2024/03/08 by Gao, Nan, Lu, Xue-Song, Zhang, Pu · 1 citation
#16E65 #16G50 #18N40 #FOS: Mathematics #Primary 16E30 #Representation Theory (math.RT) #Secondary 16E10
paper · doi:10.48550/arxiv.2403.05232
For any integer n≥ 0 and any ring R, (\mathcal PGFn, \mathcal Pn^⊥ ∩ \mathcal PGF⊥) proves to be a complete hereditary cotorsion pair in R-Mod, where \mathcal PGF is the class of PGF modules, introduced by J. Šaroch and J. Štovíček, and \mathcal PGFn is the class of R-modules of PGF dimension ≤ n. For any Artin algebra R, (\mathcal GPn, \mathcal Pn^⊥ ∩ \mathcal GP⊥) proves to be a complete and hereditary cotorsion pair in R-Mod, where \mathcal GPn is the class of modules of Gorenstein projective dimension ≤ n. These cotorsion pairs induce two chains of hereditary Hovey triples (\mathcal PGFn, \mathcal Pn^⊥, \mathcal PGF⊥) and (\mathcal GPn, \mathcal Pn^⊥, \mathcal GP⊥), and the corresponding homotopy categories in the same chain are the same. It is observed that some complete cotorsion pairs in R-Mod can induce complete cotorsion pairs in some special extension closed subcategories of R-Mod. Then corresponding results in exact categories \mathcal PGFn, \mathcal GPn, \mathcal GFn, \mathcal PGF<∞, \mathcal GP<∞ and \mathcal GF<∞, are also obtained. As a byproduct, PGF = \mathcal GP for a ring R if and only if PGF^⊥\capGPn=\mathcal Pn for some n.