Proposition 6

Suppose the A-module J injective; for every A-module N, put $D(N) = \mathrm{Hom}_A(N, J)$.

a) The homomorphisms $\theta^i(M, P) : \mathrm{Ext}_A^i(M, D(P)) \longrightarrow D(\mathrm{Tor}_i^A(M, P))$ are bijective.

b) If the ring $A$ is noetherian and the $A$-module $M$ of finite type, the homomorphisms $\rho_i(M, P) : \mathrm{Tor}_i^A(M, D(P)) \longrightarrow D(\mathrm{Ext}_A^i(M, P))$ are bijective.

a) By construction, the homomorphism $\theta(M, P)$ is bijective as soon as $\lambda(C \otimes_A P, J)$ is bijective, which is the case when $J$ is injective (A, X, p. 85, Corollary 2).

b) Choose the resolution $C$ so that each module $C_p$ is free of finite type (A, X, p. 53, Proposition 6). Then the homomorphism $\nu$ is bijective, and the same is true of $\lambda(\mathrm{Homgr}_A(C, P), J)$ since $J$ is injective; hence $\rho(M, P)$ is bijective.