Exercise 3, § 6

Let $I$ be an injective $k$-module; for every $A$-module $M$, denote by $D(M)$ the right $A$-module $\operatorname{Hom}_k(M, I)$. Let $M$ and $N$ be two left $A$-modules, $P$ a right $A$-module. Exercise prove that for every $p \geqslant 0$, the $k$-modules $\operatorname{Ext}_A^p(P, D(M))$ and $D(\operatorname{Tor}_p^A(P, M))$ are isomorphic; if $A$ is left noetherian and if $M$ is a finitely generated $A$-module, the $k$-modules $\operatorname{Tor}_p^A(D(N), M)$ and $D(\operatorname{Ext}_A^p(M, N))$ are isomorphic (use exercise 2, b) and exercise 6, b), p. 188).