Remark

Let $S$ be a multiplicative subset of $A$ and $M$ a finitely generated Macaulay $A$-module. Then $S^{-1}M$ is a Macaulay $S^{-1}A$-module. In fact, let $q \in \mathrm{Spec}(S^{-1}A)$; let $i_A^S : A \to S^{-1}A$ denote the canonical homomorphism and put $p = (i_A^S)^{-1}(q)$. The ring $(S^{-1}A)_q$ is identified with $A_p$ (II, § 2, No. 5, prop. 11), and the $A_p$-module $(S^{-1}M)_q$ with the $A_p$-module $M_p$ (II, § 2, No. 7, prop. 20), which is Macaulay by the corollary.