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Let $G$ be a group, and let $V$ be a finite dimensional $\mathbb Q$-linear representation of $G.$ By extension of scalars we obtain the $\mathbb Q_l$-linear representation $V_l=V\otimes\mathbb Q_l.$

Question: Is the natural morphism ´$$ H^1(G,V)\to\prod_lH^1(G,V_l) $$ injective? Stated in this generality, the answer is possibly negative. But are there some contexts (e.g. G is a $\mathbb Q$-algebraic group and $V$ is an algebraic representation etc.) in which the answer is yes?

And what about other degrees $H^i$?

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If $G$ satisfies the usual universal coefficient theorem (for example if $G$ is a finitely generated abstract group), doesn't $H^1(G,V_l) \cong H^1(G,V) \otimes \mathbb{Q}_l$ hold ? –  Ralph Sep 11 '11 at 17:19
Thank you Ralph for giving the answer. I was wondering where I saw $(V^G)\otimes L=V_L^G$ for an extension $L/k,$ maybe in some of Milne´s notes. This should work by taking derived functors. –  shenghao Sep 11 '11 at 17:36

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