Corollary

Let B be an integral noetherian ring and let A be a regular subring of B. Suppose that the A-module B is finitely generated. In order that the ring B be a Macaulay ring, it is necessary and sufficient that the A-module B be projective.

If the ring B is a Macaulay ring, the A-module B is projective by Prop. 5.

Suppose conversely that the A-module B is projective. The A-module A is Macaulay (Cor. 2 of Prop. 4); the A-module B is a direct factor of a free A-module of finite type, hence is Macaulay (§ 2, No. 1, Example 2). One then applies Cor. 1 of Prop. 8 of § 2, No. 6.