26. CJM 2000 (vol 52 pp. 225)
 Alonso Tarrío, Leovigildo; Jeremías López, Ana; Souto Salorio, María José

Localization in Categories of Complexes and Unbounded Resolutions
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 Kinjective 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.
Categories:18E30, 18E15, 18E35 

27. CJM 1999 (vol 51 pp. 294)
 Enochs, Edgar E.; Herzog, Ivo

A Homotopy of Quiver Morphisms with Applications to Representations
It is shown that a morphism of quivers having a certain path
lifting property has a decomposition that mimics the decomposition
of maps of topological spaces into homotopy equivalences composed
with fibrations. Such a decomposition enables one to describe the
right adjoint of the restriction of the representation functor
along a morphism of quivers having this path lifting property.
These right adjoint functors are used to construct injective
representations of quivers. As an application, the injective
representations of the cyclic quivers are classified when the base
ring is left noetherian. In particular, the indecomposable
injective representations are described in terms of the injective
indecomposable $R$modules and the injective indecomposable
$R[x,x^{1}]$modules.
Categories:18A40, 16599 

28. CJM 1999 (vol 51 pp. 3)
 Allday, C.; Puppe, V.

On a Conjecture of Goresky, Kottwitz and MacPherson
We settle a conjecture of Goresky, Kottwitz and MacPherson related
to Koszul duality, \ie, to the correspondence between differential
graded modules over the exterior algebra and those over the
symmetric algebra.
Keywords:Koszul duality, HirschBrown model Categories:13D25, 18E30, 18G35, 55U15 

29. CJM 1998 (vol 50 pp. 1048)
 Goerss, P. G.; Jardine, J. F.

Localization theories for simplicial presheaves
Most extant localization theories for spaces, spectra and diagrams
of such can be derived from a simple list of axioms which are verified
in broad generality. Several new theories are introduced, including
localizations for simplicial presheaves and presheaves of spectra at
homology theories represented by presheaves of spectra, and a theory
of localization along a geometric topos morphism. The
$f$localization concept has an analog for simplicial presheaves, and
specializes to the $\hbox{\Bbbvii A}^1$local theory of
MorelVoevodsky. This theory answers a question of Soul\'e concerning
integral homology localizations for diagrams of spaces.
Categories:55P60, 19E08, 18F20 
