Lemma 64.15.8 (03UC)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 64: The Trace Formula
Section 64.15: Preliminaries and sorites
Lemma 64.15.8 (cite)

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Lemma 64.15.8. 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 64.15.6. $\square$

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