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

Abstract

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 $R=A[X_1,\ldots,X_n]$ is a standard $\mathbb{N}^n$-graded polynomial ring over $A$, i.e., $\operatorname{deg} A=\underline{0}\in \mathbb{N}^n$ and $\operatorname{deg}(X_j)=e_j\in \mathbb{N}^n$. Let $I$ be a $\mathfrak{C}$-monomial ideal of $R$ and let $M:= H^i_I(R)=\bigoplus_{\underline{u}\in \mathbb{Z}^n}M_{\underline{u}}$. Recently, the second author and S. Roy [2025, J. Algebra 681, 1-21] proved that for a fixed $\underline{u}\in\mathbb{Z}^n$, the Bass numbers $\mu_i(\mathfrak{p},M_{\underline{u}})$ are finite for each prime ideal $\mathfrak{p}$ in $A$ and for every $i\geq 0$. Let for a subset of $U$ of $\mathcal{S}=\{1, \ldots, n\}$, define a block to be the set $\displaystyle\mathcal{B}(U)=\{\underline{u} \in \mathbb{Z}^n \mid u_i \geq 0 \mbox{ if } i \in U \mbox{ and } u_i \leq -1 \mbox{ if } i \notin U \}$. 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 $i\geq 0$, the set of Bass numbers $\{\mu_i(\mathfrak{p},M_{\underline{u}})\mid \underline{u}\in \mathbb{Z}^n\}$ is constant on $\mathcal{B}(U)$ for each subset $U$ of $\{1, \ldots, n\}$. 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.