Related papers: $R$-linear triangulated categories and stability c…
Categorical resolutions of singularities are a replacement of resolution of singularities within the realm of triangulated categories. They allow the study of the derived category of a singular variety $X$ via a triangulated category that…
In this paper we give an example of a triangulated category, linear over a field of characteristic zero, which does not carry a DG-enhancement. The only previous examples of triangulated categories without a model have been constructed by…
Given a triangulated category $\mathcal{C}$, we construct a partial compactification, denoted $\mathcal{A}\mathrm{Stab}(\mathcal{C})$, of the quotient of its stability manifold by $\mathbb{C}$. The purpose of…
We give an elementary introduction to the theory of triangulated categories covering their axioms, homological algebra in triangulated categories, triangulated subcategories, and Verdier localization. We try to use a minimal set of axioms…
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].
Recently, Anno, Bezrukavnikov and Mirkovic have introduced the notion of a "real variation of stability conditions" (which is related to Bridgeland's stability conditions), and construct an example using categories of coherent sheaves on…
Two left noetherian rings $R$ and $S$ are said to be {\it singularly equivalent} if their singularity categories are equivalent as triangulated categories. The aim of this paper is to give a necessary condition for two commutative…
We propose a general method to construct new triangulated categories, relative stable categories, as additive quotients of a given one. This construction enhances results of Beligiannis, particularly in the tensor-triangular setting. We…
We provide examples of an explicit submanifold in Bridgeland stabilities space of a local Calabi-Yau, and propose a new variant of definition of stabilities on a triangulated category, which we call a "real variation of stability…
Consider a 2-Calabi--Yau triangulated category with a Bridgeland stability condition. We devise an effective procedure to reduce the phase spread of an object by applying spherical twists. Using this, we give new proofs of the following…
In this paper we give necessary and sufficient conditions for a functor to be representable in a strongly generated triangulated category which has a linear action by a graded ring, and we discuss some applications and examples.
The orthogonal groups are a series of simple Lie groups associated to symmetric bilinear forms. There is no analogous series associated to symmetric trilinear forms. We introduce an infinite dimensional group-like object that can be viewed…
These notes attempt to give a short survey of the approach to support theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the material necessary to…
In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider the subcategory of extensions X*Y. We give conditions for X*Y to be triangulated and use them to provide tools for constructing stable…
In this paper we relate triangulated category structures to the cohomology of small categories and define initial obstructions to the existence of an algebraic or topological enhancement. We show that these obstructions do not vanish in an…
We give a characterisation of those local not necessary commutative rings, for which the category of projective modules admits a triangulation with the identity as translation functor. By "admits a triangulation" we mean that the category…
We study stability conditions induced by functors between triangulated categories. Given a finite group acting on a smooth projective variety we prove that the subset of invariant stability conditions embeds as a closed submanifold into the…
We develop a unified approach for identifying spaces of stability conditions of triangulated categories arising from weighted marked surfaces with moduli spaces of quadratic differentials. This approach is based on the notion of a perverse…
We prove that any non-commutative smooth projective variety with a Bridgeland stability condition of dimension less than $\frac{6}{5}$ must be a smooth projective curve. As a consequence, we deduce the non-existence of such categories with…
The main goal of this paper is to prove the following: for a triangulated category $ \underline{C}$ and $E\subset \operatorname{Obj} \underline{C}$ there exists a cohomological functor $F$ (with values in some abelian category) such that…