Homological Dimensions in Cotorsion Pairs

Abstract

Two classes $\mathcal{A}$ and $\mathcal{B}$ of modules over a ring $R$ are said to form a cotorsion pair $(\mathcal{A},\mathcal{B})$ if $\mathcal{A}={\mathrm{KerExt}}_R^1(-,\mathcal{B})$ and $\mathcal{B}={\mathrm{KerExt}}_R^1(\mathcal{A},-)$. We investigate relative homological dimensions in cotorsion pairs. This can be applied to study the big and the little finitistic dimension of $R$. We show that $\Findim R<\infty$ if and only if the following dimensions are finite for some cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathrm{Mod}R$: the relative projective dimension of $\A$ with respect to itself, and the $\mathcal{A}$-resolution dimension of the category $\mathcal{P}$ of all $R$-modules of finite projective dimension. Moreover, we obtain an analogous result for $\findim R$, and we characterize when $\Findim R=\findim R.$