Proposition 10

Under the conditions of Proposition 9, for an $\varepsilon$-derivation $d' : B \to F$ to be $A$-linear for the left (resp. right) $A$-module structures on $B$ and $\rho^*(F)$, it is necessary and sufficient that $d'$ be zero on the subalgebra $\rho(A)$ of $B$.

We perform the proof for left A-module structures. For $a \in A, b \in B$,

and hence if $d' \circ \rho = 0$, $d'$ is linear for the left A-module structures on B and $\rho^*(F)$. Conversely, if this is so, in particular (no. 5, Proposition 3).

In particular let $D_K(B, F)$ denote the K-module of derivations of B into F (no. 2); those among these derivations which are A-linear, in other words those which are zero on $\rho(A)$, form a sub-K-module of $D_K(B, F)$, denoted by $D_{A,\rho}(B, F)$ or simply $D_A(B, F)$ (obviously $D_K(B, F) = D_{K,\phi}(B, F)$, where $\phi : K \to B$ is the homomorphism defining the K-algebra structure on B).

Now let A, B, C be three graded K-algebras, $\rho : A \to B, \sigma : B \to C$ two graded algebra homomorphisms and G a graded (C, C)-bimodule; if $D_A(B, G)$, $D_B(C, G)$ and $D_A(C, G)$ denote the respective K-modules $D_{A,\rho}(B, \sigma_*(G))$, $D_{B,\sigma}(C, G)$ and $D_{A,\sigma \circ \rho}(C, G)$, $D_B(C, G)$ is clearly a sub-K-module of $D_A(C, G)$ since $\sigma(\rho(A)) \subset \sigma(B)$.