Homological Dimensions in Cotorsion Pairs

TL;DR

This paper explores the relationship between homological dimensions within cotorsion pairs and classical finitistic dimensions of rings, providing criteria for their finiteness and equality.

Contribution

It establishes new equivalences linking finitistic dimensions to relative homological dimensions in cotorsion pairs, advancing understanding of module theory.

Findings

Finite finitistic dimension characterized by cotorsion pair dimensions

Conditions for equality of big and little finitistic dimensions

New criteria for finiteness of homological dimensions in module categories

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 $\mathcal A={\rm Ker Ext}^1_R(-,\mathcal B)$ and $\mathcal B={\rm Ker Ext}^1_R(\mathcal A,-)$. 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 $\Findim R<\infty$ 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 $\Findim R=\findim R.$