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October 3, 20250 citationsOpen Access

Finiteness of homological dimensions of Ext modules

KKKaito KimuraYamagata University

Key Points

  • The main finding reveals that finiteness in projective dimension of Ext modules relates directly to dimensions of M and N.
  • Key evidence indicates that if Ext^i has finite projective dimension for 0 ≤ i ≤ Rfd_R M, then dimensions of M and N are closely tied.
  • Assessment focuses on finitely generated R-modules, particularly examining the implications of homological dimensions.
  • These findings highlight a critical relationship, suggesting that the homological behavior of one module influences the other.

Abstract

Let R be a commutative Noetherian local ring and let M and N be nonzero finitely generated R-modules. In this paper, we investigate how the finiteness of the homological dimension of Ext modules between M and N affects that of M and N. One of our main result states that if ExtⁱR (M, N) has finite projective dimension for any 0 i RfdR M, where RfdR M is the (large) restricted flat dimension of M, then M has finite projective or injective dimension if and only if N does.

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Cite This Study

Kaito Kimura (2025) studied this question.

synapsesocial.com/papers/68e02f40f0e39f13e7fa28bdhttps://doi.org/10.48550/arxiv.2508.12843
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