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Localization in Categories of Complexes and Unbounded Resolutions

Published online by Cambridge University Press:  20 November 2018

Leovigildo Alonso Tarrío ,
Ana Jeremías López  and
María José Souto Salorio
Leovigildo Alonso Tarrío
Affiliation:
Departamento de Álxebra, Facultade de Matemáticas, Universidade de Santiago de Compostela, E-15771 Santiago de Compostela, Spain email: lalonso@zmat.usc.es
Ana Jeremías López
Affiliation:
Departamento de Álxebra, Facultade de Matemáticas, Universidade de Santiago de Compostela, E-15771 Santiago de Compostela, Spain email: jeremias@zmat.usc.es
María José Souto Salorio
Affiliation:
Facultade de Informática, Campus de Elviña, Universidade da Coruña, E-15071 A Coruña, Spain email: mariaj@udc.es
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Abstract

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In this paper we show that for a Grothendieck category $\mathcal{A}$ and a complex $E$ in $\mathbf{C}(\mathcal{A})$ there is an associated localization endofunctor $\ell$ in $\mathbf{D}(\mathcal{A})$. This means that $\ell$ is idempotent (in a natural way) and that the objects that go to 0 by $\ell$ are those of the smallest localizing (= triangulated and stable for coproducts) subcategory of $\mathbf{D}(\mathcal{A})$ that contains $E$. As applications, we construct $\text{K}$-injective resolutions for complexes of objects of $\mathcal{A}$ and derive Brown representability for $\mathbf{D}(\mathcal{A})$ from the known result for $\mathbf{D}(R-\mathbf{mod})$, where $R$ is a ring with unit.

Keywords

18E3018E1518E35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 2 , 01 April 2000 , pp. 225 - 247
Copyright
Copyright © Canadian Mathematical Society 2000

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