Localization in Categories of Complexes and Unbounded Resolutions

Abstract

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

Structural theorem for gr-injective modules over gr-noetherian G-graded commutative rings and local cohomology functors

“…Since the gr-injective hull E gr (M ) of M is a finite direct sum of indecomposable gr-injective modules, we have already shown that F (E gr (M )) = Γ V F (E gr (M )). Thus, using Lemma 1.1, we have [22,Theorem 5.4] or [11,Proposition B.2], every complex of graded R-module has a K-injective resolution. For any gr-injective complex J • ∈ K(I), RΓ V (J • ) = Γ V (J • ) is the subcomplex of J • consisting of gr-injective modules supported in V .…”

The bar derived category of a curved dg algebra

“…we denote by SV or V [1] the graded k-module with (SV) p = V p+1 for all p ∈ Z. We call SV the suspension or the shift of V. The shift extends to an automorphism of the category of graded k-modules, with inverse denoted by S −1 .…”

Right triangulated categories: As extriangulated categories, aisles and co-aisles

“…For the case of t-structures, there are related works [3,34,35] that give various characterisations of aisles. We note that our approach differs in the sense that we look to give characterisations intrinsic to the right triangulated category, that is, the aisle, rather than properties of the aisle related to the ambient triangulated category.…”