Cotorsion pairs and model structures induced by resolution dimensions relative to Frobenius pairs

Abstract

We explore some methods to obtain cotorsion pairs and model category structures from a Frobenius pair. Specifically, we consider a type of Frobenius pair $(\mathcal{X},\omega)$ in an abelian category $\mathfrak{C}$ for which there exists a nonnegative integer $m$ such that every object in the right orthogonal complement $\mathcal{X}^{\perp_1}$ of $\mathcal{X}$ under $\text{Ext}^1_{\mathfrak{C}}(-,\sim)$ has a special $\omega^\wedge_m$-precover and a special $[\omega^\wedge_m]^{\perp_1}$-preenvelope, where $\omega^\wedge_m$ denotes the class of objects in $\mathfrak{C}$ with resolution dimension relative to $\omega$ at most $m$. The cotorsion pairs and model structures obtained from these Frobenius pairs will involve the classes $\mathcal{X}$, $\omega$, $\mathcal{X}^\wedge_m$, $\omega^\wedge_m$, and their orthogonal complements, in the description of its cofibrant, fibrant and trivial objects. As applications of our results, we obtain several cotorsion pairs and model structures related to relative and absolute Gorenstein homological dimensions.