Open loci of graded modules

Let $A={\oplus }_{i\in \text{nn}}{A}_i$ be an excellent homogeneous Noetherian graded ring and let $M={\oplus }_{n\in \text{zz}}{M}_n$ 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 $\text{Spec}(A_0)$. In particular, the Cohen-Macaulay locus $U_{CM}^0=\{\text{p}\in \text{Spec}(A_0)\mid M_{\text{p}}isCohen-Macaulay\}$ is an open subset of $\text{Spec}(A_0)$. 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 $M\mathrm{/}{I}^{n}M$ are eventually stable.