Gorenstein model structures and generalized derived categories
James Gillespie and Mark Hoveyu=
In a previous paper, 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 homotopy category of $R$. Here we show that if such
a ring $R$ has finite global dimension, the graded ring $R[x]/(x^2)$
is Gorenstein and the three associated Gorenstein model structures on
$R[x]/(x^2)$-Mod, the category of graded $R[x]/(x^2)$-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
$\Z[x]/(x^2)$ 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.