Characterizing local rings via homological dimensions and regular sequences

Let (R,m)(R,m) be a Noetherian local ring of depth dd and CC a semidualizing RR-complex. Let MM be a finite RR-module and tt an integer between 0 and dd. If the GCGC-dimension of M/aMM/aM is finite for all ideals aa generated by an RR-regular sequence of length at most d−td−t then either the GCGC-dimension of MM is at most tt or CC is a dualizing complex. Analogous results for other homological dimensions are also given.