Bull. Korean Math. Soc. 2023; 60(1): 23-32
Online first article January 25, 2023 Printed January 31, 2023
https://doi.org/10.4134/BKMS.b210703
Copyright © The Korean Mathematical Society.
Zhicheng Wang
Northwest Normal University
Let $\mathcal{S}$ be a Serre class in the category of modules and $\mathfrak{a}$ an ideal of a commutative Noetherian ring $R$. We study the containment of Tor modules, Koszul homology and local homology in $\mathcal{S}$ from below. With these results at our disposal, by specializing the Serre class to be Noetherian or zero, a handful of conclusions on Noetherianness and vanishing of the foregoing homology theories are obtained. We also determine when $\mathrm{Tor}_{s+t}^R(R/\mathfrak{a},X)\cong\mathrm{Tor}_{s}^R(R/\mathfrak{a},\mathrm{H}_{t}^\mathfrak{a}(X))$.
Keywords: Homology, Serre class, complex
MSC numbers: 13D45, 13D02
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