Related papers: Generation time for biexact functors and Koszul ob…
A triangulated category $\mathcal{T}$ whose suspension functor $\Sigma$ satisfies $\Sigma^m \simeq \mathrm{Id}_{\mathcal{T}}$ as additive functors is called an $m$-periodic triangulated category. Such a category does not have a tilting…
This paper deals with computing topological invariants such as connected components, boundary surface genus, and homology groups. For each input data set, we have designed or implemented algorithms to calculate connected components,…
This work establishes a condition that determines when strong generation in the bounded derived category of a Noetherian $J\textrm{-}2$ scheme is preserved by the derived pushforward of a proper morphism. Consequently, we can produce upper…
We call a triangulated category \emph{hereditary} provided that it is equivalent to the bounded derived category of a hereditary abelian category, where the equivalence is required to commute with the translation functors. If the…
The main result of this paper is that there is an additive equivalence between $\overline{\mathcal{C}}_n$, the Paquette-Yildirim completion of the discrete cluster categories of Dynkin type $A_{\infty}$, and the perfect derived category of…
We consider the finite generation property for cohomology algebra of pointed finite tensor categories via de-equivariantization and exact sequence of finite tensor categories. As a result, we prove that all coradically graded pointed finite…
Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there…
Given a certain triangulation of a punctured surface with boundary, we construct a new triangulated surface without punctures which covers it. This new surface is naturally equipped with an action of a group of order two, and its quotient…
We analyze the BGG Category $\mathcal{O}$ over a large class of generalized Weyl algebras (henceforth termed GWAs). Given such a "triangular" GWA for which Category $\mathcal{O}$ decomposes into a direct sum of subcategories, we study in…
We explain why every non-trivial exact tensor functor on the triangulated category of mixed motives over a field F has zero kernel, if one assumes "all" motivic conjectures. In other words, every non-zero motive generates the whole category…
It is suggested that generations are linked to the need of calculating curvature of space via a deformed or discrete calculus. Quantization would limit the deformation, building three generations, and not four, as other interpretation could…
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 paper we investigate homologically finite-dimensional objects in the derived category of a given small dg-enhanced triangulated category. Using these we define reflexivity, hfd-closedness, and the Gorenstein property for…
For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture…
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of…
In this paper we study triangular matrix categories using the theory of recollements of abelian categories. Given a triangular matrix category we construct two canonical recollements. We show that if certain funtors of these recollements…
This paper systematically develops a notion of regular sequences in the context of $R$-linear triangulated categories for a graded-commutative ring $R$. The notion has equivalent characterizations involving Koszul objects and local…
The mapping class group of an orientable surface, which records its symmetries up to isotopy, plays a central role in low-dimensional topology. This chapter explores the foundational problem of determining minimal generating sets for these…
In this paper we characterize when a recollement of compactly generated triangulated categories admits a ladder of some height going either upwards or downwards. As an application, we show that the derived category of the preprojective…
In representation theory of algebras, there exist two types of mutation pairs: rigid type (cluster-tilting mutations by Iyama-Yoshino) and simple-minded type (mutations of simple-minded systems by Sim\~oes-Pauksztello). It is known that…