathbb Z, \operatorname{GL}_k(R_n))$ for a certain ring $R_n$" is easier to read. (Also, $\mathbb Z_\ell$ is $\ell$-adic integers, right?) $\endgroup$
    – LSpice
    Commented Aug 26, 2020 at 17:22
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    $\begingroup$ Thanks, I edited the title to incorporate your suggestion. And yes $\mathbb Z_\ell$ is the ring of $\ell$-adic ring of integers. $\endgroup$
    – Asvin
    Commented Aug 26, 2020 at 17:30

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