G-Dimension, Complete Intersection Dimension, Complexity, and Relative Homological Algebra,

It is shown that the G-dimension and the complete intersection dimension are relative projective dimensions. Relative Auslander–Buchsbaum formulas are discussed. New cohomology theories, called complexity cohomology, are constructed. The new theories play the same role in identifying rings (and modules) with prescribed complexity as Tate–Vogel cohomology does in identifying modules of finite projective dimension.