The dual of Brown representability for homotopy categories of complexes

The dual of Brown representability for some derived categories

“…Note that the derived category of quasi-coherent sheaves over a nice enough scheme (including the projective space of finite dimension over a commutative ring with one) fulfills our hypotheses, hence its dual must satisfy Brown representability. This paper continues the work from [14], [12], [11], [13] and [3]. First, in [11] we generalize the idea of [17] by showing that Brown representability hold for triangulated categories with coproducts which are deconstructible in the sense of Lemma 2.1 below.…”

“…Next in [14] it was observed that if the homotopy category of complexes K(A) over an additive category A satisfies Brown representability, then every object in A must be a direct factor of an arbitrary direct product of a fixed object. To prove the converse, in the paper [12] it is dualized the approach in [11]. One of the main results in [12] says that a triangulated category T with products satisfies the dual of Brown representability, provided that there is a set of objects S, such that every object in T is S-cofiltered, that is it can be written as a homotopy limit of an inverse tower with the property that the mapping cone of all connecting morphisms are direct factors of direct products of objects in S. This result is applied in [13] to the homotopy category of projective modules over a ring.…”

“…To prove the converse, in the paper [12] it is dualized the approach in [11]. One of the main results in [12] says that a triangulated category T with products satisfies the dual of Brown representability, provided that there is a set of objects S, such that every object in T is S-cofiltered, that is it can be written as a homotopy limit of an inverse tower with the property that the mapping cone of all connecting morphisms are direct factors of direct products of objects in S. This result is applied in [13] to the homotopy category of projective modules over a ring. Here we show that, under suitable hypotheses on an abelian category A, every complex has a homotopically injective resolution, therefore the derived category of A is equivalent to the category of these homotopically injective complexes.…”

The dual of the homotopy category of projective modules satisfies Brown representability

“…The first ingredient in the proof of the main theorem of this paper is contained in [6]. Here, we recall it briefly.…”

Constructing cogenerators in triangulated categories and Brown representability

“…This paper belongs to a suite of works concerned with this subject. In [7] is given a criterion for Brown representability for contravariant functors which is dualized in [8], where it is shown that if T o is deconstractible then T o satisfies Brown representability (see Proposition 1.1 here). The paper [9] applies this result for derived categories of Grothendieck categories which are AB4-n, in the sense that the n + 1-th derived functor of the direct product functor is exact.…”