Corollary 3

Every presentable Macaulay ring (in particular, every complete local Macaulay ring) has a dualizing module.

For let indeed $R$ be a regular ring and $A$ a Macaulay ring quotient of $R$. The $R$-module $A$ is a Macaulay module ($§ 2$, No. 5, example 5), and $R$ has a dualizing module (No. 1, example 2); hence the same is true of $A$ by cor. 1. Moreover, it has already been observed that a complete local noetherian ring is presentable ($§ 4$, No. 4, prop. 6, c)).

More generally, every Macaulay ring quotient of a Gorenstein ring has a dualizing module. Conversely, one can prove that a local Macaulay ring which has a dualizing module is a quotient of a Gorenstein local ring (exercise 1).