Corollary

If the $A$-module $M$ is Macaulay, the homomorphism $\beta_M$ is bijective.

If $M$ is Macaulay, the $A$-module $H^i(D(M))$ is zero for $i \neq c$ (No. 1, cor. of prop. 3), so that the canonical injection $j : D'(M) \to D(M)$ is a homologism; hence the morphism of complexes $D(j) : D(D(M)) \to D(D'(M))$ is a homologism (A, X, p. 86, prop. 4). Thus $H_0(D(j))$ is bijective; on the other hand $\alpha_M$ is bijective by th. 1, whence the corollary.

When the $A$-module $M$ is of finite length, the $A$-module $\mathrm{Ext}_A^c(M, \Omega)$ identifies with the Matlis dual of $M$ ($cf.$ § 8, No. 5, example 3 and th. 3), and one recovers prop. 4 of § 8, No. 4.