Gorenstein Model Structures and Generalized Derived Categories

In [Hov02], the second author introduced the Gorenstein projective and Gorenstein injective model structures on R-Mod, the category of R-modules, where R is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce; the Gorenstein flat model structure. The homotopy category with respect to each of these is called the stable module category of R. If such a ring R has finite global dimension, the graded ring R[x]/(x2) is Gorenstein and the three associated Gorenstein model structures on R[x]/(x2)-Mod, the category of graded R[x]/(x2)-modules, are nothing more than the usual projective, injective and flat model structures on Ch(R), the category of chain complexes of R-modules. Although these correspondences only recover these model structures on Ch(R) when R has finite global dimension, we can set R = Z and use general techniques from model category theory to lift the projective model structure from Ch(Z) to Ch(R) for an arbitrary ring R. This shows that homological algebra is a special case of Gorenstein homological algebra. Moreover, this method of constructing and lifting model structures carries through when Z[x]/(x2) is replaced by many other graded Gorenstein rings (or Hopf algebras, which lead to monoidal model structures). This gives us a natural way to generalize both chain complexes over a ring R as well as the derived category of R and we give some examples of such generalizations.