Exercise 10, § 8

Let $A$ be a noetherian ring, $M, P, T$ be $A$-modules; suppose that $M$ is of finite type, $T$ of finite injective dimension, and that the modules $\mathrm{Ext}_A^i(M, P)$ and $\mathrm{Ext}_A^i(P, T)$ are zero for $i > 0$. Prove that the canonical homomorphism $M \otimes_A \mathrm{Hom}_A(P, T) \to \mathrm{Hom}_A(\mathrm{Hom}_A(M, P), T)$ which maps $m \otimes h$ to the homomorphism $u \mapsto h(u(m))$ is an isomorphism (reason as in the proof of the prop. 6, replacing $J$ by an injective resolution of $T$).