Proposition 6

Tag 0660, permanent. Cite it as [Bourbaki, Tag 0660].

Algebra, chapter III, § 9. Norms and traces

Let $A$ be a commutative $K$ -algebra admitting a finite basis over $K$ and $V$ an $A$ -module admitting a finite basis over $A$ . Then $V$ admits a finite basis over $K$ and for every $A$ -endomorphism $u$ of $V$ , if $u_K$ is the mapping $u$ considered as a $K$ -endomorphism of $V$ ,

\begin{align*} \operatorname{Tr}(u_K) &= \operatorname{Tr}_{A/K}(\operatorname{Tr}(u)), \qquad \det(u_K) = N_{A/K}(\det(u)) \\ \operatorname{Pc}(u_K; X) &= N_{A[X]/K[X]}(\operatorname{Pc}(u; X)). \end{align*}

(24)

Let $(a_i)_{1 \leq i \leq m}$ be a basis of $A$ over $K$ and $(e_j)_{1 \leq j \leq n}$ a basis of $V$ over $A$ ; then $(a_i e_j)$ is a basis of $V$ over $K$ (II, § 1, no. 13, Proposition 25). On the other hand the third of formulae (24) can be deduced from the second applied to the endomorphism $X - \bar{u}$ of the $A[X]$ -module $A[X] \otimes_A V$ (§ 8, no. 10). It will therefore suffice to show the first two formulae in (24). We shall first establish the following lemma:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Proposition 6

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