Abstract: In 1967 Auslander and Bridger introduced f.g. Gorenstein projective modules and an associated homological dimension denoted G-dim (which is defined in terms of resolutions of f.g. Gorenstein projective modules). They proved that every f.g. projective module is also Gorenstein projective, and that G-dim shares many important properties with the classical projective dimension.
In the nineties, Enochs and Jenda introduced general (meaning not necessarily f.g) Gorenstein projective modules as well as the notions of Gorenstein injective and Gorenstein flat modules. These classes of modules are the foundation of what is now commenly known as "Gorenstein homological algebra".
It has turned out that Gorenstein homological algebra behaves like ordinary homological algebra in many aspects. In the talk I will present examples of what this means, and in particular I shall prove certain Gorenstein versions of the classical Auslander-Buchsbaum formula for projective dimension. We will also see how Gorenstein homological algebra relates to the so-called Auslander categories for a dualizing complex, when the latter exists.
The main material to be presented in this talk is is joint work with Lars Winther Christensen (University of Nebraska-Lincoln), and is to appear in the Canadian Journal of Mathematics.