Relative weak global Gorenstein dimension, AB-contexts and model structures

Abstract

In this paper we introduce and study the weak Gorenstein global dimension of a ring $R$ with respect to a left $R$-module $C$. We provide several characterizations of when this homological invariant is bounded. Two main applications are given: first, we prove that the weak Gorenstein global dimension of $R$ relative to a semidualizing $(R,S)$-bimodule $C$ can be computed either by the ${\rm G_C}$-flat dimension of the left $R$-modules or right $S$-modules, just like the (absolute) weak global dimension. As a consequence, a new argument for solving Bennis' conjecture is obtained. As a second application, we give a concrete description of the weak equivalences in the ${\rm G_C}$-flat model structure recently found by the authors. In order to prove this result, an interesting connection between abelian model structures and AB-weak contexts is proved. This connection leads to a result that can be applied to obtain abelian model structures with a simpler description of trivial objects.