Example 4

If $A$ is a Macaulay ring, the same is true of $S^{-1}A$ for every multiplicative subset $S$ of $A$ (No. 1, Remark). Conversely, if the ring $A_m$ is a Macaulay ring for every maximal ideal $m$ of $A$, then the ring $A$ is Macaulay (No. 1, Def. 1).