Gorenstein model structures and generalized derived categories

<jats:title>Abstract</jats:title><jats:p>In a paper from 2002, Hovey introduced the Gorenstein projective and Gorenstein injective model structures on <jats:italic>R</jats:italic>-Mod, the category of <jats:italic>R</jats:italic>-modules, where <jats:italic>R</jats:italic> 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 <jats:italic>R</jats:italic>. If such a ring <jats:italic>R</jats:italic> has finite global dimension, the graded ring <jats:italic>R</jats:italic>[<jats:italic>x</jats:italic>]/(<jats:italic>x</jats:italic><jats:sup>2</jats:sup>) is Gorenstein and the three associated Gorenstein model structures on <jats:italic>R</jats:italic>[<jats:italic>x</jats:italic>]/(<jats:italic>x</jats:italic><jats:sup>2</jats:sup>)-Mod, the category of graded <jats:italic>R</jats:italic>[<jats:italic>x</jats:italic>]/(<jats:italic>x</jats:italic><jats:sup>2</jats:sup>)-modules, are nothing more than the usual projective, injective and flat model structures on Ch(<jats:italic>R</jats:italic>), the category of chain complexes of <jats:italic>R</jats:italic>-modules. Although these correspondences only recover these model structures on Ch(<jats:italic>R</jats:italic>) when <jats:italic>R</jats:italic> has finite global dimension, we can set <jats:italic>R</jats:italic> = ℤ and use general techniques from model category theory to lift the projective model structure from Ch(ℤ) to Ch(<jats:italic>R</jats:italic>) for an arbitrary ring <jats:italic>R</jats:italic>. 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[<jats:italic>x</jats:italic>]/(<jats:italic>x</jats:italic><jats:sup>2</jats:sup>) 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 <jats:italic>R</jats:italic> and the derived category of <jats:italic>R</jats:italic> and we give some examples of such generalizations.</jats:p>