Some applications of a lemma by Hanes and Huneke

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.