Homological Dimensions in Cotorsion Pairs

Details Homological Dimensions in Cotorsion Pairs Homological Dimensions in Cotorsion Pairs

2008-08-11

arxiv.org/abs/0808.1585v1

Mathematics

Rings and Algebras

Scientific paper

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 $\Findim R=\findim R.$

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