ABSTRACT
Examples of Gorenstein categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 12.3 Cotorsion pairs from Gorenstein homological dimensions . . . . . . . 282
Another cogenerating set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 12.4 Hovey’s model structures on Gorenstein categories . . . . . . . . . . . . . 288 12.5 The Gorenstein n-injective model structure . . . . . . . . . . . . . . . . . . . . . 289
Characterizing Gorenstein homological dimensions . . . . . . . . . . . . . . 290 Cotorsion pairs from Gorenstein homological dimensions . . . . . . . 291 Gorenstein-injective dimensions and cogenerating sets . . . . . . . . . . 293
12.6 Homotopy categories in Gorenstein homological algebra . . . . . . . . 294 12.7 Gorenstein-homological dimensions of chain complexes . . . . . . . . . 295
Matlis Theorem for chain complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Characterizing Gorenstein homological dimensions . . . . . . . . . . . . . . 298
12.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
12.1 Introduction In [Hov02, Theorem 8.6], M. Hovey constructs a unique Abelian model
structure on ModpRq (with R a Gorenstein ring) where the cofibrant objects are the G-projective modules and the trivial objects are the modules with finite projective dimension. There is also another model structure with the same trivial objects such that the G-injective modules form the class of fibrant objects. Hovey’s method consists in proving that the classes of GProjpRq and GInjpRq are the left and right halves, respectively, of two cotorsion pairs cogenerated by a set. We will present Hovey’s results in the context of Gorenstein categories, and provide other ways to obtain these model structures, by using G-projective precovers and G-injective preenvelopes of objects with finite Gorenstein homological dimensions (See Section 11.3).
