Proposition 11

Under the above conditions, there is an exact sequence of K-homomorphisms

where u is the canonical injection and v the homomorphism $d \mapsto d \circ \sigma$ (Proposition 9).

The kernel of v is the set of derivations $d : C \to G$ such that $d(\sigma(b)) = 0$ for all $b \in B$, which is precisely the image of u.