Chains of model structures arising from modules of finite Gorenstein dimension

Nan Gao, Xue-Song Lu, Pu Zhang

Published 2024-03-08Version 1

Let $n$ be a non-negative integer. For any ring $R$, the pair \ $(\mathcal {PGF}_n, \ \mathcal P_n^\perp \cap \mathcal {PGF}^{\perp})$ proves to be a complete and 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 Artin algebra $R$, it is proved that \ $(\mathcal {GP}_n, \ \mathcal P_n^\perp \cap \mathcal P^{<\infty})$ is a complete and hereditary cotorsion pair in $R$-Mod, where $\mathcal {GP}_n$ is the class of modules of Gorenstein projective dimension $\le n$, and $\mathcal P^{<\infty}$ is the class of modules of finite projective dimension. The two chains of cotorsion pairs induce two chains of hereditary Hovey triples \ $(\mathcal {PGF}_n, \ \mathcal P_n^\perp, \ \mathcal {PGF}^{\perp})$ and \ $(\mathcal {GP}_n, \ \mathcal P_n^\perp, \ \mathcal P^{<\infty})$, and the corresponding abelian model structures on $R$-Mod in the same chain have the same homotopy category, up to triangle equivalence. The corresponding results in exact categories $\mathcal {PGF}_n$, \ $\mathcal {GP}_n$, \ $\mathcal {GF}_n$ and in $\mathcal {PGF}^{<\infty}$, $\mathcal {GP}^{<\infty}$ 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 non-negative integer $n$.