Corollary

Let $M$ be a graded $K$-module of type $\Delta$; the $K$-algebras $T(M)$, $S(M)$ and $\bigwedge(M)$ are given the corresponding graduations of type $\Delta' = \Delta \times \mathbf{Z}$ ($§ 5$, no. 5, Proposition 7, $§ 6$, no. 6, Proposition 10 and $§ 7$, no. 7, Proposition 11). On the other hand $M$ is given the graduation of type $\Delta'$ such that $M_{\alpha, 1} = M_{\alpha}$ for all $\alpha \in \Delta$ and $M_{\alpha, n} = \{0\}$ for $\alpha \in \Delta$ and $n \neq 1$. Let $\varepsilon'$ be a commutation factor over $\Delta'$.

(i) Let E be a graded (left and right) $\mathbf{T}(M)$-bimodule of type $\Delta'$; for all $\delta \in \Delta$ and every integer $n \in \mathbf{Z}$, every graded K-linear mapping $f : M \to E$ of degree $\delta_1' = (\delta, n)$ can be extended uniquely to an $\varepsilon'$-derivation $d : \mathbf{T}(M) \to E$ of degree $\delta'$.

(ii) Let E be a graded $S(M)$-module of type $\Delta'$; for a graded K-linear mapping $f : M \to E$ of degree $\delta'$ to be extendable to an $\varepsilon'$-derivation $d : S(M) \to E$ of degree $\delta'$, it is necessary and sufficient that, for every ordered pair $(x, y)$ of homogeneous elements of M,

The $\varepsilon'$-derivation d is then unique.

(iii) Let E be a graded (left and right) $\Lambda(M)$-bimodule of type $\Delta'$; for a graded K-linear mapping $f : M \to E$ of degree $\delta'$ to be extendable to an $\varepsilon'$-derivation

of degree $\delta'$, it is necessary and sufficient that, for every homogeneous element x of M,

The $\varepsilon'$-derivation d is then unique.

Remark 2 of no. 2 is applied with one of the external B-module laws on E (with B equal to $\mathbf{T}(M)$, $S(M)$ or $\Lambda(M)$) modified; the external law thus modified is still, by (1) (no. 1), a B-module law and the B-module law thus obtained on E is still compatible with the other B-module structure. It then suffices to apply Proposition 14 with $A = K$ and $d_0 = 0$.

Example (1). In the application of Proposition 14 note that if $d_0 = 0$ condition (29) means simply that $d_1$ is A-linear. If we take in particular $E = B$ and the (B, B)-bimodule structure derived from the ring structure on B, conditions (30) and (31) are automatically satisfied when $d_1$ is taken to be the composition of an endomorphism s of M and the canonical injection $M \to B$; this is obvious for (30) since $S(M)$ is commutative and for (31) this follows from the fact that x and $s(x)$ are of degree 1 in $\Lambda(M)$. It is therefore seen that every endomorphism s of M can be extended uniquely to a derivation $D_s$ of $\mathbf{T}(M)$ (resp. $S(M)$, resp. $\Lambda(M)$), which is of degree 0. Moreover, for two endomorphisms s, t of M,

for both sides are derivations of $\mathbf{T}(M)$ (resp. $S(M)$, resp. $\Lambda(M)$) which are equal to $[s, t]$ on M.

The expression for $D_s$ is obtained using formula (21) of no. 5, which gives respectively, for $x_1, x_2, \ldots, x_n$ in $M$,

In the case of the algebra $\Lambda(M)$, there is the following interpretation of $D_s$: