The Stacks project

Lemma 63.15.5. Let $P$ be a finite projective $A[G]$-module and $M$ a $\Lambda [G]$-module, finite projective as a $\Lambda $-module. Then $P \otimes _ A M$ is a finite projective $\Lambda [G]$-module, for the structure induced by the diagonal action of $G$.

Proof. For any $\Lambda [G]$-module $N$ one has

\[ \mathop{\mathrm{Hom}}\nolimits _{\Lambda [G]}\left(P \otimes _ A M, N\right)= \mathop{\mathrm{Hom}}\nolimits _{A[G]}\left(P, \mathop{\mathrm{Hom}}\nolimits _{\Lambda }(M, N)\right) \]

where the $G$-action on $\mathop{\mathrm{Hom}}\nolimits _{\Lambda }(M, N)$ is given by $(g\cdot \varphi )(m) = g \varphi (g^{-1} m) $. Now it suffices to observe that the right-hand side is a composition of exact functors, because of the projectivity of $P$ and $M$. $\square$

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