CLASSIFICATION OF SUBCATEGORIES IN ABELIAN CATEGORIES AND TRIANGULATED CATEGORIES ATHESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES AND RESEARCH IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS UNIVERSITY OF REGINA By Yong Liu Regina, Saskatchewan September 2016 c Copyright 2016: Yong Liu UNIVERSITY OF REGINA FACULTY OF GRADUATE STUDIES AND RESEARCH SUPERVISORY AND EXAMINING COMMITTEE Yong Liu, candidate for the degree of Doctor of Philosophy in Mathematics, has presented a thesis titled, Classification of Subcategories in Abelian Categories and Triangulated Categories, in an oral examination held on September 8, 2016. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. External Examiner: Dr. Henning Krause, University of Bielefeld Supervisor: Dr. Donald Stanley, Department of Mathematics and Statistics Committee Member: Dr. Allen Herman, Department of Mathematics and Statistics Committee Member: *Dr. Fernando Szechtman, Department of Mathematics and Statistics Committee Member: Dr. Yiyu Yao, Department of Computer Science Chair of Defense: Dr. Renata Raina-Fulton, Department of Chemistry and Biochemistry *Not present at defense Abstract Two approaches for classifying subcategories of a category are given. We examine the class of Serre subcategories in an abelian category as our first target, using the concepts of monoform objects and the associated atom spectrum [13]. Then we generalize this idea to give a classification of nullity classes in an abelian category, using premonoform objects instead to form a new spectrum so that there is a bijection between the collection of nullity classes and that of closed and extension closed subsets of the spectrum. Additionally, we impose a natural sheaf structure induced by the center of a category on the atom spectrum, over which the sheaves of modules over the structure sheaf are also discussed. The second approach is enlightened by the lattice structure implicitly shown in the s- tatements of classification of the subcategories in an abelian category. We introduce a new concept of classifying space of subcategories, those subcategories satifying finitely many closure operations, in an either abelian or triangulated category. We show that a class of subcategories is classified by a topological space if these subcategories are primely gener- ated. Many well-known results fit into our framework, such as Neeman’s classification [19] of localizing subcategories of the derived category D(R) of a commutative Noetherian ring R, etc. i Acknowledgements I would like to thank my supervisor Don for leading me into the field of classification of various subcategories, and for his support and patience as well. During my PhD program, Adam, Ivo and Soumen helped me a lot for problem discussion etc. I also appreciate Allen, Fernando and Yiyu for useful comments on my thesis. ii Contents Abstract i Acknowledgements ii Table of Contents iii 1 Introduction 1 2 Abelian category and its localization 8 2.1 Abelian category and its Serre subcategories . 9 2.2 The calculus of fractions . 13 2.3 Quotient category . 18 2.4 Adjoint functor . 21 2.5 Kan extension . 25 2.6 Localization functor . 32 3 Monoform and premonoform objects 42 3.1 Monoform objects . 43 iii 3.2 Premonoform objects . 47 3.2.1 Definitions and properties . 47 3.2.2 A premonoform but not monoform object . 50 3.3 Equivalence relation, support and topology . 53 3.4 Classification of nullity classes . 55 4 A structure sheaf on the atom spectrum 65 4.1 Atom spectrum and a structure sheaf R+ ................... 66 4.2 Sheaf of modules . 75 4.2.1 A module structure over R+ ..................... 75 4.2.2 Examples from quiver representations . 78 5 Classifying space of subcategories and its application 86 5.1 Classifying space . 87 5.1.1 Complete distributive lattices . 87 5.1.2 Points and topology . 91 5.1.3 Examples of classifying spaces . 95 5.2 Generally prime objects and classification . 97 5.3 Irreducible vs. prime, and functoriality . 102 5.4 Primely generated subcategories . 106 5.4.1 Serre subcategories in a noetherian abelian category . 106 5.4.2 Thick subcategories in Dperf (R) . 107 5.4.3 Localizing subcategories in D(R) . 108 iv 5.4.4 Localizing subcategories in a stable homotopy category . 111 5.4.5 An example . 113 Bibliography 115 v Chapter 1 Introduction We are motivated by the problem of classifying t-structures (see [4]) in a derived cat- egory Dperf (R) of perfect complexes over a ring R. It is known that each t-structure cor- responds bijectively to an aisle, a full subcategory closed under positive suspensions and extensions such that the inclusion functor admits a right adjoint. Stanley’s idea [26] of classifying these aisles is to find an invariant given by perversity functions on the prime spectrum SpecR. However, it is efficient not to deal with aisles directly but a generalized notion of aisles called nullity classes, also known as torsion theories when in an abelian category. As a first step, I attempted to do and obtained a c Generalization of Stanley’s method into the case when the derived category is Dfg(R): c Here Dfg(R) is a full triangulated subcategory of D(R) consisting of those objects repre- sented by cofibrant objects with each degree finitely generated, within which the nullity classes have invariants given by perversity functions as well, see [17] for detail. In a general setting of a category and its subcategories, we focus on establishing a theory 1 of invariants to distinguish those subcategories of the same type, say those subcategories closed under finite coproducts, nullity classes (Definition 3.1.7) and Serre subcategories (Definition 2.1.3) etc. This is successfully fulfilled historically in many different ways, for instance A. Neeman’s classification (see [19] and [21]) of (co)localizing subcategories of D(R) in terms of subsets of the prime spectrum SpecR, P. Balmer’s classification [3] of radical thick tensor ideals of a tensor triangulated category T via closed subsets of the Hochster dual of spectrum of T , Benson’s classification [5] of localizing tensor ideals in the homotopy category K(Inj kG) by subsets of SpecH∗(G; k) and so on. Inspired by them and also R. Kanda’s paper [13] on the atom spectrum of an abelian category, we approach our problem of classifying nullity classes (or torsion classes) by looking into a new spectrum SpecA of the abelian category A, a topological space with equivalence classes of premonoform objects, shown as Corollary 3.4.12, so that Theorem 1 (Result A). There is an order preserving bijection fNullity classes in Ag !∼ fClosed and extension closed subsets of SpecAg: Here the premonoform object is a slightly weaker notion of monoform object (see [28] e.g.), the latter is the same as strongly uniform module when the abelian category A is the category of modules over a ring. As a side issue and difficult one, it is always interesting to find the corresponding elementary and optimal objects to build subcategories of a certain type, see [27] for example. There is a common idea to develop the theory even further by considering the (atom) spectrum of an abelian category A together with the structure sheaf R induced by the 2 centers of its quotients by subcategories as a ringed space, hence the study of the sheaf of modules naturally and so on. We found a similar idea later in A. Rosenberg’s papers [24] and [23] when he was attempting to reconstruct a scheme X from the spectrum of its derived category D(X) of coherent sheaves. See also Rouquier’s paper [25] on the example of coherent sheaves on separated schemes. First we start from the simple example when the category A is abelian with atom spectrum ASpecA, which consists of atom equivalence classes of monoform objects (Definition 3.1.1). With certain set theoretical assumptions, in fact, the Freyd-Mitchell Embedding Theorem always allows us to obtain a concretization of an abstract abelian category, though not the best choice. We show as Proposition 4.1.16 and Proposition 4.2.5 that Theorem 2 (Result B). Let A be an abelian category and ASpecA its atom spectrum. Then (1) there is a natural structure sheaf R of rings on ASpecA induced by the centers of A and its localization (or quotient, see [7]) with respect to Serre subcategories S, making ASpecA a locally ringed space. (2) there is an associated sheaf of modules over R, i.e. the existence of a faithfully exact functor ∼: A!R-mod, provided the concretization A! Ab is exact. As a special case, when A is the category of R-modules with R a commutative Noetherian ring with identity, the construction in (1) coincides with the one given by the affine scheme (SpecR; OSpecR). Notice that such a module structure depends on the choice of concretiza- tion F : A! Ab, and the restriction map of its sheaf of modules relies on the existence of a localization functor LS : A!A. There is a more general approach to develop this 3 theory which we consider as a new project by studying the category of coherent sheaves. Apart from Result A, we have another method to classify subcategories (of certain type) by first collecting the target subcategories and ordering them via inclusions thus giving a poset Φ. In fact, Φ can be equipped with a complete distributive lattice structure naturally. We then define the classifying space K(Φ) of subcategories of the given type to be those subcategories which cannot be generated by their proper subcategories of the same type, i.e.