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In this paper we show that for a Grothendieck category
and a complex
there is an associated localization endofunctor
. This means that
is idempotent (in a natural way) and that the objects that go to 0 by
are those of the smallest localizing (= triangulated and stable for coproducts) subcategory of
. As applications, we construct
-injective resolutions for complexes of objects of
and derive Brown representability for
from the known result for
is a ring with unit.
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