Proposition 16

Let $A$ be a commutative $K$-algebra, $M_i$ ($1 \leq i \leq n$) and $P$ $A$-modules and $H$ the $A$-module of $A$-multilinear mappings of $M_1 \times M_2 \times \cdots \times M_n$ into P. Suppose that there is given a K-derivation $d_0 : A \to A$ of the algebra A, for each i, a K-linear mapping $d_i : M_i \to M_i$ and a K-linear mapping $D : P \to P$, so that, for $1 \leq i \leq n$, $(d_0, d_i, d_i)$ is a K-derivation of $(A, M_i, M_i)$ into itself and $(d_0, D, D)$ is a K-derivation of $(A, P, P)$ into itself. Then there exists a K-linear mapping $D' : H \to H$ such that $(d_0, D', D')$ is a K-derivation of $(A, H, H)$ into itself and

(37) $D(f(x_1, \ldots, x_n))$

for all $x_i \in M_i$ for $1 \leq i \leq n$ and $f \in H$.

We show that for $f \in H$, the mapping $D'f$ of $M_1 \times M_2 \times \cdots \times M_n$ into P defined by (37) is A-multilinear. For $a \in A$,

and by hypothesis

and $d_1(ax_1) = (d_0a)x_1 + a.d_1x_1$, which gives

and linearity in each of the $x_i$ is proved similarly. On the other hand,

in other words

which establishes the proposition.