On reduced G-perfection and horizontal linkage

Abstract

In this paper, we contribute to the theory of reduced G-perfection and horizontal linkage of modules over a commutative, Noetherian (typically, local) ring R, in the general setting where properties and operations are considered relatively to a semidualizing module C. We investigate when reduced G_C-perfection is preserved by relative Auslander transpose, and how to numerically characterize horizontally linked modules. Moreover, we show how to produce reduced ${\mathrm{G}}_C$-perfect modules that are also C-k-torsionless ($k\ge 0$ is an integer) but fail to be ${\mathrm{G}}_C$-perfect, and we illustrate that, unlike the usual grade, the relative reduced grade depends on the choice of C.

Keywords:

• Auslander transpose
• Gorenstein dimension
• horizontal linkage
• reduced G-perfect module
• semidualizing module

2020 Mathematics Subject Classification:

• Primary: 13D05; 13D02; 13D07; 13C40
• Secondary: 13C15

Acknowledgments

The authors also wish to thank the anonymous referee for helpful comments and suggestions, particularly for correcting an inaccuracy in the proof of Corollary 3.14.

Disclosure statement

No potential competing interest was reported by the author(s).

Additional information

Funding

The first author was partially supported by the CNPq-Brazil grants 301029/2019-9 and 406377/2021-9. The second author was supported by a CAPES (Brazil) Doctoral Scholarship.