Proposition 4

Let A be a noetherian local ring, M a nonzero finitely generated A-module, and (x₁, ..., x_r) a sequence of elements of mA secant for M. The following conditions are equivalent:

(i) the A-module M is Macaulay ; (ii) the sequence (x₁, ..., x_r) is M-regular and the A-module M/(x₁M + ... + x_rM) is Macaulay.

Suppose that the sequence (x₁, ..., x_r) is M-regular. One then has

(§ 1, No. 4, Prop. 7), whence the implication (ii) ⇒ (i).

Suppose the A-module M Macaulay, and prove (ii) by induction on r. The assertion is obvious if r = 0. If r ≥ 1, the A-module N = M/(x₁M + ... + x_{r−1}M) is Macaulay by the induction hypothesis, and one has dim(N/x_rN) = dim(N) − 1 since the sequence (x₁, ..., x_r) is secant; the homothety (x_r)_N is therefore injective (Lemma 1), and N/x_rN is Macaulay (No. 1, Example 3), whence (ii).