Proposition 10

Let $\rho : A \to B$ be a homomorphism of noetherian rings making $B$ into a flat $A$-module. The following conditions are equivalent:

(i) $B$ is a Macaulay ring; (ii) for every prime ideal $q$ of $B$, the rings $A_{\rho^{-1}(q)}$ and $\kappa(\rho^{-1}(q)) \otimes_A B$ are Macaulay rings; (iii) for every maximal ideal $n$ of $B$, the rings $A_{\rho^{-1}(n)}$ and $\kappa(\rho^{-1}(n)) \otimes_A B$ are Macaulay rings.

If moreover $B$ is faithfully flat over $A$, these conditions imply that $A$ is a Macaulay ring.

This is the special case $M = A$, $N = B$ of prop. 9.