Homological dimension based on a class of Gorenstein flat modules

Modules of finite Gorenstein flat dimension and approximations

“…[26,Theorem 4.12]. The homological dimension theory, which is based on the class 𝙿𝙶𝙵(𝑅), is studied in [14].…”

“…As a consequence of the existence of these approximations for modules in 𝙶𝙵𝚕𝚊𝚝(𝑅), we obtain characterizations of the numerical values of the PGF-dimension (i.e., of the relative homological dimension, which is based on the class of PGF-modules [14]) and the Gorenstein flat dimension, in terms of the vanishing of certain Ext-groups. These characterizations are analogous to the corresponding characterization of the Gorenstein projective dimension for modules of finite Gorenstein projective dimension.…”

“…), 𝙿𝙶𝙵(𝑹) ⟂ )In this section, we use the approximation sequences obtained in Theorem 2.1 to characterize the PGF-modules and, more generally, the PGF-dimension of modules (which is introduced in[14]) within 𝙶𝚏𝚕𝚊𝚝(𝑅), in terms of the vanishing of certain Ext-groups. We obtain a hereditary Hovey triple in 𝙶𝚏𝚕𝚊𝚝(𝑅), such that the homotopy category of the associated exact model structure is triangulated equivalent to the stable category of PGF-modules.…”