We give a generalized version of the Freyd conjecture and a way to think about a possible proof. The essential point is to describe an elementary formal reduction of the question that holds in any triangulated category. There are no new results, but at least one known example drops out very easily. In algebraic topology, the generating hypothesis, or Freyd conjecture, is a long-standing conject...

A recollement of triangulated categories describes one such category as being “glued together” from two others. This paper gives a precise criterion for the existence of a recollement of the derived category of a Differential Graded Algebra in terms of two other such categories.

We give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products (with parameter t = 0) and the derived category of coherent sheaves on a crepant resolution of the spectrum of the centre of these algebras.

We show that K1(E) of an exact category E agrees with K1(DE) of the associated triangulated derivator DE under the hypothesis of the GilletWaldhausen theorem. More generally we show that K1(C(E)) of the category of bounded complexes in E always coincides with K1(DE).

Let M be a non–compact, connected manifold of dimension ≥ 1. Let D(sheaves/M) be the unbounded derived category of chain complexes of sheaves of abelian groups on M . We prove that D(sheaves/M) is not a compactly generated triangulated category, but is well generated. 2000 Mathematics Subject Classification: 18E30 55P99 55U35

We present a novel notion of stable objects in a triangulated category. This Postnikov-stability is preserved by equivalences. We show that for the derived category of a projective variety this notion includes the case of semistable sheaves. As one application we compactify a moduli space of stable bundles using genuine complexes via Fourier-Mukai transforms. MSC 2000: 14F05, 14J60, 14D20

For every stable model category M with a certain extra structure, we produce an associated model structure on the pro-category pro-M and a spectral sequence, analogous to the Atiyah-Hirzebruch spectral sequence, with reasonably good convergence properties for computing in the homotopy category of pro-M. Our motivating example is the category of pro-spectra. The extra structure referred to above...

We develop a suitable version of the stable module category of a finite group G over an arbitrary commutative ring k. The purpose of the construction is to produce a compactly generated triangulated category whose compact objects are the finitely presented kG-modules. The main idea is to form a localisation of the usual version of the stable module category with respect to the filtered colimits...

For every stable model category M with a certain extra structure, we produce an associated model structure on the pro-category pro-M and a spectral sequence, analogous to the Atiyah-Hirzebruch spectral sequence, with reasonably good convergence properties for computing in the homotopy category of pro-M. Our motivating example is the category of pro-spectra. The extra structure referred to above...

We develop an Auslander-Reiten theory for triangulated categories which is based on Brown’s representability theorem. In a fundamental article [3], Auslander and Reiten introduced almost split sequences for the category of finitely generated modules over an artin algebra. These are short exact sequences which look almost like split exact sequences, but many authors prefer to call them Auslander...