Open loci of graded modules

Let $A=\bigoplus_{i\in \mathbb{N}}A_i$ be an excellent homogeneous Noetherian graded ring and let $M=\bigoplus_{n\in \mathbb{Z}}M_n$ be a finitely generated graded $A$-module. We consider $M$ as a module over $A_0$ and show that the $(S_k)$-loci of $M$ are open in $\operatorname{Spec}(A_0)$. In particular, the Cohen-Macaulay locus $U^0_{CM}=\{{\mathfrak{p}}\in \operatorname{Spec}(A_0) \mid M_\mathfrak{p} \;$   is Cohen-Macaulay$\}$ is an open subset of $\operatorname{Spec}(A_0)$. We also show that the $(S_k)$-loci on the homogeneous parts $M_n$ of $M$ are eventually stable. As an application we obtain that for a finitely generated Cohen-Macaulay module $M$ over an excellent ring $A$ and for an ideal $I\subseteq A$ which is not contained in any minimal prime of $M$, the $(S_k)$-loci for the modules $M/I^nM$ are eventually stable.