Corollary 4

Let $\rho : A \to B$ be a local homomorphism of noetherian local rings. Suppose that $A$ is regular and that $\rho$ makes $B$ into a finitely generated $A$-module. Then $dp_A(B) \geq \dim(A) - \dim(B)$. For $B$ to be a Macaulay ring, it is necessary and sufficient that one have $dp_A(B) = \dim(A) - \dim(B)$. For $B$ to be a Macaulay ring of dimension equal to $\dim(A)$, it is necessary and sufficient that the $A$-module $B$ be free.

In fact, one has $\dim(B) = \dim_A(B)$ (VIII, § 2, No. 3, th. 1); moreover, $B$ is a Macaulay ring if and only if it is a Macaulay $A$-module (§ 2, No. 6, prop. 8). It is therefore enough to apply corollaries 1 and 2.