Remark

Corollary 4 makes it possible to characterize Macaulay local rings in several important cases. Let $A$ be a noetherian local ring; it is a Macaulay ring if and only if the same is true of $\hat{A}$ (§ 2, No. 7, cor. 4 of prop. 9). Henceforth suppose the local ring $A$ complete, and put $d = \dim(A)$.

a) Suppose that $A$ possesses a subfield. Then it possesses a coefficient subfield $K$ (IX, § 3, No. 3), and there exists a formal power series algebra $E = K[[T_1, \ldots, T_n]]$ and a surjective homomorphism of $K$-algebras $E \to A$ (loc. cit.); there also exists a formal power series algebra $E' = K[[T_1, \ldots, T_d]]$ and an injective local homomorphism of $K$-algebras $E' \to A$ such that $A$ is a finite algebra over $E'$ (loc. cit.). The following properties are equivalent :

(i) $A$ is a Macaulay ring; (ii) $dp_E(A) = n - d$; (iii) $A$ is a free $E'$-module.

b) Suppose that the residue field of $A$ is of characteristic $p > 0$. There exists a $p$-ring of length $+\infty$, with residue field $\kappa_A$ (IX, § 2, No. 3, prop. 5). Let $C$ be such a ring; there exists a formal power series algebra $E = C[[T_1, \ldots, T_n]]$ and a surjective homomorphism $\rho : E \to A$ (IX, § 2, No. 5, th. 3). The following properties are equivalent:

(i) $A$ is a Macaulay ring; (ii) $dp_E(A) = n + 1 - d$.

Suppose moreover that $p1_A$ is not a zero divisor in $A$; then there exists a formal power series algebra $E' = K[[T_1, \ldots, T_{d-1}]]$ and an injective local homomorphism of $K$-algebras $E' \to A$ such that $A$ is a finite algebra over $E'$ (loc. cit.). The local ring $E'$ is regular, of dimension $n + 1$ (VIII, § 5, No. 5, example 2). The preceding conditions are also equivalent to

(iii) $A$ is a free $E'$-module.

For analogous results in the case of modules, see § 5, No. 5.