Lemma 64.15.5. 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
Comments (0)