in Demazure-Gabriel "Introduction to Algebraic Geometry and Algebraic Groups" II, §3, 3.7 (Proposition), I do not understand why the implication (iii) -> (i) follows from the result 3.3: The statement in 3.7 is "All representations of the group scheme G are semi-simple" -> "for all representations, the n-th Hochschild cohomology group is 0 for $n>0$".
The proof only stats that this follows directly from the statement "the functor $V \rightarrow H^n(G,V)$ is the derived functor of $V \rightarrow V^G$.
I have no idea why this is a complete proof. Any help and or idea is appreciated!
If every $G$-representation is semi-simple, then every short exact sequence of $G$-representations splits. This implies that the functor $V\mapsto V^G$ is exact (in fact, every additive functor is exact). Therefore, the higher derived functors $V\mapsto H^n(G,V)$ vanish for $n>0$.