Related papers: Calculating dimension of triangulated categories: …
We define a dimension for a triangulated category. We prove a representabilityTheorem for a certain class of functors on finite dimensional triangulatedcategories. We study the dimension of the boundedderived category of an algebra or a…
In this paper we study the representation dimension as well as the derived dimension of the path algebra of an artin algebra over a finite and acyclic quiver.
Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for…
In this paper, we compute triangular spectrum (as defined by P. Balmer) of two classes of tensor triangulated categories which are quite common in algebraic geometry. One of them is the derived category of $G$-equivariant sheaves on a…
The global dimension of a triangulated category is defined to be the infimum value of the global dimensions of stability conditions on the triangulated category. In this paper, we study the global dimension of the derived category of an…
In this paper, we explore when a locally finite triangulated category has dimension zero or finite representation type. We also study generation of derived categories by orthogonal subcategories.
We introduce the notion of composition series of triangulated categories, which generalizes full exceptional sequences. The lengths of composition series yield invariants for triangulated categories. We study composition series of derived…
In this note we discuss three notions of dimension for triangulated categories: Rouquier dimension, diagonal dimension and Serre dimension. We prove some basic properties of these dimensions, compare them and discuss open problems.
These notes are meant to provide a rapid introduction to triangulated categories. We start with the definition of an additive category and end with a glimps of tilting theory. Some exercises are included.
In our previous papers we introduced categorical invariants, which are, roughly speaking, sets of triangulated subcategories in a given triangulated category and their quotients. Here is extended the list of examples, where these sets are…
This paper introduces the concept of the dimension of a triangulated category with respect to a fixed full subcategory. For the bounded derived category of an abelian category, upper bounds of the dimension with respect to a contravariantly…
Let A be a path A-infinity-algebra over a positively graded quiver Q. It is proved that the derived category of A is triangulated equivalent to the derived category of kQ, which is viewed as a dg algebra with trivial differential. The main…
In this paper, we study metric completions of triangulated categories in a representation-theoretic context. We provide a concrete description of completions of bounded derived categories of hereditary finite dimensional algebras of finite…
The aim of this note is to answer several open problems arising from the geometric description of the $m$-cluster categories of type $A_n$ and their realization in terms of the $m$-th power of a translation quiver. In particular, we give a…
In this paper we prove that the dimension of the bounded derived category of coherent sheaves on a smooth quasi-projective curve is equal to one. We also discuss dimension spectrums of these categories.
We study tensor network varieties associated with the triangular graph, with a focus on the case where one of the physical dimensions is 2. This allows us to interpret the tensors as pencils of matrices. We provide a complete…
The representation dimension of an artin algebra as introduced by M.Auslander in his Queen Mary Notes is the minimal possible global dimension of the endomorphism ring of a generator-cogenerator. The paper is based on two texts written in…
We compute several types of dimension for the bounded derived categories of coherent sheaves of orbifold curves. This completes the calculation of these dimensions for derived categories of noncommutative curves in the sense of Reiten-van…
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…
We extend Ng's characterisation of torsion pairs in the 2-Calabi-Yau triangulated category generated by a 2-spherical object to the characterisation of torsion pairs in the w-Calabi-Yau triangulated category, $T_w$, generated by a…