On reducing homological dimensions over noetherian rings

Araya, Tokuji; Takahashi, Ryo

doi:10.1090/proc/15785

On reducing homological dimensions over noetherian rings
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by Tokuji Araya and Ryo Takahashi PDF

Proc. Amer. Math. Soc. 150 (2022), 469-480 Request permission

Abstract:

Let $\Lambda$ be a left and right Noetherian ring. First, for $m,n\in \mathbb {N}\cup \{\infty \}$, we give equivalent conditions for a given $\Lambda$-module to be $n$-torsionfree and have $m$-torsionfree transpose. Using them, we investigate totally reflexive modules and reducing Gorenstein dimension. Next, we introduce homological invariants for $\Lambda$-modules which we call upper reducing projective and Gorenstein dimensions. We provide an inequality of upper reducing projective dimension and complexity when $\Lambda$ is commutative and local. Using it, we consider how upper reducing projective dimension relates to reducing projective dimension, and the complete intersection and AB properties of a commutative Noetherian local ring.

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