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. (C) 2000 Academic Press. [References: 17]
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