DOCSLIB.ORG

A Product of Injective Modules Is Injective

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Product of Injective Modules Is Injective MATH 131B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 9 4.3. proof of lemmas. There are four lemmas to prove: (1) Aproductofinjectivemodulesisinjective. (2) HomR(M, RR^) ⇠= HomZ(M,Q/Z)(isomorphismofleftR-modules) (3) Q/Z is an injective Z-module. (4) M M ^^. ✓ Assuming (3) T = Q/Z is injective, the other three lemmas are easy: Proof of (4) Lemma 4.5. AnaturalembeddingM M ^^ is given by the evaluation ! map ev which sends x M to ev : M ^ T which is evaluation at x: 2 x ! evx(')='(x). This is easily seen to be additive: evx+y(')='(x + y)='(x)+'(y)=evx(')+evy(')=(evx + evy)('). The right action of R on M gives a left action of R on M ^ which gives a right action of R on HomZ(M ^,T)sothat(evxr)(')=evx(r'). This implies that evaluation x evx is a right R-module homomorphism: 7! evxr(')='(xr)=(r')(x)=evx(r')=(evxr)('). Finally, we need to show that ev is a monomorphism. In other words, for every nonzero element x M we need to find some additive map ' : M T so that evx(')='(x) =0. To do this2 take the cyclic group C generated by x ! 6 C = kx : k Z { 2 } This is either Z or Z/n.Inthesecondcaseletf : C T be given by ! k f(kx)= + Z Q/Z n 2 This is nonzero on x since 1/n is not an integer: f(x)= 1 + =0+ .IfC then n Z Z ⇠= Z let f : C T be given by 6 ! k f(kx)= + Z 2 1 Then, f(x)= 2 + Z is nonzero in Q/Z.SinceT is Z-injective, f extends to an additive map f : M T .So,ev is nonzero since ev (f)=f(x)=f(x) =0.So,ev : M M ^^ ! x x 6 ! is a monomorphism. ⇤ Proof of (1): products of injectives are injective. Suppose that J↵ are injective. Then we want to show that Q = J↵ is injective. Let p↵ : Q J↵ be the projection map. Suppose that A B and f : A Q is any morphism. Then! we want to extend f to B. ⇢ Q ! Since each J↵ is injective each composition f↵ = p↵ f : A J↵ extends to a morphism f : B J .I.e.,f A = f for all ↵.Bydefinitionoftheproductthere◦ ! ↵ ! ↵ ↵| ↵ exists a unique morphism f : B Q = J so that p f = f for each ↵.So, ! ↵ ↵ ◦ ↵ p f A = f A = f : A J ↵ ◦ | Q↵| ↵ ! ↵ The uniquely induced map A J is f.Therefore,f is an extension of f to B as ! ↵ required. ⇤ Q 10 MATH 131B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA Proof of (2): HomR(M, RR^) ⇠= HomZ(M,Q/Z). To prove this we will give a 1-1 correspondence and show that it (the correspondence) is additive. We also show that this map is an isomorphism of left R-modules. If f Hom (M, R^) then f : M R^,anR-linear map, which means that for 2 R R ! R each x M we get f(x) RR^ = HomZ(R, Q/Z), i.e., f(x):R Q/Z.Furthermore, for r 2R, x M,wehave2 f(xr)=f(x)r or f(xr)(s)=f(x)(rs).! In particular we can 2 2 evaluate this at 1 R.Thisgives'(f):M Q/Z by the formula 2 ! '(f)(x):=f(x)(1) This defines a mapping ' : HomR(M, RR^) Hom (M,Q/Z) ! Z We need to show that this is additive, i.e., '(f + g)='(f)+'(g) This is an easy calculation which follows from the way that f + g is defined, namely, addition of function is defined “pointwise” which means that (f + g)(x)=f(x)+g(x) by definition. So, x M, 8 2 '(f + g)(x)=(f + g)(x)(1) = [f(x)+g(x)](1) = f(x)(1) + g(x)(1) = '(f)(x)+'(g)(x)=['(f)+'(g)](x). We would also like to know that ' is left R-linear, i.e., we want to show '(rf)=r'(f). This follows from the fact that 1r = r = r1! '(rf)(x)=[rf(x)](1) = f(x)(1r) [r'(f)](x)='(f)(xr)=f(xr)(1) = f(x)(r1) 1 Finally we need to show that ' is a bijection. To do this we find the inverse '− = . For any additive g : M Q/Z let (g):M RR^ be given by (g)(x):R Q/Z: ! ! ! [ (g)(x)](r):=g(xr) Since this is additive in all three variables, is additive and (g)isadditive.Wealso need to check that (g)isahomomorphismsofrightR-modules, i.e., that (g)(xr)= (g)(x)r. This is an easy calculation: [ (g)(xr)](s)=g((xr)s)=g(xrs) [ (g)(x)r](s)=[ (g)(x)](rs)=g(x(rs)) The verification that is the inverse of ' is also straightforward: For all f 2 HomR(M, RR^)wehave ('(f))(x)(r)='(f)(rx)=f(rx)(1) = [rf(x)](1) = f(x)(1r)=f(x)(r) So, ('(f)) = f.Similarly,forallg Hom (M,Q/Z)wehave: 2 Z '( (g))(x)=[ (g)(x)](1) = g(x1) = g(x) So, '( (g)) = g. ⇤ I will do the remaining Lemma (3) (injectivity of Q/Z)nexttime. MATH 131B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 11 4.4. Examples. First, a trivial example: 4.4.1. fields. Suppose that R = k is a field. Since all short exact sequences of vector spaces split, we have: Theorem 4.7. In both categories mod-k and Mod-k all objects are both projective and injective. 4.4.2. polynomial ring. Let R = Z[t], the integer polynomial ring in one generator. This is a commutative Noetherian ring. It has dimension 2 since a maximal tower of prime ideal is given by 0 (t) (t, 2) ⇢ ⇢ These ideals are prime since the quotient of R by these ideals are domains (i.e., have no zero divisors): R/0=Z[t],R/(t)=Z are domains and R/(t, 2) = Z/(2) = Z/2Z is a field, making (2,t)intoamaximalideal. Proposition 4.8. A Z[t]-module M is the same as an abelian group together with an endomorphism M M given by the action of t. A homomorphism of Z[t]-modules f : M N is an additive! homomorphism which commutes with the action of t. ! Proof. IwillusethefactthatthestructureofanR-module on an additive group M is the same as a homomorphism of rings ' : R End(M). When R = Z[t], this homomorphism is given by its value on t since '!(f(t)) = f('(t)). For example, if f(t)=2t2 + 3 then '(f(t)) = '(2t2 +3)=2'(t) '(t)+3id = f('(t)) ◦ M Therefore, ' is determined by '(t) End (M)whichisarbitrary. 2 Z ⇤ For R = Z[t], what do the injective R-modules look like? We know that Q = RR^ is injective. What does that look like? Q = HomZ(Z[t], Q/Z) But Z[t]isafreeabeliangrouponthegenerators1,t,t2,t3, .Therefore,anelement ··· f Q, f : Z[t] Q/Z is given uniquely by the sequence 2 ! f(1),f(t),f(t2),f(t3), Q/Z ···2 Multiplication by t shifts this sequence to the left since (ft)(ti) f(tti)=f(ti+1). This proves the following. − Theorem 4.9. The injective Z[t]-module Q = Z[t]^ is isomorphic to the additive group of all sequences (a0,a1,a2, ) of elements ai Q/Z with the action of t given by shifting to the left and dropping the··· first coordinate. I.e.,2 t(a ,a ,a , )=(a ,a , ). 0 1 2 ··· 1 2 ··· The word “isomorphism” is correct here because these are not the same set..
Load more

Recommended publications
  • On Sandwich Theorem for Delta-Subadditive and Delta-Superadditive Mappings
    Results Math 72 (2017), 385–399 c 2016 The Author(s). This article is published with open access at Springerlink.com 1422-6383/17/010385-15 published online December 5, 2016 Results in Mathematics DOI 10.1007/s00025-016-0627-7 On Sandwich Theorem for Delta-Subadditive and Delta-Superadditive Mappings Andrzej Olbry´s Abstract. In the present paper, inspired by methods contained in Gajda and Kominek (Stud Math 100:25–38, 1991) we generalize the well known sandwich theorem for subadditive and superadditive functionals to the case of delta-subadditive and delta-superadditive mappings. As a conse- quence we obtain the classical Hyers–Ulam stability result for the Cauchy functional equation. We also consider the problem of supporting delta- subadditive maps by additive ones. Mathematics Subject Classification. 39B62, 39B72. Keywords. Functional inequality, additive mapping, delta-subadditive mapping, delta-superadditive mapping. 1. Introduction We denote by R, N the sets of all reals and positive integers, respectively, moreover, unless explicitly stated otherwise, (Y,·) denotes a real normed space and (S, ·) stands for not necessary commutative semigroup. We recall that a functional f : S → R is said to be subadditive if f(x · y) ≤ f(x)+f(y),x,y∈ S. A functional g : S → R is called superadditive if f := −g is subadditive or, equivalently, if g satisfies g(x)+g(y) ≤ g(x · y),x,y∈ S. If a : S → R is at the same time subadditive and superadditive then we say that it is additive, in this case a satisfies the Cauchy functional equation a(x · y)=a(x)+a(y),x,y∈ S.
    [Show full text]
  • Notes on Elementary Linear Algebra
    Notes on Elementary Linear Algebra Adam Coffman Contents 1 Real vector spaces 2 2 Subspaces 6 3 Additive Functions and Linear Functions 8 4 Distance functions 10 5 Bilinear forms and sesquilinear forms 12 6 Inner Products 18 7 Orthogonal and unitary transformations for non-degenerate inner products 25 8 Orthogonal and unitary transformations for positive definite inner products 29 These Notes are compiled from classroom handouts for Math 351, Math 511, and Math 554 at IPFW. They are not self-contained, but supplement the required texts, [A], [FIS], and [HK]. 1 1 Real vector spaces Definition 1.1. Given a set V , and two operations + (addition) and · (scalar multiplication), V is a “real vector space” means that the operations have all of the following properties: 1. Closure under Addition: For any u ∈ V and v ∈ V , u + v ∈ V . 2. Associative Law for Addition: For any u ∈ V and v ∈ V and w ∈ V ,(u + v)+w = u +(v + w). 3. Existence of a Zero Element: There exists an element 0 ∈ V such that for any v ∈ V , v + 0 = v. 4. Existence of an Opposite: For each v ∈ V , there exists an element of V , called −v ∈ V , such that v +(−v)=0. 5. Closure under Scalar Multiplication: For any r ∈ R and v ∈ V , r · v ∈ V . 6. Associative Law for Scalar Multiplication: For any r, s ∈ R and v ∈ V ,(rs) · v = r · (s · v). 7. Scalar Multiplication Identity: For any v ∈ V ,1· v = v. 8. Distributive Law: For all r, s ∈ R and v ∈ V ,(r + s) · v =(r · v)+(s · v).
    [Show full text]
  • Equivariant Spectra and Mackey Functors
    Equivariant spectra and Mackey functors Jonathan Rubin August 15, 2019 Suppose G is a finite group, and let OrbG be the orbit category of G. By Elmendorf's theorem (cf. Talk 2.2), there is an equivalence between the homotopy theory of G-spaces, and the homotopy theory of topological presheaves over OrbG . In this note, we introduce Mackey functors and explain Guillou and May's version of Elmendorf's theorem for G-spectra. This result gives an algebraic perspective on equivariant spectra.1 1 Mackey functors Spectra are higher algebraic analogues of abelian groups. Thus, we begin by con- sidering the fixed points of G-modules, and then we turn to G-spectra in general. Suppose pM, `, 0q is a G-module. By neglect of structure, M is a G-set, and H G therefore its fixed points M – Set pG{H, Mq form a presheaf over OrbG . Spelled out, we have an inclusion MK Ðâ MH for every inclusion K Ă H Ă G of subgroups, ´1 and an isomorphism gp´q : MH Ñ MgHg for every element g P G and subgroup H Ă G. The extra additive structure on M induces extra additive structure on the fixed points of M. Suppose H Ă G is a subgroup. By adjunction, an H-fixed point G{H Ñ M is equivalent to a G-linear map ZrG{Hs Ñ M, and thus MH – homZrGspZrG{Hs, Mq is an abelian group. More interestingly, for any inclusion of subgroups K Ă H Ă G, there is a G-linear map ZrG{Hs Ñ ZrG{Ks that rep- H resents the element H{K rK P ZrG{Ks .
    [Show full text]
  • Modules and Categories∗ Lenny Taelman
    Modules and Categories∗ Lenny Taelman ∗Preliminary version, May 22, 2018 Contents Foreword 7 Prerequisites 7 Other sources 7 Acknowledgements 7 Chapter 1. Modules over a ring 9 1. Left and right modules 9 2. First examples 10 3. Homomorphisms, submodules and quotient modules 12 4. Products, direct sums and free modules 13 Exercises 17 Chapter 2. Exact sequences 21 1. Exact sequences 21 2. The Five Lemma and the Snake Lemma 22 3. Split short exact sequences 24 Exercises 26 Chapter 3. Finitely generated modules over a PID 31 1. Introduction 31 2. Review of principal ideal domains 31 3. Free modules of finite rank over a PID 32 4. Structure of finitely generated modules over a PID 33 5. Application to Jordan normal form 35 Exercises 38 Chapter 4. Categories 41 1. Definition 41 2. Big examples 42 3. Small examples 42 4. Isomorphisms, monomorphisms, epimorphisms 44 3 4 CONTENTS 5. Initial and final objects 45 Exercises 47 Chapter 5. Functors 51 1. Definition of a functor 51 2. Many examples 51 3. Contravariant functors 54 4. Functors with multiple arguments 55 5. Morphisms of functors 56 6. Equivalences of categories 57 Exercises 61 Chapter 6. Tensor product 65 1. Tensor product of a right and a left module 65 2. Tensor products and bimodules 68 3. Tensor product as a functor 70 4. The adjunction 72 5. Restriction and extension of scalars 73 Exercises 75 Chapter 7. Adjoint functors 79 1. Adjoint pairs of functors 79 2. Many examples 80 3. Yoneda and uniqueness of adjoints 83 Exercises 86 Chapter 8.
    [Show full text]
  • What Is...A Shtuka?, Volume 50, Number 1
    ?WHAT IS... a Shtuka? David Goss Shtuka is a Russian word colloquially meaning thought, motivated much early work of O. Ore, “thing”. Spelled chtouca in the French literature, a E. H. Moore, and others. mathematical shtuka is, roughly speaking, a special Drinfeld Modules kind of module with a Frobenius-linear endomor- To define a Drinfeld module we need an algebra A phism (as explained below) attached to a curve over which will play the same role in the characteristic a finite field. Shtukas came from a fundamental p theory as the integers Z play in classical arith- analogy between differentiation and the p-th power metic. For simplicity of exposition we now set mapping in prime characteristic p. We will follow A = Fp[T ], the ring of polynomials in one indeter- both history and analogy in our brief presentation minate T. Let L be as above. A Drinfeld A-module here, with the hope that the reader will come to some ψ of rank d over L [Dr1] is an Fp-algebra injection appreciation of the amazing richness and beauty of ψ: A → L{τ} such that the image of a ∈ A, denoted characteristic p algebra. ψa(τ), is a polynomial in τ of degree d times the degree of a with d>0 . Note that ψ is uniquely de- Additive Polynomials termined by ψ (τ) and therefore d is a positive in- Let L be a field in characteristic p (so L is some ex- T teger. Moreover, there is a homomorphism ı from tension field of the finite field F = Z/(p)).
    [Show full text]
  • Arxiv:1910.06312V3 [Math.PR] 12 Oct 2020
    DETERMINANTAL PROBABILITY MEASURES ON GRASSMANNIANS ADRIEN KASSEL AND THIERRY LÉVY Abstract. We introduce and study a class of determinantal probability measures generalising the class of discrete determinantal point processes. These measures live on the Grassmannian of a real, complex, or quaternionic inner product space that is split into pairwise orthogonal finite-dimensional subspaces. They are determined by a positive self-adjoint contraction of the inner product space, in a way that is equivariant under the action of the group of isometries that preserve the splitting. Contents Introduction 1 1. Overview 3 2. Invariant measures on Grassmannians8 3. Determinantal linear processes (DLP) 16 4. Geometry of DLP 24 5. The point of view of the exterior algebra 35 6. Changing coefficient field and the quaternion case 43 Concluding remarks 52 References 53 Introduction Determinantal point processes (DPP) are an extensively studied class of random locally finite subsets of a nice measured topological space, for example of a Polish space endowed with a Borel measure. There is a discrete theory and a continuous theory of DPP, corresponding to the cases where this Borel measure is atomic or diffuse; these two cases are conceptually identical, but differ slightly in their presentation and methods. The goal of this paper is to introduce a generalisation of the discrete theory of DPP, which we call determinantal linear processes (DLP), where instead of a random collection of points drawn from a ground set, we consider a random collection of linear subspaces drawn from blocks of a arXiv:1910.06312v3 [math.PR] 12 Oct 2020 Date: October 13, 2020.
    [Show full text]
  • Topology Proceedings
    Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT °c by Topology Proceedings. All rights reserved. Topology Proceedings Volume 24, Summer 1999, 547–581 BANACH SPACES OVER NONARCHIMEDEAN VALUED FIELDS Wim H. Schikhof Abstract In this survey note we present the state of the art on the theory of Banach and Hilbert spaces over complete valued scalar fields that are not isomor- phic to R or C. For convenience we treat ‘classical’ theorems (such as Hahn-Banach Theorem, Closed Range Theorem, Riesz Representation Theorem, Eberlein-Smulian’s˘ Theorem, Krein-Milman The- orem) and discuss whether or not they remain valid in this new context, thereby stating some- times strong negations or strong improvements. Meanwhile several concepts are being introduced (such as norm orthogonality, spherical complete- ness, compactoidity, modules over valuation rings) leading to results that do not have -or at least have less important- counterparts in the classical theory. 1. The Scalar Field K 1.1. Non-archimedean Valued Fields and Their Topologies In many branches of mathematics and its applications the valued fields of the real numbers R and the complex numbers C play a Mathematics Subject Classification: 46S10, 47S10, 46A19, 46Bxx Key words: non-archimedean, Banach space, Hilbert space 548 Wim H. Schikhof fundamental role. For quite some time one has been discussing the consequences of replacing in those theories R or C -at first- by the more general object of a valued field (K, ||) i.e. a com- mutative field K, together with a valuation ||: K → [0, ∞) satisfying |λ| =0iffλ =0,|λ+µ|≤|λ|+|µ|, |λµ| = |λ||µ| for all λ, µ ∈ K.
    [Show full text]
  • Quasi-Homomorphisms
    FUNDAMENTA MATHEMATICAE * (200*) Quasi-homomorphisms by F´elix Cabello S´anchez (Badajoz) Abstract. We study the stability of homomorphisms between topological (abelian) groups. Inspired by the “singular” case in the stability of Cauchy’s equation and the technique of quasi-linear maps we introduce quasi-homomorphisms between topological groups, that is, maps ω : G → H such that ω(0) = 0 and ω(x + y) − ω(x) − ω(y) → 0 (in H) as x,y → 0 in G. The basic question here is whether ω is approximable by a true homomorphism a in the sense that ω(x) − a(x) → 0 in H as x → 0 in G. Our main result is that quasi-homomorphisms ω : G → H are approximable in the following two cases: • G is a product of locally compact abelian groups and H is either R or the circle group T. • G is either R or T and H is a Banach space. This is proved by adapting a classical procedure in the theory of twisted sums of Banach spaces. As an application, we show that every abelian extension of a quasi-Banach space by a Banach space is a topological vector space. This implies that most classical quasi-Banach spaces have only approximable (real-valued) quasi-additive functions. Introduction and statement of the main result. This paper deals with the stability of homomorphisms on topological abelian groups. Our methods (and results) lie on the frontiers of stability theory, extension of topological groups, and homology of topological linear spaces. As a motivation, consider an additive function on the line, that is, a map a : R → R satisfying (1) a(s + t)= a(s)+ a(t) (s,t ∈ R).
    [Show full text]
  • An Introduction to the Theory of Hilbert Spaces
    An Introduction to the Theory of Hilbert Spaces Eberhard Malkowsky Abstract These are the lecture notes of a mini–course of three lessons on Hilbert spaces taught by the author at the First German–Serbian Summer School of Modern Mathematical Physics in Sokobanja, Yu- goslavia, 13–25 August, 2001. The main objective was to present the fundamental definitions and results and their proofs from the theory of Hilbert spaces that are needed in applications to quantum physics. In order to make this paper self–contained, Section 1 was added; it contains well–known basic results from linear algebra and functional analysis. 1 Notations, Basic Definitions and Well–Known Results In this section, we give the notations that will be used throughout. Fur- thermore, we shall list the basic definitions and concepts from the theory of linear spaces, metric spaces, linear metric spaces and normed spaces, and deal with the most important results in these fields. Since all of them are well known, we shall only give the proofs of results that are directly applied in the theory of Hilbert spaces. 352 Throughout, let IN, IR and C| denote the sets of positive integers, real and complex numbers, respectively. For n ∈ IN , l e t IR n and C| n be the sets of n–tuples x =(x1,x2,...,xn) of real and complex numbers. If S is a set then |S| denotes the cardinality of S.Wewriteℵ0 for the cardinality of the set IN. In the proof of Theorem 4.8, we need Theorem 1.1 The Cantor–Bernstein Theorem If each of two sets allows a one–to–one map into the other, then the sets are of equal cardinality.
    [Show full text]
  • COMMUTING MAPS on SOME SUBSETS THAT ARE NOT CLOSED UNDER ADDITION a Dissertation Submitted to Kent State University in Partial F
    COMMUTING MAPS ON SOME SUBSETS THAT ARE NOT CLOSED UNDER ADDITION A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Willian V. Franca August 2013 Dissertation written by Willian V. Franca B.S., Universidade Federal do Rio de Janeiro Janeiro (UFRJ), Rio de Janeiro, Brazil, 2005 M.S., Universidade Federal do Rio de Janeiro Janeiro (UFRJ), Rio de Janeiro, Brazil, 2008 Ph.D., Kent State University, 2013 Approved by |||||||||||||| Chair, Doctoral Dissertation Committee Dr. Mikhail Chebotar |||||||||||||| Member, Doctoral Dissertation Committee Dr. Richard Aron |||||||||||||| Member, Doctoral Dissertation Committee Dr. Artem Zvavitch |||||||||||||| Member, Outside Discipline Dr. Sergey Anokhin |||||||||||||| Member, Graduate Faculty Representative Dr. Arden Ruttan Accepted by |||||||||||||| Chair, Department of Mathematical Sciences Dr. Andrew Tonge |||||||||||||| Associate Dean, College of Arts and Sciences Dr. Raymond A. Craig ii TABLE OF CONTENTS ACKNOWLEDGEMENTS ..............................v INTRODUCTION ...................................1 1 Preliminaries .....................................3 1.1 Additive Commuting Maps . .3 1.2 Commuting Traces of Multiadditive Maps . .4 2 Additive Commuting Maps on The Set of Invertible or Singular Matrices 6 2.1 Additive Commuting Maps on The Set of Singular Matrices . .6 2.2 Additive Commuting Maps on The Set of Invertible Matrices . .7 3 Commuting Traces of Multiadditive Maps on Invertible or Singular Ma- trices .......................................... 11 3.1 Commuting Traces on The Set of Invertible Matrices . 11 3.2 Commuting Traces on The Set of Singular Matrices . 15 3.3 Applications . 16 3.4 Herstein's Conjecture . 18 4 Commuting Traces of Multiadditive Maps on The Set of Rank-k Matrices 20 4.1 Additive Commuting Maps on The Set of Rank-k Matrices .
    [Show full text]
  • Categorifying Measure Theory: a Roadmap 11
    CATEGORIFYING MEASURE THEORY: A ROADMAP G. RODRIGUES Abstract. A program for categorifying measure theory is outlined. Contents 1. Introduction 2 Contents 6 Acknowledgements 7 2. Measurable bundles of Hilbert spaces 7 3. Towardscategorifiedmeasuretheory 11 3.1. Hilb as a categorified ring 12 3.2. Measurability of bundles 15 3.3. Linearity of the direct integral 20 3.4. Continuity of the direct integral 22 3.5. Universalpropertyofthedirectintegral 24 4. From Hilbert to Banach spaces 28 4.1. The failure of the Radon-Nikodym property 33 5. Foregrounding:measurealgebrasandintegrals 37 5.1. The Banach algebra L∞(Ω) 40 5.2. The Bochner integral 43 5.3. TheStonespaceofaBooleanalgebraandweakintegrals. 46 6. Banach 2-spaces 51 6.1. Presheaf categories 53 6.2. Free Banach 2-spaces over sets 59 6.3. Categorifiedmeasuretheory:thediscretecase 62 7. Categorified measures and integrals 65 7.1. Banach sheaves 65 arXiv:0912.4914v1 [math.CT] 24 Dec 2009 7.2. Cosheaves and inverters 72 7.3. The cosheafification functor 79 7.4. The spectral measure of a cosheaf 81 References 87 This work was supported by the Programa Operacional Ciˆencia e Inova¸c˜ao 2010, financed by the Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT / Portugal) and cofinanced by the European Community fund FEDER, in part through the research project Quantum Topology POCI / MAT / 60352 / 2004. 1 2 G. RODRIGUES 1. Introduction Begin, ephebe, by perceiving the idea Of this invention, this invented world, The inconceivable idea of the sun. You must become an ignorant man again And see the sun again with an ignorant eye And see it clearly in the idea of it.
    [Show full text]
  • THE GROTHENDIECK GROUP K0 There Are Several Ways to Construct
    CHAPTER II THE GROTHENDIECK GROUP K0 There are several ways to construct the “Grothendieck group” of a mathematical object. We begin with the group completion version, because it has been the most historically important. After giving the applications to rings and topological spaces, we discuss λ-operations in 4. In sections 6 and 7 we describe the Grothendieck group of an “exact category,”§ and apply it to the K-theory of schemes in 8. This construction is generalized to the Grothendieck group of a “Waldhausen category”§ in 9. § 1. The Group Completion of a monoid § 0 Both K0(R) and K (X) are formed by taking the group completion of an abelian monoid—the monoid P(R) of finitely generated projective R-modules and the monoid VB(X) of vector bundles over X, respectively. We begin with a description of this construction. Recall that an abelian monoid is a set M together with an associative, commu- tative operation + and an “additive” identity element 0. A monoid map f: M N is a set map such that f(0) = 0 and f(m + m′)= f(m)+ f(m′). The most famous→ example of an abelian monoid is N = 0, 1, 2,... , the natural numbers with additive identity zero. If A is an abelian group{ then not} only is A an abelian monoid, but so is any additively closed subset of A containing 0. The group completion of an abelian monoid M is an abelian group M −1M, together with a monoid map [ ]: M M −1M which is universal in the sense that, for every abelian group A and every→ monoid map α: M A, there is a unique abelian group homomorphismα ˜: M −1M A such that→α ˜([m]) = α(m) for all m M.
    [Show full text]