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arXiv · 2305.12251

Numerical aspects of complexes of finite homological dimensions

Majid Rahro Zargar·Mohsen Gheibi
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Abstract

Let $(R,\fm)$ be a local ring, and let $C$ be a semidualizing complex. We establish the equality $r_R(Z) = ν(\Ext^{g-\inf C}_R(Z,C))μ^{\depth C}_R(\mathfrak{m}, C)$ 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}.

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BibTeXRIS

Majid Rahro Zargar, Mohsen Gheibi. 2023-05-20. Numerical aspects of complexes of finite homological dimensions. https://arxiv.org/abs/2305.12251

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