Chains of model structures arising from modules of finite Gorenstein dimension

TL;DR

The paper develops chains of complete hereditary cotorsion pairs and hereditary Hovey triples arising from modules of finite Gorenstein dimension, notably PGF_n and GP_n, and extends these structures to exact subcategories. By leveraging induced cotorsion-pair constructions, it produces multiple nontrivial abelian/exact model structures whose homotopy categories all stabilize to the stable PGF/GP category $\mathcal{PGF}/\mathcal{P}$, with explicit kernel calculations and conditions for projectivity. It also analyzes finitistic dimensions, showing key equalities among finitistic projective/flat/Gorenstein dimensions and obtaining analogous results for $\mathcal{PGF}^{<\infty}$, $\mathcal{GP}^{<\infty}$ and $\mathcal{GF}^{<\infty}$ under finiteness conditions. A central theme is when PGF coincides with GP, providing equivalent formulations in terms of intersections $\mathcal{PGF}^{\perp}\cap\mathcal{GP}_n=\mathcal{P}_n$ for some or all $n$, and showing how these equivalences influence the derived-like categories in both abelian and exact settings. Overall, the work unifies and systematizes the construction of multiple, nontrivial, and triangulated homotopy categories arising from Gorenstein-related dimensions in a broad non-Noetherian arena.

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. Šaroch and J. Štovíček, 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 $n$.

Theorems & Definitions (57)

• Lemma 1.1
• proof
• Corollary 1.2
• proof
• Theorem 1.3
• Corollary 1.4
• proof
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3