Exercise 4, § 5

Let $P$ be a complex of projective A-modules, zero on the right, and let $n$ be an integer such that $H_i(P) = 0$ for $i < n$. Let $M$ be a left a-module, $N$ a right a-module. Now prove that one has $H^i(\mathrm{Homgr}_A(P, M)) = 0$ and $H_i(N \otimes_A P) = 0$ for $i < n$, and that the canonical homomorphisms

are bijective.