Homological Algebra in -Abelian Categories
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Homological Algebra
Homological Algebra Donu Arapura April 1, 2020 Contents 1 Some module theory3 1.1 Modules................................3 1.6 Projective modules..........................5 1.12 Projective modules versus free modules..............7 1.15 Injective modules...........................8 1.21 Tensor products............................9 2 Homology 13 2.1 Simplicial complexes......................... 13 2.8 Complexes............................... 15 2.15 Homotopy............................... 18 2.23 Mapping cones............................ 19 3 Ext groups 21 3.1 Extensions............................... 21 3.11 Projective resolutions........................ 24 3.16 Higher Ext groups.......................... 26 3.22 Characterization of projectives and injectives........... 28 4 Cohomology of groups 32 4.1 Group cohomology.......................... 32 4.6 Bar resolution............................. 33 4.11 Low degree cohomology....................... 34 4.16 Applications to finite groups..................... 36 4.20 Topological interpretation...................... 38 5 Derived Functors and Tor 39 5.1 Abelian categories.......................... 39 5.13 Derived functors........................... 41 5.23 Tor functors.............................. 44 5.28 Homology of a group......................... 45 1 6 Further techniques 47 6.1 Double complexes........................... 47 6.7 Koszul complexes........................... 49 7 Applications to commutative algebra 52 7.1 Global dimensions.......................... 52 7.9 Global dimension of -
Homotopy Equivalences and Free Modules Steven
Topology, Vol. 21, No. 1, pp. 91-99, 1982 0040-93831821010091-09502.00/0 Printed in Great Britain. Pergamon Press Ltd. HOMOTOPY EQUIVALENCES AND FREE MODULES STEVEN PLOTNICKt:~ (Received 15 December 1980) INTRODUCTION IN THIS PAPER, we are concerned with the problem of determining the group of self-homotopy equivalences of some familiar spaces. We have in mind the following spaces: Let ~- be a finite group acting freely and cellulary on a CW-complex with the homotopy type of S 2m+1. Removing a point from S2m+1/Tr yields a 2m-complex X. We will (theoretically) describe the self-homotopy equivalences of X. (I use the word theoretically since the results is given in terms of units in the integral group ring Z Tr and certain projective modules over ZTr, and these are far from being completely understood.) Once this has been done, it will be fairly straightforward to describe the self-homotopy equivalences of $2~+1/7r and of the 2m-complex which results from removing more than one point from s2m+l/Tr (i.e. X v v $2~). As one might expect, the more points one removes, the easier it is to realize automorphisms of ~r by homotopy equivalences. It seems quite nice that this can be phrased as a question in Ko(Z~). The results can be summarised as follows. First, some standard facts, definitions, and notation: If ~r has order n and acts freely on S :m+~, then H:m+1(~r; Z)m Zn. Thus, a E Aut ~r acts on H2m+l(~r) via multiplication by an integer r~, where (r~, n) = 1. -
Arxiv:2001.09075V1 [Math.AG] 24 Jan 2020
A topos-theoretic view of difference algebra Ivan Tomašić Ivan Tomašić, School of Mathematical Sciences, Queen Mary Uni- versity of London, London, E1 4NS, United Kingdom E-mail address: [email protected] arXiv:2001.09075v1 [math.AG] 24 Jan 2020 2000 Mathematics Subject Classification. Primary . Secondary . Key words and phrases. difference algebra, topos theory, cohomology, enriched category Contents Introduction iv Part I. E GA 1 1. Category theory essentials 2 2. Topoi 7 3. Enriched category theory 13 4. Internal category theory 25 5. Algebraic structures in enriched categories and topoi 41 6. Topos cohomology 51 7. Enriched homological algebra 56 8. Algebraicgeometryoverabasetopos 64 9. Relative Galois theory 70 10. Cohomologyinrelativealgebraicgeometry 74 11. Group cohomology 76 Part II. σGA 87 12. Difference categories 88 13. The topos of difference sets 96 14. Generalised difference categories 111 15. Enriched difference presheaves 121 16. Difference algebra 126 17. Difference homological algebra 136 18. Difference algebraic geometry 142 19. Difference Galois theory 148 20. Cohomologyofdifferenceschemes 151 21. Cohomologyofdifferencealgebraicgroups 157 22. Comparison to literature 168 Bibliography 171 iii Introduction 0.1. The origins of difference algebra. Difference algebra can be traced back to considerations involving recurrence relations, recursively defined sequences, rudi- mentary dynamical systems, functional equations and the study of associated dif- ference equations. Let k be a commutative ring with identity, and let us write R = kN for the ring (k-algebra) of k-valued sequences, and let σ : R R be the shift endomorphism given by → σ(x0, x1,...) = (x1, x2,...). The first difference operator ∆ : R R is defined as → ∆= σ id, − and, for r N, the r-th difference operator ∆r : R R is the r-th compositional power/iterate∈ of ∆, i.e., → r r ∆r = (σ id)r = ( 1)r−iσi. -
Derived Functors and Homological Dimension (Pdf)
DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION George Torres Math 221 Abstract. This paper overviews the basic notions of abelian categories, exact functors, and chain complexes. It will use these concepts to define derived functors, prove their existence, and demon- strate their relationship to homological dimension. I affirm my awareness of the standards of the Harvard College Honor Code. Date: December 15, 2015. 1 2 DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION 1. Abelian Categories and Homology The concept of an abelian category will be necessary for discussing ideas on homological algebra. Loosely speaking, an abelian cagetory is a type of category that behaves like modules (R-mod) or abelian groups (Ab). We must first define a few types of morphisms that such a category must have. Definition 1.1. A morphism f : X ! Y in a category C is a zero morphism if: • for any A 2 C and any g; h : A ! X, fg = fh • for any B 2 C and any g; h : Y ! B, gf = hf We denote a zero morphism as 0XY (or sometimes just 0 if the context is sufficient). Definition 1.2. A morphism f : X ! Y is a monomorphism if it is left cancellative. That is, for all g; h : Z ! X, we have fg = fh ) g = h. An epimorphism is a morphism if it is right cancellative. The zero morphism is a generalization of the zero map on rings, or the identity homomorphism on groups. Monomorphisms and epimorphisms are generalizations of injective and surjective homomorphisms (though these definitions don't always coincide). It can be shown that a morphism is an isomorphism iff it is epic and monic. -
Projective Dimensions and Almost Split Sequences
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 271 (2004) 652–672 www.elsevier.com/locate/jalgebra Projective dimensions and almost split sequences Dag Madsen 1 Department of Mathematical Sciences, NTNU, NO-7491 Trondheim, Norway Received 11 September 2002 Communicated by Kent R. Fuller Abstract Let Λ be an Artin algebra and let 0 → A → B → C → 0 be an almost split sequence. In this paper we discuss under which conditions the inequality pd B max{pd A,pd C} is strict. 2004 Elsevier Inc. All rights reserved. Keywords: Almost split sequences; Homological dimensions Let R be a ring. If we have an exact sequence ε :0→ A → B → C → 0ofR-modules, then pd B max{pdA,pd C}. In some cases, for instance if the sequence is split exact, equality holds, but in general the inequality may be strict. In this paper we will discuss a problem first considered by Auslander (see [5]): Let Λ be an Artin algebra (for example a finite dimensional algebra over a field) and let ε :0→ A → B → C → 0 be an almost split sequence. To which extent does the equality pdB = max{pdA,pd C} hold? Given an exact sequence 0 → A → B → C → 0, we investigate in Section 1 what can be said in complete generality about when pd B<max{pdA,pd C}.Inthissectionwedo not make any assumptions on the rings or the exact sequences involved. In all sections except Section 1 the rings we consider are Artin algebras. -
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ASIAN J. MATH. © 1997 International Press Vol. 1, No. 2, pp. 330-417, June 1997 009 K-THEORY FOR TRIANGULATED CATEGORIES 1(A): HOMOLOGICAL FUNCTORS * AMNON NEEMANt 0. Introduction. We should perhaps begin by reminding the reader briefly of Quillen's Q-construction on exact categories. DEFINITION 0.1. Let £ be an exact category. The category Q(£) is defined as follows. 0.1.1. The objects of Q(£) are the objects of £. 0.1.2. The morphisms X •^ X' in Q(£) between X,X' G Ob{Q{£)) = Ob{£) are isomorphism classes of diagrams of morphisms in £ X X' \ S Y where the morphism X —> Y is an admissible mono, while X1 —y Y is an admissible epi. Perhaps a more classical way to say this is that X is a subquotient of X''. 0.1.3. Composition is defined by composing subquotients; X X' X' X" \ ^/ and \ y/ Y Y' compose to give X X' X" \ s \ s Y PO Y' \ S Z where the square marked PO is a pushout square. The category Q{£) can be realised to give a space, which we freely confuse with the category. The Quillen if-theory of the exact category £ was defined, in [9], to be the homotopy of the loop space of Q{£). That is, Ki{£)=ni+l[Q{£)]. Quillen proved many nice functoriality properties for his iT-theory, and the one most relevant to this article is the resolution theorem. The resolution theorem asserts the following. 7 THEOREM 0.2. Let F : £ —> J be a fully faithful, exact inclusion of exact categories. -
Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1
Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1: A directed set I is a set with a partial order ≤ such that for every i; j 2 I there is k 2 I such that i ≤ k and j ≤ k. Let R be a ring. A directed system of R-modules indexed by I is a collection of R modules fMi j i 2 Ig with a R module homomorphisms µi;j : Mi ! Mj for each pair i; j 2 I where i ≤ j, such that (i) for any i 2 I, µi;i = IdMi and (ii) for any i ≤ j ≤ k in I, µi;j ◦ µj;k = µi;k. We shall denote a directed system by a tuple (Mi; µi;j). The direct limit of a directed system is defined using a universal property. It exists and is unique up to a unique isomorphism. Theorem 2 (Direct limits). Let fMi j i 2 Ig be a directed system of R modules then there exists an R module M with the following properties: (i) There are R module homomorphisms µi : Mi ! M for each i 2 I, satisfying µi = µj ◦ µi;j whenever i < j. (ii) If there is an R module N such that there are R module homomorphisms νi : Mi ! N for each i and νi = νj ◦µi;j whenever i < j; then there exists a unique R module homomorphism ν : M ! N, such that νi = ν ◦ µi. The module M is unique in the sense that if there is any other R module M 0 satisfying properties (i) and (ii) then there is a unique R module isomorphism µ0 : M ! M 0. -
Van Kampen Colimits As Bicolimits in Span*
Van Kampen colimits as bicolimits in Span? Tobias Heindel1 and Pawe lSoboci´nski2 1 Abt. f¨urInformatik und angewandte kw, Universit¨atDuisburg-Essen, Germany 2 ECS, University of Southampton, United Kingdom Abstract. The exactness properties of coproducts in extensive categories and pushouts along monos in adhesive categories have found various applications in theoretical computer science, e.g. in program semantics, data type theory and rewriting. We show that these properties can be understood as a single universal property in the associated bicategory of spans. To this end, we first provide a general notion of Van Kampen cocone that specialises to the above colimits. The main result states that Van Kampen cocones can be characterised as exactly those diagrams in that induce bicolimit diagrams in the bicategory of spans Span , C C provided that C has pullbacks and enough colimits. Introduction The interplay between limits and colimits is a research topic with several applica- tions in theoretical computer science, including the solution of recursive domain equations, using the coincidence of limits and colimits. Research on this general topic has identified several classes of categories in which limits and colimits relate to each other in useful ways; extensive categories [5] and adhesive categories [21] are two examples of such classes. Extensive categories [5] have coproducts that are “well-behaved” with respect to pullbacks; more concretely, they are disjoint and universal. Extensivity has been used by mathematicians [4] and computer scientists [25] alike. In the presence of products, extensive categories are distributive [5] and thus can be used, for instance, to model circuits [28] or to give models of specifications [11]. -
LOOP SPACES for the Q-CONSTRUCTION Charles H
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 52 (1988) l-30 North-Holland LOOP SPACES FOR THE Q-CONSTRUCTION Charles H. GIFFEN* Department of Mathematics, University of Virginia, Charlottesville, VA 22903, U.S. A Communicated by C.A. Weibel Received 4 November 1985 The algebraic K-groups of an exact category Ml are defined by Quillen as KJ.4 = TT,+~(Qkd), i 2 0, where QM is a category known as the Q-construction on M. For a ring R, K,P, = K,R (the usual algebraic K-groups of R) where P, is the category of finitely generated projective right R-modules. Previous study of K,R has required not only the Q-construction, but also a model for the loop space of QP,, known as the Y’S-construction. Unfortunately, the S-‘S-construction does not yield a loop space for QfLQ when M is arbitrary. In this paper, two useful models of a loop space for Qu, with no restriction on the exact category L&, are described. Moreover, these constructions are shown to be directly related to the Y’S- construction. The simpler of the two constructions fails to have a certain symmetry property with respect to dualization of the exact category mm. This deficiency is eliminated in the second construction, which is somewhat more complicated. Applications are given to the relative algebraic K-theory of an exact functor of exact categories, with special attention given to the case when the exact functor is cofinal. -
Abelian Categories
Abelian Categories Lemma. In an Ab-enriched category with zero object every finite product is coproduct and conversely. π1 Proof. Suppose A × B //A; B is a product. Define ι1 : A ! A × B and π2 ι2 : B ! A × B by π1ι1 = id; π2ι1 = 0; π1ι2 = 0; π2ι2 = id: It follows that ι1π1+ι2π2 = id (both sides are equal upon applying π1 and π2). To show that ι1; ι2 are a coproduct suppose given ' : A ! C; : B ! C. It φ : A × B ! C has the properties φι1 = ' and φι2 = then we must have φ = φid = φ(ι1π1 + ι2π2) = ϕπ1 + π2: Conversely, the formula ϕπ1 + π2 yields the desired map on A × B. An additive category is an Ab-enriched category with a zero object and finite products (or coproducts). In such a category, a kernel of a morphism f : A ! B is an equalizer k in the diagram k f ker(f) / A / B: 0 Dually, a cokernel of f is a coequalizer c in the diagram f c A / B / coker(f): 0 An Abelian category is an additive category such that 1. every map has a kernel and a cokernel, 2. every mono is a kernel, and every epi is a cokernel. In fact, it then follows immediatly that a mono is the kernel of its cokernel, while an epi is the cokernel of its kernel. 1 Proof of last statement. Suppose f : B ! C is epi and the cokernel of some g : A ! B. Write k : ker(f) ! B for the kernel of f. Since f ◦ g = 0 the map g¯ indicated in the diagram exists. -
Homological Algebra Over the Representation Green Functor
Homological Algebra over the Representation Green Functor SPUR Final Paper, Summer 2012 Yajit Jain Mentor: John Ullman Project suggested by: Mark Behrens February 1, 2013 Abstract In this paper we compute Ext RpF; F q for modules over the representation Green functor R with F given to be the fixed point functor. These functors are Mackey functors, defined in terms of a specific group with coefficients in a commutative ring. Previous work by J. Ventura [1] computes these derived k functors for cyclic groups of order 2 with coefficients in F2. We give computations for general cyclic groups with coefficients in a general field of finite characteristic as well as the symmetric group on three elements with coefficients in F2. 1 1 Introduction Mackey functors are algebraic objects that arise in group representation theory and in equivariant stable homotopy theory, where they arise as stable homotopy groups of equivariant spectra. However, they can be given a purely algebraic definition (see [2]), and there are many interesting problems related to them. In this paper we compute derived functors Ext of the internal homomorphism functor defined on the category of modules over the representation Green functor R. These Mackey functors arise in Künneth spectral sequences for equivariant K-theory (see [1]). We begin by giving definitions of the Burnside category and of Mackey functors. These objects depend on a specific finite group G. It is possible to define a tensor product of Mackey functors, allowing us to identify ’rings’ in the category of Mackey functors, otherwise known as Green functors. We can then define modules over Green functors. -
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.