Numerical aspects of complexes of finite homological dimensions

Let $(R,\text{fm})$ be a local ring, and let $C$ be a semidualizing complex. We establish the equality $r_R(Z)=\nu ({\text{Ext}}_R^{g-\inf C}(Z,C)){\mu }_R^{\text{depth}C}(\mathfrak{m},C)$ for a homologically finite and bounded complex $Z$ with finite $\text{GC}$-dimension $g$. Additionally, we prove that if ${\text{Ext}}^i(M,N)=0$ for sufficiently large $i$, while ${\text{id}}_R{\text{Ext}}^i(M,N)$ remains finite for all $i$, then both ${\text{pd}}_{R}M$ and ${\text{id}}_{R}N$ are finite when $M$ and $N$ are finitely generated $R$-modules. These findings extend the recent results of Ghosh and Puthenpurakal \cite{Ghosh}, addressing their questions as presented in \cite[Question 3.9]{Ghosh} and \cite[Question 4.2]{Ghosh}.