No abstract

We study equivalences induced by a complex P, consisting of projectives and concentrated in degrees −1 and 0, which is silting in the derived category D(R) of a ring R.

Abstract. We say that a projective class in a triangulated category with coproducts is perfect if the corresponding ideal is closed under coproducts of maps. We study perfect projective classes and the associated phantom and cellular towers. Given a perfect generating projective class, we show that every object is isomorphic to the homotopy colimit of a cellular tower associated to that object. Using this result and the Neeman's Freyd-style representability theorem we give a new proof of Brown Representability Theorem.

We show that for the homotopy category K(Ab) of complexes of abelian groups, both Brown representability and Brown representability for the dual fail. We also provide an example of a localizing subcategory of K(Ab) for which the inclusion into K(Ab) does not have a right adjoint.Comment: 9 pages; version 2: minor changes, references added and updated, Remark 11 on the existence of product generators adde

We prove that a triangulated category which is the underlying category of a stable derivator has a filtered enhancement, providing an affirmative answer to a conjecture in [3].