Abstract view

# Localization in Categories of Complexes and Unbounded Resolutions

Published:2000-04-01
Printed: Apr 2000
• Leovigildo Alonso Tarrío
• Ana Jeremías López
• María José Souto Salorio
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## Abstract

In this paper we show that for a Grothendieck category $\A$ and a complex $E$ in $\CC(\A)$ there is an associated localization endofunctor $\ell$ in $\D(\A)$. 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 $\D(\A)$ that contains $E$. As applications, we construct K-injective resolutions for complexes of objects of $\A$ and derive Brown representability for $\D(\A)$ from the known result for $\D(R\text{-}\mathbf{mod})$, where $R$ is a ring with unit.
 MSC Classifications: 18E30 - Derived categories, triangulated categories 18E15 - Grothendieck categories 18E35 - Localization of categories