Ext-projectives in suspended subcategories

Abstract

Let H be a hereditary abelian k-category with tilting object and D b (H) denote the bounded derived category of H. This paper is devoted to a study of suspended subcategories of D b (H) by means of their Ext-projectives.The concept of a t-structure in a triangulated category T was introduced in the early eighties in [10]. It was meant as a technique for constructing various abelian subcategories of T ("hearts" of the t-structures) and is helpful for understanding the structure of T . Our motivation for their study comes from the representation theory of Artin algebras, which involves the derived category as an essential tool. There, t-structures are useful due to their relationship with tilting theory (see, for instance, [12,17]). In [17], Keller and Vossieck exhibited a bijection between t-structures in a given triangulated category T and contravariantly finite suspended subcategories of T , called aisles.Our objective in this paper is to study t-structures and aisles, from the point of view of the Ext-projectives. The concept of Ext-projective in a subcategory of a module category was introduced by Auslander and Smalø in their study of relative almost split sequences [7]. Since then, they were useful in various contexts (see [1,4,11]). In our situation, it follows from [17,11] that the Ext-projectives in an aisle U are the projectives in the heart of the t-structure determined by U. Moreover, if U is generated by the Ext-projectives, then the heart of the t-structure is a module category. This explains our interest in them.

From m-clusters to m-noncrossing partitions via exceptional sequences

“…By [AST,Cor. 3.2] (see [KV] in the Dynkin case) the smallest suspended subcategory U (M ) containing M is a torsion class.…”

“…The following connection with exceptional sequences is a special case of [AST,Theorem 2.3]. We include the sketch of a proof for convenience.…”

Bounded t-Structures on the Bounded Derived Category of Coherent Sheaves over a Weighted Projective Line

“…115771329) and also by the Fundamental Research Funds for the Central Unviersities. 2 One can argue as follows for this simple fact. By [8, Lemma 3.2.3], we have an injection Ext 2  ( , ) ↪ Ext 2  ( , ) for , ∈ .…”

Derived equivalences for hereditary Artin algebras

“…One attractive possibility is to take the right perpendicular category E ⊥ on an Ext-projective object E ∈ U. It is shown in [4] that the subcategory U ∩ E ⊥ is again an aisle in E ⊥ , but in general it will not be closed under the Serre functor in E ⊥ .…”

“…By Schur's Lemma, End S is a skew field and the embedding thick S → D b A has a left and a right adjoint as described in §2. 4.…”