Theorem 1

Let A be a noetherian local ring, M a nonzero finitely generated A-module, d the dimension of M, x = (x₁, ..., x_d) a sequence of elements of mA secant for M, and J the ideal generated by it. The following conditions are equivalent:

(i) the A-module M is Macaulay ;

(ii) the sequence $x$ is M-regular ; (iii) the sequence $x$ is completely secant for $M$ (A, X, p. 157, Def. 2) ; (iv) the multiplicity (VIII, § 7, No. 1, Def. 1) $e_J(M)$ of $M$ relative to the ideal $J$ is equal to the length of the A-module $M/JM$ ; (v) for each integer $i$ such that $1 \leq i \leq d$, the A-module $M/(x_1M + \ldots + x_{i-1}M)$ is pure.

The equivalence of (ii) and (iii) follows from A, X, p. 160, Cor. 1 to Th. 1. Since the A-module $M/JM$ is of finite length (VIII, § 3, No. 2, Th. 1), the equivalence of (iii) and (iv) follows from VIII, § 4, No. 3, Prop. 4 and No. 4, Th. 3. It remains to prove the equivalence of (i), (ii), and (v).

(i) $\Rightarrow$ (v) : if $M$ is Macaulay, each of the modules $M/(x_1M + \ldots + x_{i-1}M)$ is Macaulay (Prop. 4), hence pure.

(v) $\Rightarrow$ (ii) : this follows from Lemma 1 applied to each of the modules $M/(x_1M + \ldots + x_{i-1}M)$.

(ii) $\Rightarrow$ (i) : this follows from Prop. 4, since $M/JM$ is of finite length, hence Macaulay.