Chains of model structures arising from modules of finite Gorenstein dimension

Abstract

For any integer $n\ge 0$ and any ring $R$, \ $(\mathcal {PGF}_n, \ \mathcal P_n^\perp \cap \mathcal {PGF}^{\perp})$ proves to be a complete hereditary cotorsion pair in $R$-Mod, where $\mathcal {PGF}$ is the class of PGF modules, introduced by J. \v{S}aroch and J. \v{S}\'{t}ov\'{i}\v{c}ek, and \ $\mathcal {PGF}_n$ is the class of $R$-modules of PGF dimension $\le n$. For any Artin algebra $R$, \ $(\mathcal {GP}_n, \ \mathcal P_n^\perp \cap \mathcal {GP}^{\perp})$ proves to be a complete and hereditary cotorsion pair in $R$-Mod, where $\mathcal {GP}_n$ is the class of modules of Gorenstein projective dimension $\le n$. These cotorsion pairs induce two chains of hereditary Hovey triples \ $(\mathcal {PGF}_n, \ \mathcal P_n^\perp, \ \mathcal {PGF}^{\perp})$ and \ $(\mathcal {GP}_n, \ \mathcal P_n^\perp, \ \mathcal {GP}^{\perp})$, 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 $\mathcal {PGF}_n$, \ $\mathcal {GP}_n$, \ $\mathcal {GF}_n$, \ $\mathcal {PGF}^{<\infty}$, \ $\mathcal {GP}^{<\infty}$ and $\mathcal {GF}^{<\infty}$, are also obtained. As a byproduct, $\mathcal{PGF} = \mathcal {GP}$ for a ring $R$ if and only if $\mathcal{PGF}^\perp\cap\mathcal{GP}_n=\mathcal P_n$ for some $n$.