Resolving Resolution Dimensions

Abstract

Let $\mathcal{A}$ be an abelian category. A subcategory $\mathcal{X}$ of $\mathcal{A}$ is called a resolving subcategory if $\mathcal{X}$ is closed under extensions and kernels of epimorphisms and contains all projective objects of $\mathcal{A}$. In this paper, we consider the $\mathcal{X}$-resolution dimensions and special $\mathcal{X}$-precovers for a resolving subcategory $\mathcal{X}$ of $\mathcal{A}.$ Many results in Araya et al. (J Math Kyoto Univ 45:287–306, 2005), Auslander and Bridger (Mem Am Math Soc(94), 1969), Avramov and Martsinkovsky (Proc Lond Math Soc 85:393–440, 2002), Christensen (2000), Christensen et al. (J Algebra 302:231–279, 2006), Holm (J Pure Appl Algebra 189:167–193, 2004), Holm and Jørgensen (J Pure Appl Algebra 205:423–445, 2006), Sather-Wagstaff et al. (Algebr Represent Theor 14:403–428, 2011), Takahashi and White (Math Scand 106:5–22, 2010), White (J Commut Algebra 2:111–137, 2010) and Xu (1996) are generalized.