Gorenstein Homological Dimensions and Abelian Model Structures

Abstract

We construct new complete cotorsion pairs in the categories of modules and chain complexes over a Gorenstein ring $R$, from the notions of Gorenstein homological dimensions, in order to obtain new Abelian model structures on both categories. If $r$ is a positive integer, we show that the class of modules with Gorenstein-projective (or Gorenstein-flat) dimension $\leq r$ forms the left half of a complete cotorsion pair. Analogous results also hold for chain complexes over $R$. In any Gorenstein category, we prove that the class of objects with Gorenstein-injective dimension $\leq r$ is the right half of a complete cotorsion pair. The method we use in each case consists in constructing a cogenerating set for each pair. Later on, we give some applications of these results. First, as an extension of some results by M. Hovey and J. Gillespie, we establish a bijective correspondence between the class of differential graded $r$-projective complexes and the class of modules over $R[x] / (x^2)$ with Gorenstein-projective dimension $\leq r$, provided $R$ is left and right Noetherian with finite global dimension. The same correspondence is also valid for the (Gorenstein-)injective and (Gorenstein-)flat dimensions.