Ring homomorphisms and local rings with quasi-decomposable maximal ideal

Abstract

The notion of local rings with quasi-decomposable maximal ideal was formally introduced by Nasseh and Takahashi. In separate works, the authors of the present paper showed that such rings have rigid homological properties; for instance, they are both Ext- and Tor-friendly. One point of this paper is to further explore the homological properties of these rings and also introduce new classes of such rings from a combinatorial point of view. Another point is to investigate how far some of these homological properties can be pushed along certain diagrams of local ring homomorphisms.

KEYWORDS:

• Cohen-Macaulay
• complete intersection
• decomposable maximal ideal
• dualizing complex
• Ext-friendly
• fiber product
• Gorenstein
• hypersurface
• injective dimension
• local ring homomorphism
• projective dimension
• quasi-decomposable maximal ideal
• semidualizing complexes
• Tor-friendly
• totally reflexive

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D02
• 13D05
• 13D07
• 13H10
• 18A30

Acknowledgments

We are grateful to the referee for reading the paper carefully and for giving us valuable suggestions.

Notes

1 Another generalization of the result of Nasseh and Yoshino [43, Theorem 3.1] to the differential graded homological algebra setting is found in [12, Theorem 4.1].

2 The notions of dualizing module and canonical module agree when R is Cohen-Macaulay.

Additional information

Funding

K. A. Sather-Wagstaff was supported by the National Science Foundation under Grant No. EES-2243134. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. R. Takahashi was partly supported by JSPS Grants-in-Aid for Scientific Research 23K03070.