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We contribute to the theory of G-dimension relative to a semidualizing module C , in connection to the properties of a module being totally C -reflexive, C - k -torsionless, and C - k -syzygy, where k ≥ 0 is an integer. We extend several known results, and for an integer q ≥ 0 we introduce the class of C - q -Gorenstein rings. We also consider C -duals and initiate the study of C -valued derivation modules over a local ring R of low depth; if C = R and R is a three-dimensional Gorenstein domain, we find a bound for the number of generators and we propose a question when C is general. [ABSTRACT FROM AUTHOR]

Our main goal in this note is to use a version of a lemma by Hanes and Huneke to provide characterizations of when certain one-dimensional reduced local rings are regular. This is of interest in view of the long-standing Berger's Conjecture (the ring is predicted to be regular if its universally finite differential module is torsion-free), which in fact we show to hold under suitable additional conditions, mostly toward the G-regular case of the conjecture. Furthermore, applying the same lemma to a Cohen-Macaulay local ring which is locally Gorenstein on the punctured spectrum but of arbitrary dimension, we notice a numerical characterization of when an ideal is strongly non-obstructed and of when a given semidualizing module is free. [ABSTRACT FROM AUTHOR]

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 GC-perfection is preserved by relative Auslander transpose, and how to numerically characterize horizontally linked modules. Moreover, we show how to produce reduced G C -perfect modules that are also C-k-torsionless ( k ≥ 0 is an integer) but fail to be G C -perfect, and we illustrate that, unlike the usual grade, the relative reduced grade depends on the choice of C. [ABSTRACT FROM AUTHOR]