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Localization in Categories of Complexes and Unbounded Resolutions

  • Leovigildo Alonso Tarrío (a1), Ana Jeremías López (a2) and María José Souto Salorio (a3)

Abstract

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.

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References

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