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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 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.

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, Hirsch-Brown 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 Morel-Voevodsky. This theory answers a question of Soul\'e concerning integral homology localizations for diagrams of spaces.

Categories:55P60, 19E08, 18F20
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