Homological dimensions in cotorsion pairs

Abstract

Given a ring $R$, two classes $\mathcal A$ and $\mathcal B$ of $R$-modules are said to form a cotorsion pair $(\mathcal A, \mathcal B)$ in $\operatorname{Mod}R$ if $\mathcal A=\operatorname {Ker}\operatorname{Ext}^1_R(-,\mathcal B)$ and $\mathcal B=\operatorname{Ker}\operatorname{Ext}^1_R(\mathcal A,-)$. 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 $\operatorname{Findim} R<\infty$ if and only if the following dimensions are finite for some cotorsion pair $(\mathcal A, \mathcal B)$ in $\operatorname{Mod}R$: the relative projective dimension of $\mathcal A$ 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 $\operatorname{findim} R$, and we characterize when $\operatorname{Findim} R=\operatorname{findim} R.$