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Let $K$ be a local field, and $\bar{K}$ its algebraic closure. Let $\mathcal{C}$ be the category of (continuous) $G=\text{Gal}(\bar{K}/K)$-modules. Let $M$ be a finite $G$-module. For any injective module $I$ of $\mathcal{C}$,

is $H^r(G, \text{Hom}_{\mathbb{Z}}(M,~I))=0$ for all $r>0$?

(Here on $\text{Hom}_{\mathbb{Z}}(M,~A)$, $g \in G$ acts as $(g\phi)(x)=g(\phi(g^{-1}(x)))$.)

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  • $\begingroup$ Just to clarify, are you really considering all $G$-modules? No topology involved, just some abelian group with a $G$-action? $\endgroup$
    – user94041
    Feb 24, 2017 at 16:41
  • $\begingroup$ Sorry, I changed a bit. Thank you very much. $\endgroup$
    – user1225
    Feb 24, 2017 at 16:49
  • $\begingroup$ With the new formulation, I would be very surprised if $\mathcal{C}$ had any non-zero injectives. $\endgroup$
    – David Loeffler
    Feb 24, 2017 at 17:04
  • $\begingroup$ Thanks David, I changed again, which seems better than before. $\endgroup$
    – user1225
    Feb 24, 2017 at 17:30
  • $\begingroup$ If this "exercise" (for which no motivation has been given) is now rendered moot due to the answer to your other question at mathoverflow.net/questions/263088/… (which seems almost surely to be what the motivation was), perhaps the above question should be edited to reflect that the exercise is no longer of interest to you (if I understand correctly the motivation). $\endgroup$
    – nfdc23
    Feb 25, 2017 at 23:14

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