Definition 1

Let $A$ be a noetherian ring and $M$ a finitely generated $A$-module. One says that $M$ is Macaulay or is a Macaulay module if, for every maximal ideal $m \in \mathrm{Supp}(M)$, one has $\mathrm{prof}_{A_m}(M_m) = \dim_{A_m}(M_m)$.

From the above, it amounts to the same thing to say that one has $\mathrm{prof}_{A_m}(M_m) \geq \dim_{A_m}(M_m)$ for every maximal ideal $m$ of $A$. Let $A$ be a noetherian local ring; in order that a nonzero finitely generated $A$-module be Macaulay, it is necessary and sufficient that its depth be equal to its dimension.