DOCSLIB.ORG

Quotient Categories and Grothendieck's Splitting

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quotient Categories and Grothendieck's Splitting QUOTIENT CATEGORIES AND GROTHENDIECK'S SPLITTING THEOREM Zachary Murphy Supervisor: Associate Professor Daniel Chan School of Mathematics and Statistics UNSW Sydney October 2017 Submitted in partial fulfillment of the requirements of the degree of Bachelor of Science (Advanced Mathematics) with Honours Plagiarism statement I declare that this thesis is my own work, except where acknowledged, and has not been submitted for academic credit elsewhere. I acknowledge that the assessor of this thesis may, for the purpose of assessing it: • Reproduce it and provide a copy to another member of the University; and/or, • Communicate a copy of it to a plagiarism checking service (which may then retain a copy of it on its database for the purpose of future plagiarism check- ing). I certify that I have read and understood the University Rules in respect of Student Academic Misconduct, and am aware of any potential plagiarism penalties which may apply. By signing this declaration I am agreeing to the statements and conditions above. Signed: Date: i Acknowledgements When I finished high school and decided to study pure mathematics at university, I thought I knew a lot of maths. Over the four years of my degree, I have gradually come to the realisation that I know very little, and nothing has made that more apparent than researching this project under my supervisor, Associate Professor Daniel Chan. I have learnt so much maths from him this year that I never knew existed, and for that he has my most sincere and profound thanks. His unwavering support, patience and enthusiasm made this honours year a joy. I would also like to thank my high school maths teacher Bruce Smith, who was the first person to pique my interest in mathematics and without whom I would never have considered studying it at the tertiary level. I would also like to thank my friends for their friendship, support and banter throughout this trying year. Finally, sincere thanks goes to my family, in particular my parents, who have put up with me for the longest. Without their support, encouragement and proof-reading, I would never have made it this far. Zachary Murphy, 27 October 2017. ii Abstract At first glance, quotient categories might seem like a strange notion to define, but in actuality they turn out to be quite useful constructions. They arise naturally in geometry as a way of studying spaces which may not be distinguishable using more elementary algebraic methods. Specifically, quotient categories provide a way to construct the category of quasi-coherent sheaves on a space. This category encodes much of the geometry of the case, and in particular contains as a subcategory the category of vector bundles, which have been important objects of study in algebraic geometry since the start of the 20th century. In 1957, Grothendieck proved an important structure theorem for vector bundles on the projective line. This thesis is intended firstly as an expository work on the theory of quotient categories and how they interact with geometry, and secondly we offer an alternative approach to Grothendieck's Splitting Theorem. iii Contents Chapter 1 Introduction 1 1.1 Preface . .1 1.2 Introduction . .1 1.3 Structure of Thesis . .4 1.4 Assumed Knowledge . .5 Chapter 2 Abelian Categories 6 2.1 Additive Categories . .6 2.2 Abelian Categories . .9 Chapter 3 The Quotient of an Abelian Category 16 3.1 Localisation . 16 3.2 Quotient Categories . 20 3.3 Localising Subcategories . 28 Chapter 4 Homological Algebra 31 4.1 Chain Complexes and Homology . 31 4.2 The Hom and Tensor Functors . 32 4.3 Ext and Tor Functors . 34 4.4 Hilbert's Syzygies Theorem . 40 Chapter 5 Grothendieck's Splitting Theorem 43 5.1 The Category of Graded k[x; y]-modules . 43 5.2 Torsion in Graded Modules . 45 5.3 The Splitting Theorem . 48 Chapter 6 Applications to Geometry 56 6.1 The Projective Line . 56 6.2 Quasi-Coherent Sheaves . 57 6.3 Vector Bundles . 63 v Chapter 7 Conclusion 67 7.1 Summary . 67 7.2 Generalisations and Further Work . 67 References 69 vi Chapter 1 Introduction 1.1 Preface Category theory is often referred to (endearingly or otherwise) as \abstract non- sense", and the theory of quotient categories is unsurprisingly, no exception. How- ever, the main goal of this thesis is to motivate them and to give the reader a sense of how these are actually natural and useful constructions. Category theory is an essential tool in studying geometry, so while this is first and foremost an exposition of the theory of quotient categories, the motivation comes from the geometry of both affine and projective space, and as such some parts of this thesis have a rather geometric flavour. This thesis attempts to motivate the theory of quotient categories by presenting a proof of Grothendieck's famous Splitting Theorem for vector bundles. This is a theorem which is purely geometric in statement and nature, but we attempt to reinterpret it and consequently prove it in a purely algebraic way. The pieces which fit together to form this proof draw from aspects of category theory of course, but also from homological algebra. As far as we can tell, the proof of the Splitting Theorem presented in this essay has not appeared in print before. However it is something that experts in the field are most likely aware of|a \folklore result", so to speak. As such, much of the information I have included in this thesis, particularly in the final two chapters, I have gleaned from conversations with my supervisor, Associate Professor Daniel Chan, which is why the reference list may not be as long as one might expect. 1.2 Introduction Vector bundles are very important objects of study in geometry, because they turn out to capture quite a lot of the geometry and invariants of certain algebraic vari- eties and manifolds. In some ways the correspondence between vector bundles and 1 varieties or manifolds can be likened to that between modules and rings, and this analogy becomes even clearer when we expand the category of vector bundles to the category of quasi-coherent sheaves. Rougly speaking, a vector bundle on a space is a way of assigning a vector space to each point, in a consistent manner. Two basic examples of 1-dimensional vector bundles on the circle S1 are the cylinder and the M¨obiusband. If the base space has some extra structure, the assignation of the vector spaces should of course be compatible with that structure: for example, for complex manifolds, we would like the function to be holomorphic. In 1957, Alexander Grothendieck [7], proved that any holomorphic vector bundle on the complex projective line is isomorphic to a direct sum of 1-dimensional holo- morphic vector bundles, or line bundles. Hazewinkel and Martin [8] then proved in 1982 that the same result holds for algebraic vector bundles on the projective line over any field k. The statement of this theorem is thus: If E is an algebraic vector 1 bundle on Pk, then ∼ E = O(n1) ⊕ · · · ⊕ O(nr); where O(n) is the nth twisted line bundle. In the case of the real projective line, 1 is homeommorphic to S1, and it can PR be shown that the cylinder and the M¨obiusband are the only two examples of line bundles (up to vector bundle isomorphism), so in this case the classification is rather simple1. Classically, Grothendieck's Splitting Theorem is proved geometrically, using coho- mology of line bundles. Our approach in this paper is to reinterpret the statement of the theorem using quotient categories. Via a canonical identification, vector bundles on a space V can be considered ex- amples of quasi-coherent sheaves on V . These are algebraic objects which form an abelian category. This category, denoted QCoh(V ), encodes all the geometric information about the space X, and so in Grothendieck's philosophy, is considered the quintessential object of study in algebraic geometry. The reason we embed the vector bundles into this larger category is precisely so that we obtain the properties of an abelian category. This is the incentive for developing quotient categories, since this construction gives a way of determining the category QCoh(V ) for a certain class of varieties V . Quotients of abelian categories were first treated in detail by Gabriel in his famous masters thesis Des Cat´egoriesAb´elienne [6]. This notion of quotient achieves the 1Of course, simple in the sense that there are few examples of vector bundles up to isomorphism. The proof is still non-trivial. 2 same goal as quotients in other areas of mathematics|it makes the category in question \smaller" in some way. The way it achieves this however, seems counter productive in the sense that the resulting category has more things in it. What happens is that instead of identifying objects in the way that one identifies elements in a quotient group, isomorphisms are added to the category, which has the ultimate effect of reducing the number of isomorphism classes, even though the class of objects remains the same. This is consistent with the usual philosophy of category theory that things should only be considered unique up to isomorphism. 1 As it turns out, the category QCoh(Pk) can be obtained by taking the category of graded modules over a polynomial ring in two variables and quotienting out by the subcategory of m = (x; y)-torsion modules. In this case, torsion free noetherian objects satisfy a structure theorem: ∼ M = R(n1) ⊕ · · · ⊕ R(nr); (y) where R(n) denotes the nth degree shift of R. This version of Grothendieck's Splitting Theorem is of algebraic importance as well|as a generalisation of the usual structure theorem for finitely generated mod- ules over a principal ideal domain (PID).
Load more

Recommended publications
  • Arxiv:1403.7027V2 [Math.AG] 21 Oct 2015 Ytnoigit Iebnlson Bundles Line Into Tensoring by N Ytednsyfoundation
    ON EQUIVARIANT TRIANGULATED CATEGORIES ALEXEY ELAGIN Abstract. Consider a finite group G acting on a triangulated category T . In this paper we investigate triangulated structure on the category T G of G-equivariant objects in T . We prove (under some technical conditions) that such structure exists. Supposed that an action on T is induced by a DG-action on some DG-enhancement of T , we construct a DG-enhancement of T G. Also, we show that the relation “to be an equivariant category with respect to a finite abelian group action” is symmetric on idempotent complete additive categories. 1. Introduction Triangulated categories became very popular in algebra, geometry and topology in last decades. In algebraic geometry, they arise as derived categories of coherent sheaves on algebraic varieties or stacks. It turned out that some geometry of varieties can be under- stood well through their derived categories and homological algebra of these categories. Therefore it is always interesting and important to understand how different geometrical operations, constructions, relations look like on the derived category side. In this paper we are interested in autoequivalences of derived categories or, more gen- eral, in group actions on triangulated categories. For X an algebraic variety, there are “expected” autoequivalences of Db(coh(X)) which are induced by automorphisms of X or by tensoring into line bundles on X. If X is a smooth Fano or if KX is ample, essentially that is all: Bondal and Orlov have shown in [6] that for smooth irreducible projective b variety X with KX or −KX ample all autoequivalences of D (coh(X)) are generated by automorphisms of X, twists into line bundles on X and translations.
    [Show full text]
  • Classification of Subcategories in Abelian Categories and Triangulated Categories
    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.
    [Show full text]
  • N-Quasi-Abelian Categories Vs N-Tilting Torsion Pairs 3
    N-QUASI-ABELIAN CATEGORIES VS N-TILTING TORSION PAIRS WITH AN APPLICATION TO FLOPS OF HIGHER RELATIVE DIMENSION LUISA FIOROT Abstract. It is a well established fact that the notions of quasi-abelian cate- gories and tilting torsion pairs are equivalent. This equivalence fits in a wider picture including tilting pairs of t-structures. Firstly, we extend this picture into a hierarchy of n-quasi-abelian categories and n-tilting torsion classes. We prove that any n-quasi-abelian category E admits a “derived” category D(E) endowed with a n-tilting pair of t-structures such that the respective hearts are derived equivalent. Secondly, we describe the hearts of these t-structures as quotient categories of coherent functors, generalizing Auslander’s Formula. Thirdly, we apply our results to Bridgeland’s theory of perverse coherent sheaves for flop contractions. In Bridgeland’s work, the relative dimension 1 assumption guaranteed that f∗-acyclic coherent sheaves form a 1-tilting torsion class, whose associated heart is derived equivalent to D(Y ). We generalize this theorem to relative dimension 2. Contents Introduction 1 1. 1-tilting torsion classes 3 2. n-Tilting Theorem 7 3. 2-tilting torsion classes 9 4. Effaceable functors 14 5. n-coherent categories 17 6. n-tilting torsion classes for n> 2 18 7. Perverse coherent sheaves 28 8. Comparison between n-abelian and n + 1-quasi-abelian categories 32 Appendix A. Maximal Quillen exact structure 33 Appendix B. Freyd categories and coherent functors 34 Appendix C. t-structures 37 References 39 arXiv:1602.08253v3 [math.RT] 28 Dec 2019 Introduction In [6, 3.3.1] Beilinson, Bernstein and Deligne introduced the notion of a t- structure obtained by tilting the natural one on D(A) (derived category of an abelian category A) with respect to a torsion pair (X , Y).
    [Show full text]
  • Basic Category Theory and Topos Theory
    Basic Category Theory and Topos Theory Jaap van Oosten Jaap van Oosten Department of Mathematics Utrecht University The Netherlands Revised, February 2016 Contents 1 Categories and Functors 1 1.1 Definitions and examples . 1 1.2 Some special objects and arrows . 5 2 Natural transformations 8 2.1 The Yoneda lemma . 8 2.2 Examples of natural transformations . 11 2.3 Equivalence of categories; an example . 13 3 (Co)cones and (Co)limits 16 3.1 Limits . 16 3.2 Limits by products and equalizers . 23 3.3 Complete Categories . 24 3.4 Colimits . 25 4 A little piece of categorical logic 28 4.1 Regular categories and subobjects . 28 4.2 The logic of regular categories . 34 4.3 The language L(C) and theory T (C) associated to a regular cat- egory C ................................ 39 4.4 The category C(T ) associated to a theory T : Completeness Theorem 41 4.5 Example of a regular category . 44 5 Adjunctions 47 5.1 Adjoint functors . 47 5.2 Expressing (co)completeness by existence of adjoints; preserva- tion of (co)limits by adjoint functors . 52 6 Monads and Algebras 56 6.1 Algebras for a monad . 57 6.2 T -Algebras at least as complete as D . 61 6.3 The Kleisli category of a monad . 62 7 Cartesian closed categories and the λ-calculus 64 7.1 Cartesian closed categories (ccc's); examples and basic facts . 64 7.2 Typed λ-calculus and cartesian closed categories . 68 7.3 Representation of primitive recursive functions in ccc's with nat- ural numbers object .
    [Show full text]
  • Weak Subobjects and Weak Limits in Categories and Homotopy Categories Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 38, No 4 (1997), P
    CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES MARCO GRANDIS Weak subobjects and weak limits in categories and homotopy categories Cahiers de topologie et géométrie différentielle catégoriques, tome 38, no 4 (1997), p. 301-326 <http://www.numdam.org/item?id=CTGDC_1997__38_4_301_0> © Andrée C. Ehresmann et les auteurs, 1997, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE ET Volume XXXVIII-4 (1997) GEOMETRIE DIFFERENTIELLE CATEGORIQUES WEAK SUBOBJECTS AND WEAK LIMITS IN CATEGORIES AND HOMOTOPY CATEGORIES by Marco GRANDIS R6sumi. Dans une cat6gorie donn6e, un sousobjet faible, ou variation, d’un objet A est defini comme une classe d’6quivalence de morphismes A valeurs dans A, de faqon a étendre la notion usuelle de sousobjet. Les sousobjets faibles sont lies aux limites faibles, comme les sousobjets aux limites; et ils peuvent 8tre consid6r6s comme remplaqant les sousobjets dans les categories "a limites faibles", notamment la cat6gorie d’homotopie HoTop des espaces topologiques, ou il forment un treillis de types de fibration sur 1’espace donn6. La classification des variations des groupes et des groupes ab£liens est un outil important pour d6terminer ces types de fibration, par les foncteurs d’homotopie et homologie. Introduction We introduce here the notion of weak subobject in a category, as an extension of the notion of subobject.
    [Show full text]
  • Introduction to Category Theory (Notes for Course Taught at HUJI, Fall 2020-2021) (UNPOLISHED DRAFT)
    Introduction to category theory (notes for course taught at HUJI, Fall 2020-2021) (UNPOLISHED DRAFT) Alexander Yom Din February 10, 2021 It is never true that two substances are entirely alike, differing only in being two rather than one1. G. W. Leibniz, Discourse on metaphysics 1This can be imagined to be related to at least two of our themes: the imperative of considering a contractible groupoid of objects as an one single object, and also the ideology around Yoneda's lemma ("no two different things have all their properties being exactly the same"). 1 Contents 1 The basic language 3 1.1 Categories . .3 1.2 Functors . .7 1.3 Natural transformations . .9 2 Equivalence of categories 11 2.1 Contractible groupoids . 11 2.2 Fibers . 12 2.3 Fibers and fully faithfulness . 12 2.4 A lemma on fully faithfulness in families . 13 2.5 Definition of equivalence of categories . 14 2.6 Simple examples of equivalence of categories . 17 2.7 Theory of the fundamental groupoid and covering spaces . 18 2.8 Affine algebraic varieties . 23 2.9 The Gelfand transform . 26 2.10 Galois theory . 27 3 Yoneda's lemma, representing objects, limits 27 3.1 Yoneda's lemma . 27 3.2 Representing objects . 29 3.3 The definition of a limit . 33 3.4 Examples of limits . 34 3.5 Dualizing everything . 39 3.6 Examples of colimits . 39 3.7 General colimits in terms of special ones . 41 4 Adjoint functors 42 4.1 Bifunctors . 42 4.2 The definition of adjoint functors . 43 4.3 Some examples of adjoint functors .
    [Show full text]
  • Groups and Categories
    \chap04" 2009/2/27 i i page 65 i i 4 GROUPS AND CATEGORIES This chapter is devoted to some of the various connections between groups and categories. If you already know the basic group theory covered here, then this will give you some insight into the categorical constructions we have learned so far; and if you do not know it yet, then you will learn it now as an application of category theory. We will focus on three different aspects of the relationship between categories and groups: 1. groups in a category, 2. the category of groups, 3. groups as categories. 4.1 Groups in a category As we have already seen, the notion of a group arises as an abstraction of the automorphisms of an object. In a specific, concrete case, a group G may thus consist of certain arrows g : X ! X for some object X in a category C, G ⊆ HomC(X; X) But the abstract group concept can also be described directly as an object in a category, equipped with a certain structure. This more subtle notion of a \group in a category" also proves to be quite useful. Let C be a category with finite products. The notion of a group in C essentially generalizes the usual notion of a group in Sets. Definition 4.1. A group in C consists of objects and arrows as so: m i G × G - G G 6 u 1 i i i i \chap04" 2009/2/27 i i page 66 66 GROUPSANDCATEGORIES i i satisfying the following conditions: 1.
    [Show full text]
  • On Ideals and Homology in Additive Categories
    IJMMS 29:8 (2002) 439–451 PII. S0161171202011675 http://ijmms.hindawi.com © Hindawi Publishing Corp. ON IDEALS AND HOMOLOGY IN ADDITIVE CATEGORIES LUCIAN M. IONESCU Received 28 January 2001 and in revised form 26 July 2001 Ideals are used to define homological functors in additive categories. In abelian categories the ideals corresponding to the usual universal objects are principal, and the construction reduces, in a choice dependent way, to homology groups. The applications considered in this paper are: derived categories and functors. 2000 Mathematics Subject Classification: 18G50, 18A05. 1. Introduction. Categorification is by now a commonly used procedure [1, 6, 9]. The concept of an additive category generalizes that of a ring in the same way group- oids generalize the notion of groups. Additive categories were called “rings with sev- eral objects” in [14], and were studied by imitating results and proofs from noncom- mutative homological ring theory, to additive category theory. Alternatively, the addi- tive category theory may be applied, as in [15], to the ring theory. Subsequent related papers adopted the “ideal theory” point of view, for example, [5], and in [17] the prob- lem of lifting algebraic geometry to the category theory level was considered and a notion of prime spectrum of a category was defined. In this paper, we consider the Dedekind’s original aim for introducing ideals [7], and leading to the study of general rings, not only principal ideal rings (PIR). In the context of categories, we relax the requirements of an exact category for the existence of kernels and cokernels, and define homological objects in an intrinsic way, using ideals.
    [Show full text]
  • Agnieszka Bodzenta
    June 12, 2019 HOMOLOGICAL METHODS IN GEOMETRY AND TOPOLOGY AGNIESZKA BODZENTA Contents 1. Categories, functors, natural transformations 2 1.1. Direct product, coproduct, fiber and cofiber product 4 1.2. Adjoint functors 5 1.3. Limits and colimits 5 1.4. Localisation in categories 5 2. Abelian categories 8 2.1. Additive and abelian categories 8 2.2. The category of modules over a quiver 9 2.3. Cohomology of a complex 9 2.4. Left and right exact functors 10 2.5. The category of sheaves 10 2.6. The long exact sequence of Ext-groups 11 2.7. Exact categories 13 2.8. Serre subcategory and quotient 14 3. Triangulated categories 16 3.1. Stable category of an exact category with enough injectives 16 3.2. Triangulated categories 22 3.3. Localization of triangulated categories 25 3.4. Derived category as a quotient by acyclic complexes 28 4. t-structures 30 4.1. The motivating example 30 4.2. Definition and first properties 34 4.3. Semi-orthogonal decompositions and recollements 40 4.4. Gluing of t-structures 42 4.5. Intermediate extension 43 5. Perverse sheaves 44 5.1. Derived functors 44 5.2. The six functors formalism 46 5.3. Recollement for a closed subset 50 1 2 AGNIESZKA BODZENTA 5.4. Perverse sheaves 52 5.5. Gluing of perverse sheaves 56 5.6. Perverse sheaves on hyperplane arrangements 59 6. Derived categories of coherent sheaves 60 6.1. Crash course on spectral sequences 60 6.2. Preliminaries 61 6.3. Hom and Hom 64 6.4.
    [Show full text]
  • Classifying Finite Localizations of Quasicoherent Sheaves
    Algebra i analiz St. Petersburg Math. J. Tom 21 (2009), 3 Vol. 21 (2010), No. 3, Pages 433–458 S 1061-0022(10)01102-7 Article electronically published on February 26, 2010 CLASSIFYING FINITE LOCALIZATIONS OF QUASICOHERENT SHEAVES G. GARKUSHA In memory of Vera Puninskaya Abstract. Given a quasicompact, quasiseparated scheme X, a bijection between the tensor localizing subcategories of finite type in Qcoh(X) and the set of all subsets ⊆ \ ∈ Y X of the form Y = i∈Ω Yi,withX Yi quasicompact and open for all i Ω, is established. As an application, an isomorphism of ringed spaces ∼ (X, OX ) −→ (spec(Qcoh(X)), OQcoh(X)) is constructed, where (spec(Qcoh(X)), OQcoh(X)) is a ringed space associated with the lattice of tensor localizing subcategories of finite type. Also, a bijective correspon- dence between the tensor thick subcategories of perfect complexes Dper(X)andthe tensor localizing subcategories of finite type in Qcoh(X) is established. §1. Introduction In his celebrated paper [1] on Abelian categories, Gabriel proved that for any Noe- therian scheme X, the assignments ⊇D→ ⊇ →{ ∈ | ⊆ } (1.1) coh X suppX (x)andX U x coh X suppX (x) U x∈D induce bijections between (1) the set of all tensor Serre subcategories of coh X,and ⊆ ∈ (2) the set of all subsets U X of the form U = i∈Ω Yi, where, for all i Ω, Yi has a quasicompact open complement X \ Yi. As a consequence of this result, X can be reconstructed from its Abelian category, coh X, of coherent sheaves (see Buan–Krause–Solberg [2, §8]).
    [Show full text]
  • Components of the Fundamental Category II
    Appl Categor Struct (2007) 15:387–414 DOI 10.1007/s10485-007-9082-7 Components of the Fundamental Category II E. Goubault · E. Haucourt Received: 23 March 2006 / Accepted: 19 March 2007 / Published online: 20 July 2007 © Springer Science + Business Media B.V. 2007 Abstract In this article we carry on the study of the fundamental category (Goubault and Raussen, Dihomotopy as a tool in state space analysis. In: Rajsbaum, S. (ed.) LATIN 2002: Theoretical Informatics. Lecture Notes in Computer Science, vol. 2286, Cancun, Mexico, pp. 16–37, Springer, Berlin Heidelberg New York, 2002; Goubault, Homology, Homotopy Appl., 5(2): 95–136, 2003) of a partially ordered topological space (Nachbin, Topology and Order, Van Nostrand, Princeton, 1965; Johnstone, Stone Spaces, Cambridge University Press, Cambridge, MA, 1982), as arising in e.g. concurrency theory (Fajstrup et al., Theor. Comp. Sci. 357: 241–278, 2006), initiated in (Fajstrup et al., APCS, 12(1): 81–108, 2004). The “algebra” of dipaths modulo dihomotopy (the fundamental category) of such a po- space is essentially finite in a number of situations. We give new definitions of the component category that are more tractable than the one of Fajstrup et al. (APCS, 12(1): 81–108, 2004), as well as give definitions of future and past component categories, related to the past and future models of Grandis (Theory Appl. Categ., 15(4): 95–146, 2005). The component category is defined as a category of fractions, but it can be shown to be equivalent to a quotient category, much easier to portray. A van Kampen theorem is known to be available on fundamental categories (Grandis, Cahiers Topologie Géom.
    [Show full text]
  • A Categorys Quotient Category of Isomorphism Types Versus Its
    GLASNIK MATEMATICKI SERIJA III. Vol. 20 (40) - 1985 GLMAB 2 ZAGREB 1985 DRUSTVO MATEMATICARA I FIZICARA SR HRVATSKE SADRZAJ—CONTENTS Originalni znanstveni radovi /. Hafner, On lower bound of the proof length in the equivalen- tial calculus 269 N. Seifter, Fixed points, fixed ends and groups of graphs ... 271 V. Vukovic, (Nonassociative) near-rings 279 R. Brü and M. Löpiz- -Pellicer, Extensions of algebraic Jordan basis 289 M. E. Harris, A remark on the Schur's lemma Situation 293 R. Fritsch, A category's quotient category of isomorphism types versus its skeleton 297 K. R. Yacoub, On finite groups with four independent generators two of which being of odd prime order p and one of order 2 301 AI. Polonijo, On transfers and closure conditions for quasigroups 313 J. Dana, On translations of sets in Banach space 319 M. Pal, On an extension of a theorem of S. Kurepa .... 327 Z. Kominek and H. L Miller, Some remarks on a theorem-of Steinhaus 337 Af. Perovic, On the problem of radius of injectivity for the map- pings quasiconformal in the mean 345 J. Schinasand A. Meimaridou, Asymptotic behavior for difference equations which cannot be linearized around a periodic orbit .... 349 I. Aganovic and Z. Tutek, On the lower-dimensional approximations of the mixed problem for Laplace equation 355 D. lubrinii, On the existence of Solutions of — Au = g (x, u, V u) 363 K. Baron, A remark on linear functional equations in the indeter- minate case 373 M. Bajraktarevic, Sur certaines Solutions continues de T6quation foncti- pnnelle Ay> = B\p 377 B.
    [Show full text]