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Spring 2009 Homological dimensions in cotorsion pairs
Lidia Angeleri Hügel, Octavio Mendoza Hernández
Illinois J. Math. 53(1): 251-263 (Spring 2009). DOI: 10.1215/ijm/1264170849

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 $\operatorname{Findim} R=\operatorname{findim} R.$

Citation

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Lidia Angeleri Hügel. Octavio Mendoza Hernández. "Homological dimensions in cotorsion pairs." Illinois J. Math. 53 (1) 251 - 263, Spring 2009. https://doi.org/10.1215/ijm/1264170849

Information

Published: Spring 2009
First available in Project Euclid: 22 January 2010

zbMATH: 1205.16005
MathSciNet: MR2584945
Digital Object Identifier: 10.1215/ijm/1264170849

Subjects:
Primary: 16E10, 16G99

Rights: Copyright © 2009 University of Illinois at Urbana-Champaign

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Vol.53 • No. 1 • Spring 2009
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