Corollary

a) If the $A$-algebra $B$ is finite, the $B$-module $I_B = \mathrm{Hom}_A(B, I_A)$ is a Matlis $B$-module.

b) If the $B$-module $M$ is artinian, the mapping $\alpha$ is a $B$-isomorphism of $D_B(M)$ onto $D_A(M)$.

Proposition 5 applies in particular when $A$ is a field $k$, in which case one may take $I_A = k$, hence $D_k(M) = \mathrm{Hom}_k^{cont}(M, k)$ ("Macaulay duality"). It will be noted that the assumption $[\kappa_B : k] < +\infty$ is satisfied in particular when the $k$-algebra $B$ is the local ring at a maximal ideal of a $k$-algebra of finite type (A, VIII, App. 3, cor. 1).

More particularly, consider an $\mathbf{N}$-graded $k$-algebra of finite type $S$, such that $S_0$ is a field, of finite degree over $k$. One may apply Proposition 5 to the local ring $S'$ of $S$ at the maximal ideal $S_+ = \bigoplus_{n > 0} S_n$ or, what amounts to the same thing, to its completion $\widehat{S} = \prod_{n \geq 0} S_n$ (III, § 1, No. 3, lemma 2 and § 2, No. 12, example 1). The $S$-module $I_{\widehat{S}} = \mathrm{Hom}_k^{cont}(\widehat{S}, k)$ is then identified with

for $s \in S$ and $u \in S^{*gr}$, the element $su$ of $S^{*gr}$ is the interior product $s \perp u$ (A, III, p. 156 and p. 157). Take, for example, $S = k[T_1, \ldots, T_d]$, whence $\widehat{S} = k[[T_1, \ldots, T_d]]$. Let $(u_\alpha)_{\alpha \in \mathbf{N}^d}$ be the basis of the $k$-vector space $S^{*gr}$ dual to the basis $(T^\alpha)_{\alpha \in \mathbf{N}^d}$ of $S$. The structure of $S^{*gr}$ as $S$-module is then described by the formulae (A, III, p. 167)