If (A,B) and (A′,B′) are co-t-structures of a triangulated category, then (A′,B′) is called intermediate if A ⊆ A′ ⊆ ΣA. Our main results show that intermediate co-t-structures are in bijection with two-term silting subcategories, and also with support τ -tilting subcategories under some assumptions. We also show that support τ -tilting subcategories are in bijection with certain finitely gener...

It is becoming increasingly difficult for geometers and even physicists to avoid papers containing phrases like “triangulated category”, not to mention derived functors. I will give some motivation for such things from algebraic geometry, and show how the concepts are already familiar from topology. This gives a natural and simple way to look at cohomology and other scary concepts in homologica...

We consider the Berglund-Hübsch transpose of a bimodal invertible polynomial and construct a triangulated category associated to the compactification of a suitable deformation of the singularity. This is done in such a way that the corresponding Grothendieck group with the (negative) Euler form can be described by a graph which corresponds to the Coxeter-Dynkin diagram with respect to a disting...

This category was introduced in [1] and has been studied by Assem, Brüstle, Schiffler and Todorov [2], the authors [3], Thomas [4], Wralsen [5] and Zhu [6]. In particular, Thomas and Zhu have shown that it gives rise to the combinatorics of the generalized cluster complexes of Fomin and Reading [7] (defined by Tzanaki [8] for types A and B). It is known that C∆ is triangulated [1], Krull-Schmid...

It is becoming increasingly difficult for geometers and even physicists to avoid papers containing phrases like “triangulated category”, not to mention derived functors. I will give some motivation for such things from algebraic geometry, and show how the concepts are already familiar from topology. This gives a natural and simple way to look at cohomology and other scary concepts in homologica...

Given a smooth projective varietyX , we denote byD(X) the bounded derived category of coherent sheaves D(Coh(X)). All varieties we consider below are over the complex numbers. A result of Rouquier, [Ro] Théoréme 4.18, asserts that if X and Y are smooth projective varieties with D(X) ' D(Y ) (as linear triangulated categories), then there is an isomorphism of algebraic groups Aut(X)× Pic(X) ' Au...

We recall the construction, following the method of Morel and Voevodsky, of the triangulated category of étale motivic sheaves over a base scheme. We go through the formalism of Grothendieck’s six operations for these categories. We mention the relative rigidity theorem. We discuss some of the tools developed by Voevodsky to analyze motives over a base field. Finally, we discuss some long-stand...

We provide a categorical interpretation of a well-known identity from linear algebra as an isomorphism of certain functors between triangulated categories arising from finite dimensional algebras. As a consequence, we deduce that the Serre functor of a finite dimensional triangular algebra A has always a lift, up to shift, to a product of suitably defined reflection functors in the category of ...

Following Krause [Kra99], we prove Krull-Schmidt theorems for thick subcategories of various triangulated categories: derived categories of rings, noetherian stable homotopy categories, stable module categories over Hopf algebras, and the stable homotopy category of spectra. In all these categories, it is shown that the thick ideals of small objects decompose uniquely into indecomposable thick ...

Abstract We continue the work started in parts (I) and (II) of this series. In paper, we classify which continuous quivers type A are derived equivalent. Next, define new ${\mathcal {C}(A_{{\mathbb {R}},S})}$ , call weak cluster category. It is a triangulated category, it does not have structure but has weaker notion “cluster theory.” show that original category [15] localization theories to be...