Remark 3

Since the mapping $c_r$ from End(M) to L is a homogeneous polynomial mapping of degree $r$ for $0\leqslant r\leqslant m-1$, it follows from formula (6) that the mapping $u\mapsto \widetilde{u}$ from End(M) to End(M) is a homogeneous polynomial mapping of degree $m-1$.

Let B be an algebra over the ring L; suppose that B is a free L-module of rank $m\geqslant 1$, and identify L with the subring $L\cdot 1$ of B. Let $b$ be an element of B. We apply the above to the endomorphism $\gamma (b) :x\mapsto bx$ of the L-module B. Set $\gamma_r(b) =c_r(\gamma (b))$ for $0\leqslant r\leqslant m$; we have, in particular, $\gamma_m(b) = N_{B/L}(b)$ (III, §9, No. 3, p. 543). The characteristic polynomial of $b$ (loc. cit.) can be written as

(16) Pc$_{B/L}(b; X) =\sum_{r=0}^m(-1)^r\gamma_r(b) X^{m-r}$.

Since the mapping $\gamma$ from B to End$_L(B)$ is L-linear, the mapping $\gamma_r$ from B to L is a homogeneous polynomial mapping of degree $r$. In particular, the mapping $b\mapsto N_{B/L}(b)$ from B to L is a homogeneous polynomial mapping of degree $m$.

For any element $b$ of B, set

By Proposition 1, the linear mapping $\gamma (\widetilde{b})$ from B to B is the cotranspose of the mapping $\gamma (b)$; from this, we deduce

Moreover, the mapping $b\mapsto \widetilde{b}$ from B to B is a homogeneous polynomial mapping of degree $m-1$.