NUMERICAL ASPECTS OF COMPLEXES OF FINITE HOMOLOGICAL DIMENSIONS

Abstract

Let $(R,\mathfrak{𝔪})$ be a local ring, and let $C$ be a semidualizing complex. We establish the equality $r_R(Z)=\nu ({\text{Ext}}_R^{g-\text{inf}C}(Z,C)){\mu }_R^{\text{depth}C}(\mathfrak{𝔪},C)$ for a homologically finite and bounded complex $Z$ with finite ${\text{G}}_C$-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 (Algebr. Represent. Theory 27:1 (2024), 639–653), addressing the questions as presented in their Questions 3.9 and 4.2.