Characterizing local rings via homological dimensions and regular sequences

Let (R,m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If G_C-dimension of M/IM is finite for all ideals I generated by an R-regular sequence of length at most d-t then either G_C-dimension of M is at most t or C is a dualizing complex. Analogous results for other homological dimensions are also given.