Lemma 63.15.8. Let $P$ be an $A[\Gamma ]$-module, finite projective as $A[G]$-module. Let $M$ be a $\Lambda [\Gamma ]$-module, finite projective as a $\Lambda$-module. Then

$\text{Tr}_{\Lambda }^{Z_\gamma }(\gamma , P \otimes _ A M) = \text{Tr}_ A^{Z_\gamma }(\gamma , P)\cdot \text{Tr}_\Lambda (\gamma , M).$

Proof. This follows directly from Lemma 63.15.6. $\square$

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