Proposition 15

If $M$ is a free $K$-module of finite rank $n$, then, for every endomorphism $s$ of $M$, the restriction to $\Lambda^n(M)$ of the derivation $D_s$ is the homothety of ratio $\operatorname{Tr}(s)$.

Let $(e_j)_{1 \leq j \leq n}$ be a basis of $M$ and write $e = e_1 \wedge e_2 \wedge \cdots \wedge e_n$. If

the third formula in (36) gives

Example (2). In the Corollary to Proposition 14, part (iii), let $\Delta = \{0\}$, the graduation on $\Lambda(M)$ then being the usual graduation of type $\mathbf{Z}$; on the other hand take $\varepsilon(p, q) = (-1)^{pq}$. Then, for every linear form $x^* \in M^*$ on $M$, $x \mapsto \langle x, x^* \rangle$ is a graded $K$-linear mapping of degree $-1$ of $M$ into $\Lambda(M)$ satisfying relation (34); then there exists an antiderivation $i(x^*)$ of $\Lambda(M)$, of degree $-1$, such that (by virtue of formula (21) of no. 5)

and which is a special case of the inner product to be defined in § 11, no. 9, formula (68).