From triangulated categories to module categories via localisation II: calculus of fractions

We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor HomC(T, −), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equivalent to the category of finite dimensional modules over the endomorphism algebra of T in C. Introduction Let k be a field and C a skeletally small, triangulated Hom-finite k-category which is Krull-Schmidt and has Serre duality. A standard example of such a category is the bounded derived category of finite dimensional modules over a finite dimensional algebra of finite global dimension. In this case, the triangulated category is obtained from the abelian category of modules by Gabriel-Zisman (or Verdier) localisation of the quasi-isomorphisms in the bounded homotopy category of complexes of modules. Here, our approach is the other way around. Given a triangulated category C as above, we are interested in gaining information about related abelian categories. We are particularly interested in the module categories over endomorphism algebras of objects in C. An object T in C satisfying Ext(T, T ) = 0 is known as a rigid object. In this case it is known [2] that the category of finite dimensional modules over End(T ) can be obtained as a Gabriel-Zisman localisation of C, formally inverting the class S of maps which are inverted by the functor HomC(T, −). However, the class S does not admit a calculus of left or right fractions in the sense of [4, Sect. I.2] (see also [11, Sect. 3]). If T is a cluster-tilting object then, by a result of Koenig-Zhu [10, Cor. 4.4], the additive quotient C/ΣT , where Σ denotes the suspension functor of C, is equivalent to modEndC(T ) op (see also [7, Prop. 6.2] and [9, Sect. 5.1]; the case where C is 2-Calabi-Yau was proved in [9, Prop. 2.1], generalising [3, Thm. 2.2]). However, when T is rigid, this is no longer the case in general. It is natural to consider instead the quotient C/XT where XT is the class of objects in C sent to zero by the functor Date: 28 March 2011.