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Localization in categories of complexes and unbounded resolutions. (English) Zbl 0948.18008

Authors’ abstract: In this paper we show that for a Grothendieck category \({\mathcal A}\) and a complex \(E\) in \(\mathbb{C}({\mathcal A})\) there is an associated localization endofunctor \(\ell\) in \(\mathbb{D} ({\mathcal A})\). 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 \(\mathbb{D} ({\mathcal A})\) that contains \(E\). As applications, we construct \(K\)-injective resolutions for complexes of objects of \({\mathcal A}\) and derive Brown representability for \(\mathbb{D}({\mathcal A})\) from the known result for \(\mathbb{D}(R\)-mod), where \(R\) is a ring with unit.

MSC:

18E30 Derived categories, triangulated categories (MSC2010)
18E15 Grothendieck categories (MSC2010)
18E35 Localization of categories, calculus of fractions
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