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}.
Categories: math.AC