Corollary 4

Let A be a noetherian ring, J an ideal of A and M a finitely generated A-module. Denote by Ā and Ĝ the separated completions of A and M for the J-adic topology, and by S the multiplicative subset 1 + J of Λ. Consider the following conditions:

(i) the A-module M is a Macaulay module; (ii) the Ā-module Ĝ is a Macaulay module; (iii) the S^{-1}A-module S^{-1}M is a Macaulay module; (iv) for every maximal ideal m ∈ Supp(M) ∩ V(J), the A_m-module M_m is a Macaulay module; (v) for every prime ideal p ∈ Supp(M) such that p + J ≠ A, the Λ_p-module M_p is a Macaulay module and the ring κ(p) ⊗_A Ā is a Macaulay ring.

Conditions (ii) to (v) are equivalent, and are implied by (i). When the ideal J is contained in the radical of A, conditions (i) to (v) are equivalent.

We know that (i) implies (iii) (No. 1, example 3), and (iii) is identical with (i) when J is contained in the radical of A (since the elements of S are then invertible).

It is clear that (v) implies (iv) and that (iv) implies (i).

(i) $\Rightarrow$ (ii) : Let $m$ be a maximal ideal of $A$; then $m\widehat{A}$ is a maximal ideal of $\widehat{A}$ lying over $m$, and every maximal ideal of $\widehat{A}$ is obtained in this way (III, § 3, No. 4, prop. 8). The ring $\kappa(m) \otimes_A \widehat{A}$ is a field, hence a Macaulay ring; if the $A$-module $M$ is a Macaulay module, the same is true of the $\widehat{A}$-module $\widehat{M}$ by implication (iii) $\Rightarrow$ (i) of prop. 9.

(ii) $\Rightarrow$ (v) : The $\widehat{A}$-module $\widehat{M}$ is isomorphic to $M \otimes_A \widehat{A}$ (III, § 3, No. 4, th. 3); if it is a Macaulay module, it follows from prop. 9, (i) $\Rightarrow$ (ii) that $\kappa(p) \otimes_A \widehat{A}$ is a Macaulay ring for every $p \in \mathrm{Supp}(M)$, and that the $A$-module $M$ is a Macaulay module.