From triangulated categories to module categories via localization 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 admits a calculus of left and right fractions. It follows that the Gabriel–Zisman localization 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 opposite of 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 (see [6]). In this case, the triangulated category is obtained from the abelian category of modules by Gabriel–Zisman (or Verdier) localization 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 (the opposites of) 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 localization 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 Gabriel and Zisman [4, Section I.2] (see also [13, Section 3]). If T is a cluster-tilting object then, by a result of Koenig–Zhu [12, Corollary 4.4], the additive quotient C/ΣT , where Σ denotes the suspension functor of C, is equivalent to modEndC(T ) (see also [8, Proposition 6.2] and [11, Section 5.1]; the case where C is 2-Calabi–Yau was proved in [11, Proposition 2.1], generalizing [3, Theorem 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 HomC(T,−), since, in the cluster-tilting case, XT = addΣT . However, one does not obtain the module category this way, since in general C/XT is not abelian. Received 29 March 2011; revised 20 September 2011; published online 12 March 2012. 2010 Mathematics Subject Classification 16D90, 18E05, 18E30, 18E35 (primary) 13F60, 16G10 (secondary). This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/G007497/1] and by the NFR [FRINAT grant number 196600]. C 2012 London Mathematical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. LOCALIZING TRIANGULATED CATEGORIES II 153 Our approach here is to show first that C/XT is preabelian, using some arguments generalizing those of Koenig–Zhu [12]. This means that, in addition to C/XT being an additive category, every morphism in C/XT has a kernel and a cokernel. This category, in general, possesses regular morphisms which are not isomorphisms (that is, morphisms which are both monomorphisms and epimorphisms but which do not have inverses), so it cannot be abelian in general. However, we show that C/XT does have a nice property. It is integral, that is, the pullback of any epimorphism (respectively, monomorphism), is again an epimorphism (respectively, monomorphism). This allows us to apply a result of Rump [17, p. 173], which implies that (C/XT )R, the localization of the category C/XT at the class R of regular morphisms, is abelian. We assume that C is skeletally small to ensure that the localization exists. Furthermore, by the same reference, the class R admits a calculus of left and right fractions. We go on to show that the projective objects in (C/XT )R are, up to isomorphism, exactly the objects induced by the objects in the additive subcategory of C generated by T . This implies our main result: Theorem. Let C be a skeletally small, Hom-finite, Krull–Schmidt triangulated category with Serre duality, containing a rigid object T . Let XT denote the class of objects X in C such that HomC(T,X) = 0. Let R denote the class of regular morphisms in C/XT . Then R admits a calculus of left and right fractions. Let (C/XT )R denote the localization of C/XT at R. Then (C/XT )R modEndC(T ). Let S denote the class of maps in C which are inverted by HomC(T,−), and let S denote the image of this class in C/XT . Then R = S, and the localization functor LS : C → CS factors through C/XT . The main result of Buan and Marsh [2] was the construction of an equivalence G from CS to modEndC(T ), such that HomC(T,−) = GLS . Our theorem above can be seen as a refinement of this. It was noted in [2] that S does not admit a calculus of left or right fractions, thus we can observe that the advantage of passing first to the quotient C/XT is that the subsequent localization does then admit such a calculus. We note that [15] contains results obtaining abelian categories as subquotients of triangulated categories; we give an explanation of the relationship between the results obtained here and those in [15] in Section 6. We also remark that A. Beligiannis has recently informed us that, in a subsequent work using a different approach, he has been able to generalize our main result to the case of a functorially finite rigid subcategory. There are interesting parallels between our approach here and the construction of the derived category of an abelian category A. We follow [10] and [5, III.2, III.4]. The derived category of A can be defined (following Grothendieck) as the Gabriel–Zisman localization of the category C(A) of complexes over A at the class of quasi-isomorphisms. This class does not, in general, admit a calculus of left and right fractions. However, the more commonly used construction (due to Verdier) of the derived category involves passing first to the homotopy category K(A). Then C(A) is a Frobenius category and K(A) is the corresponding stable category, hence a quotient of C(A). Then the class of quasi-isomorphisms in K(A) admits a calculus of left and right fractions. Localizing at this class gives rise to the derived category of A. In Section 1, we set-up the context in which we work. In Section 2, we recall the definitions of semi-abelian and integral categories and some results of Rump [17] (see also [18]) which will be useful. In Section 3, we prove that C/XT is integral. In Section 4, we recall the Gabriel–Zisman theory of localization and calculi of fractions and also how it can be applied (following [17, Section 1]) to the case of the regular morphisms in an integral category. In Section 5, we apply this to (C/XT )R to show that it is abelian. By classifying the projective objects in (C/XT )R, we deduce the main result. In Section 6, we explain the relationship of the results here to work 154 ASLAK BAKKE BUAN AND ROBERT J. MARSH of Nakaoka [15]. In Section 7, we explain the relationship between our main result and the results in [2]. 1. Notation We first set up the context in which we work and define some notation. Let k be a field and C be a skeletally small, triangulated, Hom-finite, Krull–Schmidt k-category with suspension functor Σ. We need the skeletally small assumption to ensure that the localizations we need exist. We assume that C has a Serre duality, that is, an autoequivalence ν : C → C such that HomC(X,Y ) DHomC(Y, νX) (natural in X and Y ) for all objects X and Y in C, where D denotes the duality Homk(−, k). Let T be a rigid object in C and set Γ = EndC(T ). For a full subcategory X of C, let X⊥ = {C ∈ C | Ext(X,C) = 0 for each X ∈ X}, and define ⊥X dually. For an object X in C, let addX denote its additive closure, and let X⊥ = (addX)⊥. A rigid object T is called cluster-tilting if addT = T⊥. Let XT = (ΣT )⊥. We also recall the triangulated version of Wakamatsu’s Lemma; see, for example, [8, Section 2]; see also [10, Lemma 2.1]. Lemma 1.1. Let X be an extension-closed subcategory of a triangulated category C. (a) Suppose that X → C is a minimal right X -approximation of C and Σ−1C → Y → X → C a completion to a triangle. Then Y is in X⊥, and the map Σ−1C → Y is a left X⊥approximation of Σ−1C. (b) Suppose that C → X is a minimal left X -approximation of C and Σ−1Z → C → X → Z → ΣC a completion to a triangle. Then Z is in ⊥X , and the map Z → ΣC is a right ⊥X approximation of ΣC. Using this, we obtain: Lemma 1.2. Let T be a rigid object in C. Then the subcategory XT of C is functorially finite. Proof. This follows from combining Wakamatsu’s Lemma (Lemma 1.1) with the existence of Serre duality. 2. Preabelian categories Recall that an additive category A is said to be preabelian if every morphism has a kernel and a cokernel. In this section, we shall recall some of the theory of preabelian categories that we need in order to study C/XT . A morphism is said to be regular (or a bimorphism) if it is both an epimorphism and a monomorphism. According to [17, Section 1] a preabelian category is called left semi-abelian (respectively, right semi-abelian) if every morphism f has a factorization of the form ip where p is a cokernel and i is a monomorphism (respectively, where p is an epimorphism and i is a kernel); see [17, Section 1], where it is pointed out that in the left semi-abelian case p is necessarily coim(f) = coker(ker(f)) and in the right semi-abelian case i is necessarily im(f) = ker(coker(f)). A preabelian category is said to be semi-abelian if it is both left and right semi-abelian. We remark that pullbacks and pushouts always exist in a preabelian category. For the pullback of maps c : B → D and d : C → D, we can take the kernel of the map B C → D LOCALIZING TRIANGULATED CATEGORIES II 155 whose components are c and −d, obtaining a pullback diagram: A a