  In this paper we study homological dimensions of finitely generated modules
over commutative Noetherian local rings, called reducing homological
dimensions. We obtain new characterizations of Gorenstein and complete
intersection local rings via reducing homological dimensions. For example, we
extend a classical result of Auslander and Bridger, and prove that a local ring
is Gorenstein if and only if each finitely generated module over it has finite
reducing Gorenstein dimension. Along the way, we prove various connections
between complexity and reducing projective dimension of modules.
