Corollary

Let k be a field, Λ a noetherian local k-algebra, x = (x_1, ..., x_r) a sequence of elements of m_A, and M a finitely generated Λ-module. Let Ā and Ĝ be the completions of A and M for their $(x_1\bar{A} + ... + x_r\bar{A})$-adic topology; let us denote by u : k[T_1, ..., T_r] → Ā the unique homomorphism of k-algebras such that u(T_i) = x_i for i = 1, ..., r, and by ū : k[[T_1, ..., T_r]] → Ā the unique continuous homomorphism extending it. The following conditions are equivalent:

(i) the sequence $x$ is $M$-regular; (ii) the homomorphism $u$ makes $M$ into a flat $k[T_1, \ldots, T_r]$-module; (iii) the homomorphism $\hat{u}$ makes $\hat{M}$ into a flat $k[[T_1, \ldots, T_r]]$-module. The equivalence of (i) and (ii) follows from the equivalence of conditions (i) and (iv) of proposition 5; the equivalence of (ii) and (iii) follows from III, § 5, No. 4, prop. 4.

These results make it possible to characterize macaulay modules in two important cases. Let $A$ denote a noetherian local ring, $M$ a finitely generated $A$-module. It is equivalent to say that the $A$-module $M$ is macaulay or that the $\widehat{A}$-module $\hat{M}$ is macaulay (§ 2, No. 7, cor. 4 to prop. 8). Henceforth we shall suppose that the noetherian local ring $A$ is complete.

1) Let us suppose first that $A$ possesses a subfield; then it admits a field of representatives $k$ (IX, § 3, No. 3, Theorem 1). Let $(x_1, \ldots, x_r)$ be a maximal secant sequence for $M$; let us denote by $u : k[[T_1, \ldots, T_r]] \longrightarrow A$ the unique continuous homomorphism such that $u(T_i) = x_i$ for $i = 1, \ldots, r$. By Lemma 4 b) of IX, § 2, No. 5 and Remark 1 of VIII, § 3, No. 2, $A/\mathrm{Ann}(M)$ is a finitely generated $k[[T_1, \ldots, T_r]]$-module and consequently $M$ is a finitely generated $k[[T_1, \ldots, T_r]]$-module. This being so, the following conditions are equivalent: (i) the $k[[T_1, \ldots, T_r]]$-module $M$ is free; (ii) the $A$-module $M$ is Macaulay.

In fact, it is equivalent to say that $M$ is a Macaulay $A$-module or that the sequence $(x_1, \ldots, x_r)$ is $M$-regular (§ 2, No. 3, Theorem 1). By the above Corollary, the latter condition means that the $k[[T_1, \ldots, T_r]]$-module $M$ is flat, or again that it is free since it is finitely generated.

2) Suppose that the residue field $\kappa_A$ of $A$ is of characteristic $p > 0$ and that one has $\dim(M/pM) < \dim(M)$. Let $(x_1, \ldots, x_r)$ be a maximal secant sequence for $M/pM$, so that $(p1_A, x_1, \ldots, x_r)$ is a maximal secant sequence for $M$. Let $C$ be a $p$-ring of length $+\infty$, with residue field $\kappa_A$ (IX, § 2, No. 3, Proposition 5). There exists a homomorphism $u_0$ of $C$ into $A$ which induces the identity on the residue fields; let $u : C[[T_1, \ldots, T_r]] \longrightarrow A$ be the unique homomorphism extending $u_0$ and sending $T_i$ to $x_i$ for every $i$. It follows as above from loc. cit., No. 5, Lemma 4 that $u$ makes $M$ into a finitely generated $C[[T_1, \ldots, T_r]]$-module. The following conditions are equivalent: (i) the $C[[T_1, \ldots, T_r]]$-module $M$ is free; (ii) the $A$-module $M$ is Macaulay.

In fact condition (ii) is equivalent to saying that the sequence $(x_1, \ldots, x_r)$ is $M$-regular and that the homothety of ratio $p$ in $M/(x_1M + \ldots + x_rM)$ is injective (§ 2, No. 3, Theorem 1). Now this latter condition means that the $C$-module $M/(x_1M + \ldots + x_rM)$ is torsion-free, hence flat ($A$, X, p. 9, Example 7). Thus, having regard to Proposition 5, (iv) $\Leftrightarrow$ (i), condition (ii) is equivalent to the fact that the $C[T_1, \ldots, T_r]$-module $M$ is flat, or again (III, § 5, No. 4, Proposition 4) that the $C[[T_1, \ldots, T_r]]$-module $M$ is flat, that is to say free since it is finitely generated.