Numerical aspects of complexes of finite homological dimensions
Abstract
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}.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2023
- DOI:
- 10.48550/arXiv.2305.12251
- arXiv:
- arXiv:2305.12251
- Bibcode:
- 2023arXiv230512251R
- Keywords:
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- Mathematics - Commutative Algebra
- E-Print:
- arXiv admin note: text overlap with arXiv:2305.10244