ABSTRACT
One of the main reasons for studying homological dimensions was a result showing that a commutative noetherian ring R with residue field k is regular if and only if every R-module M has finite projective dimension. This result started a trend – proving that finiteness of a homological dimension for all modules characterizes rings with special properties. Gorenstein homological algebra is based on finitely generated modules. They generalize the classes of projective, injective, and flat modules. This chapter provides information on the G-dimension of a finitely generated module over a commutative noetherian ring. It focuses on Gorenstein projective modules – definition and properties. The dual notion that of the Gorenstein injective module as well as the Gorenstein flat modules are also considered.
