We consider a homology theory h : T → A on a triangulated category T with values in a graded abelian category A . If the functor h reflects isomorphisms, is full and is such that for any object x in A there is an object X in T with an isomorphism between h(X) and x, we prove that A is a hereditary abelian category, all idempotents in T split and the kernel of h is a square zero ideal which as a...

Given a good homology theory E and a topological space X, E∗X is not just an E∗-module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a well-behaved Hopf algebroid (A,Γ). That is, we construct the derived category Stable(Γ) of (A,Γ) as the homotopy category of a Quillen model structure on Ch(Γ), the category...

We define and study a gluing procedure for Bridgeland stability conditions in the situation when a triangulated category has a semiorthogonal decomposition. As an application, we construct stability conditions on the derived categories of Z2-equivariant sheaves associated with ramified double coverings of P. Also, we study the stability space for the derived category of Z2-equivariant coherent ...

By using only combinatorial data on two posets X and Y , we construct a set of so-called formulas. A formula produces simultaneously, for any abelian category A, a functor between the categories of complexes of diagrams over X and Y with values in A. This functor induces a triangulated functor between the corresponding derived categories. This allows us to prove, for pairs X, Y of posets sharin...

We construct a natural continuous map from the triangular spectrum of a tensor triangulated category to the algebraic Zariski spectrum of the endomorphism ring of its unit object. We also consider graded and twisted versions of this construction. We prove that these maps are quite often surjective but far from injective in general. For instance, the stable homotopy category of finite spectra ha...

In this paper we introduce the notion of a ‘generalised’ co-slicing of a triangulated category. This generalises the theory of co-stability conditions in a manner analogous to the way in which Gorodentsev, Kuleshov and Rudakov’s tstabilities generalise Bridgeland’s theory of stability conditions. As an application of this notion, we use a complete classification of ‘generalised’ co-slicings in ...

First, we show that a compact object C in a triangulated category, which satisfies suitable conditions, induces a t-structure. Second, in an abelian category we show that a complex P of small projective objects of term length two, which satisfies suitable conditions, induces a torsion theory. In the case of module categories, using a torsion theory, we give equivalent conditions for P to be a t...