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Let $G$ be a finite infinitesimal group scheme (e.g.$\mu_p,\alpha_p) $ over a perfect field $k$, how much is known about $H^1_{fppf}(k,G)$?

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What precisely do you want to know about it? –  Kestutis Cesnavicius Jan 27 '13 at 12:30
    
I know for $\mu_p$ and $\alpha_p$ these are trivial. Do I have more examples? Are these all trivial for commutative finite groups? –  stefan Jan 27 '13 at 14:04
    
Sorry, I meant for commutative finite infinitesimal groups. –  stefan Jan 27 '13 at 14:25
    
@stefan: Use the connected-etale sequence to see the triviality whenever $G$ is commutative and infinitesimal. –  user30180 Jan 27 '13 at 14:34
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