On Bass numbers of graded components of local cohomology modules supported on $\mathfrak{C}$-monomial ideals in mixed characteristic

Let $A$ be a Dedekind domain of characteristic zero such that for each height one prime ideal $\mathfrak{p}$ in $A$, the local ring $A_{\mathfrak{p}}$ has mixed characteristic with finite residue field. Suppose that $R=A[X_1,\dots ,X_n]$ is a standard ${\mathbb{N}}^n$-graded polynomial ring over $A$, i.e., $\mathrm{deg}A=\underset{‾}{0}\in {\mathbb{N}}^n$ and $\mathrm{deg}(X_j)=e_j\in {\mathbb{N}}^n$. Let $I$ be a $\mathfrak{C}$-monomial ideal of $R$ and let $M:=H_I^i(R)={⨁}_{\underset{‾}{u}\in {\mathbb{Z}}^n}{M}_{\underset{‾}{u}}$. Recently, the second author and S. Roy [2025, J. Algebra 681, 1-21] proved that for a fixed $\underset{‾}{u}\in {\mathbb{Z}}^n$, the Bass numbers ${\mu }_i(\mathfrak{p},M_{\underset{‾}{u}})$ are finite for each prime ideal $\mathfrak{p}$ in $A$ and for every $i\ge 0$. Let for a subset of $U$ of $\mathcal{S}=\{1,\dots ,n\}$, define a block to be the set ${\displaystyle \mathcal{B}(U)=\{\underset{‾}{u}\in {\mathbb{Z}}^n\mid u_i\ge 0ifi\in Uand{u}_i\le -1ifi\mathrm{\notin }U\}}$. Note that ${\bigcup }_{U\subseteq \mathcal{S}}\mathcal{B}(U)={\mathbb{Z}}^n$. In this article, the main result we establish is that for a fixed prime ideal $\mathfrak{p}$ in $A$ and $i\ge 0$, the set of Bass numbers $\{{\mu }_i(\mathfrak{p},M_{\underset{‾}{u}})\mid \underset{‾}{u}\in {\mathbb{Z}}^n\}$ is constant on $\mathcal{B}(U)$ for each subset $U$ of $\{1,\dots ,n\}$. Our idea is to prove this by carrying out a comprehensive study of the structure theorem for the graded components of $M$ when $A$ is a complete DVR of mixed characteristic with finite residue field.