Example 3

Let $M$ be a finitely generated Macaulay $A$-module and $(x_1, \ldots, x_n)$ an $M$-regular sequence of elements of $A$. Then the $A$-module $\overline{M} = M/(x_1M + \cdots + x_nM)$ is Macaulay. Let indeed $m$ be a maximal ideal of $A$ belonging to the support of $\overline{M}$; one has $x_i \in m$ for every $i$ since $x_i$ annihilates $\overline{M}$, and the canonical images of the $x_i$ in $A_m$ form an $M_m$-regular sequence of elements of $mA_m$. One consequently has ($§ 1$, No. 4, Proposition 7 and VIII, $§ 3$, No. 2, Cor. of Proposition 3) the equalities

whence our assertion.