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Localization in Categories of Complexes and Unbounded Resolutions |
Canad. J. Math. 52(2000), 225-247
Published:2000-04-01
Printed: Apr 2000
Printed: Apr 2000
- Leovigildo Alonso Tarrío
- Ana Jeremías López
- María José Souto Salorio
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 |