Lemma 3

The connecting homomorphism $\mathrm{H}(B \otimes_A C') \to \mathrm{H}(Z \otimes_A C')$ associated with the exact sequence (12) is equal to $\mathrm{H}(i \otimes 1)$.

Indeed, let $a \in \mathrm{H}(B \otimes_A C')$; since $B$ is flat, $a$ belongs to the image of $B \otimes_A Z(C')$, hence can be written as $\sum_\lambda da_\lambda \otimes b_\lambda$, where $a_\lambda \in C$, $b_\lambda \in C'$, and $db_\lambda = 0$. The image of the class of $a$ under the connecting homomorphism is, by definition, the class of $D(\sum a_\lambda \otimes b_\lambda) = \sum da_\lambda \otimes b_\lambda = (i \otimes 1)(a)$, which proves the lemma.

The associated homology exact sequence to (12) is therefore

On the other hand, since $Z$ is flat, from (II) we obtain a sequence of graded $k$-modules

finally, we have the canonical homomorphisms of No. 1

whence a diagram of graded $k$-modules, with exact rows:

which is commutative by definition of the homomorphisms $\gamma$. But since the complexes $B$ and $Z$ are split and flat, $\gamma_B$ and $\gamma_Z$ are bijective (X, p. 65, Corollary 1). From this, on the one hand, we deduce that $\gamma_C$ is injective and has image equal to $\operatorname{Ker} H(\delta \otimes 1)$, and on the other hand, that $\gamma_B \circ \partial(\mathrm{II}, H')$ is injective with image equal to $\operatorname{Im} H(\delta \otimes 1)$. The theorem follows immediately from this.