Proposition 2

Tag 065U, permanent. Cite it as [Bourbaki, Tag 065U].

Algebra, chapter III, § 9. Norms and traces

Let $A, A'$ be two $K$ -algebras, $M$ an $A$ -module and $M'$ an $A'$ -module. Suppose that $M$ and $M'$ are free $K$ -modules of respective dimensions $n$ and $n'$ and consider $M \otimes_K M'$ as an $(A \otimes_K A')$ -module ( $§ 4$ , no. 3, Example 2). Then, for $a \in A$ and $a' \in A'$ ,

\operatorname{Tr}_{M \otimes M'}(a \otimes a') = \operatorname{Tr}_M(a) \operatorname{Tr}_{M'}(a')

(16)

N_{M \otimes M'}(a \otimes a') = (N_M(a))^{n'} (N_{M'}(a'))^n.

(17)

Formula (16) follows from II, § 4, no. 4, formula (26) and formula (17) from § 8, no. 6, formula (33).

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

Cited by

Definition 2, § 9
Example 4, § 21
Example 5, § 21

Cites

II, § 4, no. 4
§ 8, no. 6