  Let $(R,\fm)$ be a local ring, and let $C$ be a semidualizing complex. We
establish the equality $r_R(Z) = \nu(\Ext^{g-\inf C}_R(Z,C))\mu^{\depth
C}_R(\mathfrak{m}, C)$ for a homologically finite and bounded complex $Z$ with
finite $\GC$-dimension $g$. Additionally, we prove that if $\Ext^i(M,N)=0$ for
sufficiently large $i$, while $\id_R\Ext^i(M,N)$ remains finite for all $i$,
then both $\pd_R M$ and $\id_R N$ 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}.
