Fundamentals of Advanced Mathematics 1: Categories, Algebraic Structures, Linear and Homological Algebra

By Henri Bourles

This precis, comprised of three volumes, of which this book is the first, exposes the mathematical elements which make up the foundations of a number of contemporary scientific methods: modern theory on systems, physics and engineering.

This first volume focuses primarily on algebraic questions: categories and functors, groups, rings, modules and algebra. Notions are introduced in a general framework and then studied in the context of commutative and homological algebra; their application in algebraic topology and geometry is therefore developed. These notions play an essential role in algebraic analysis (analytico-algebraic systems theory of ordinary or partial linear differential equations).

The book concludes with a study of modules over the main types of rings, the rational canonical form of matrices, the (commutative) theory of elemental divisors and their application in systems of linear differential equations with constant coefficients.

• Part of the New Mathematical Methods, Systems, and Applications series
• Presents the notions, results, and proofs necessary to understand and master the various topics
• Provides a unified notation, making the task easier for the reader.
• Includes several summaries of mathematics for engineers

• Language: English
• Publisher: Elsevier Science
• Release date: Jul 10, 2017
• ISBN: 9780081021125

Henri Bourles

Henri Bourlès is Full Professor and Chair at the Conservatoire National des Arts et Métiers, Paris, France.

Book preview

Fundamentals of Advanced Mathematics 1

Categories, Algebraic Structures, Linear and Homological Algebra

Henri Bourlès

New Mathematical Methods, Systems and Applications Set

coordinated by

Henri Bourlès

Table of Contents

Cover

Title page

Dedication

Copyright

Preface

List of Notations

1: Categories and Functors

Abstract

1.1 Categories

1.2 Functors

1.3 Structures

2: Elementary Algebraic Structures

Abstract

2.1 Monoids and ordered sets

2.2 Groups

2.3 Rings and algebras

3: Modules and Algebras

Abstract

3.1 Additional concepts from linear algebra

3.2 Notions of commutative algebra

3.3 Homological notions

3.4 Modules over principal ideal domains and related notions

Bibliography

Cited Authors

Index

Dedication

O récompense après une pensée.

Qu’un long regard sur le calme des dieux!

Paul Valéry

When thought has had its hour, oh how rewarding are the long vistas of celestial calm.

(Translation by Cecil Day-Lewis)

Copyright

First published 2017 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

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Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

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© ISTE Press Ltd 2017

The rights of Henri Bourlès to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

Library of Congress Cataloging in Publication Data

A catalog record for this book is available from the Library of Congress

ISBN 978-1-78548-173-4

Printed and bound in the UK and US

Preface

Henri Bourlès April 2017

The objective of this Précis in three volumes, of which this is the first, is to present the mathematical objects that make up the foundation of certain methods, not only from modern Systems Theory, but also from Mathematics, Physics and many fields of Engineering. Viewed from this perspective, these mathematical concepts are fundamental. They are advanced because they assume that the reader has mastered the most important parts of a Mathematics degree or the mathematical content taught in most advanced engineering courses. The reference works, Algebra by R. Godement [GOD 64] and Foundations of Modern Analysis by J. Dieudonné (first volume of his Treatise on Analysis [DIE 82]), together with a few notions of measure theory and integration [MAC 14], are amply sufficient as prerequisites. The original plan was only to write one single volume, and so this work could never have become an encyclopedia of mathematics. Such an endeavor would have exceeded the skills of the author by far, even excluding all mathematics discovered (or invented, according to your epistemological preference) during the last 50 years, as we shall do below - with a few exceptions. Presenting this more recent material would require too technical of a discussion. Moreover, one such encyclopedia already exists, at least in part. Although it does not cover every topic discussed between these pages, Éléments de mathématique by N. Bourbaki presents a great many others - this work represents a monumental milestone that we shall refer to abundantly. Nevertheless, it quickly became apparent that this Précis could not limit itself to being a simple formula booklet or collection of results (as had originally been imagined by the author), which would have been incapable of conveying any understanding. Hence, the decision was made to construct a coherent presentation without attempting to prove everything in the usual manner of a Treatise. Long proofs are omitted, especially when they do not aid comprehension (at least in the opinion of the author) or where they exceed the scope of the discussion. Some results are simply justified by examples. Whenever a proof is omitted, a reference is given (chosen to be as accessible as possible). A number of easy proofs are given as exercises. Others, slightly more difficult but nonetheless within the reach of any reader (assuming pencil and paper at hand), are given as starred exercises (written as exercise* and supplemented by a reference). For some ideas, we refer to well-written Wikipedia articles (although we can only guarantee the quality of the French pages). Each volume is divided into chapters, paragraphs and items; section 3.2.4 is the fourth item of the second paragraph of the third chapter. To simplify these references, larger items are subdivided into smaller ones labeled by Roman numerals in parentheses. Throughout this Précis, the three volumes are respectively labeled by [P1], [P2] and [P3].

This volume begins with Category Theory (-modules [KAS 95], [BJÖ 79], [MAI 93], [COU 95]). Mathematically, the non-commutative case includes the commutative case, and so we won’t deprive ourselves of the opportunity to explore a few notions of Commutative Algebra (localization, primary decomposition, Nullstellensatz-module theory mentioned above, see also [EHR 70], [PAL 70]). The elementary case of systems of ordinary linear differential equations with constant coefficients is discussed in section 3.4.4.

---

¹ For Bourbaki, we might say that: In the beginning there was Hilbert’s operator, as discussed in [BKI 70] (Chap. I, section 1.1). This perspective is controversial: see section 1.1.2 [FRO 83] and (more polemically) [MAT 92].

² Here, we use the mathematical meaning of the term system. In the field of automatic control, the existence of concrete examples of multidimensional systems (without boundary conditions and with distributed control variables) is not given and has been repeatedly reaffirmed, refuted and generally the subject of much controversy since 1990.

List of Notations

Standard notation

:= equal by definition

 set of non-negative integers {0, 1, 2, …}

 set of natural integers {1, 2,…}

 set of relative integers {… − 2, − 1, 0, 1, 2, …}

 set of rational numbers

 set of real numbers

:

and : x: x≥0}

 set of complex numbers

)

 set of subsets of the set X

 set of finite subsets of the set X

 complement of B in A (B ⊂ A)

∪, ∩ union, intersection

≤, ≥ order relation, dual order relation

 <  strict order relation

⊂,  inclusion, strict inclusion

idX = 1X identity mapping of X

can canonical morphism

mod. modulo

f |A restriction of f : X → Y to A ⊂ Y

δij Kronecker delta (δij = 1 if i = j, 0 otherwise)

det determinant

Tr trace

In n×n identity matrix

AT transpose of the matrix A

Categories (section 1.1)

, pp. 1–2

-morphisms from X to Y, p. 2

Set, Top category of sets, topological spaces, p. 2

Mon, Grp, Rng category of monoids, groups, rings, p. 2

KVec category of left vector spaces over the division ring K, p. 2

RMod category of left modules over the ring R, p. 2

, p. 2

A B, A B,  monomorphism, epimorphism, isomorphism, p. 3

f − 1 inverse isomorphism, p. 3

PX,QX set of morphisms with codomain X, with domain X, p. 3

u ≅  v equivalent morphisms, p. 3

 double arrow, p. 4

eq (f, g) , ker (f ) equalizer, kernel, p. 4

coeq (f, g) , coker (f ) coequalizer, cokernel, p. 4

im (f ) image, p. 5

ϕ choice function, p. 6

∏i∈I Xi product, p. 6

Eq (A, B) equipotent sets, p. 9

Card (A) cardinality of the set A, p. 10

, p. 10

 aleph, p. 10

 cardinal exponentiation, p. 11

τ Hilbert’s operator, p. 12

f − 1 (B) inverse image of B under f, p. 12

E/~ set of equivalence classes mod. ~, p. 13

Functors (section 1.2)

Ab category of abelian groups, p. 14

ComRng category of commutative rings, p. 14

, p. 14

 functorial morphism, p. 15

 functorial isomorphism, p. 15

 isomorphic functors, p. 15

, p. 15

, p. 18

pri : ∏i∈I Xi → Xi canonical projection, p. 19

XI, X(I) power, copower, p. 19

inji : Xi →  i∈I Xi canonical injection, p. 19

i∈I Xi disjoint union, p. 20

X1×Z X2, X1 Z X2 fibered product, fibered sum, p. 21

 Inductive limit, projective limit, p. 22

Structures (section 1.3)

 concrete category p. 28

 forgetful functor p. 28

A ⊆ B subobject in a concrete category p. 28

A B proper subobject p. 28

Monoids and ordered sets (section 2.1)

U(M) set of units of M, p. 34

M× set of non-zero elements of M, p. 34

[S], [x1,…, xn] monoid generated by S, generated by the elements x1, …, xn, p. 34

a | b, a b divisor, total divisor, pp. 35–36

[a, b] interval, p. 36

i∈I ai = inf{ai :i ∈ Ii∈I ai = sup{ai : i ∈ I}, p. 36

x x⊥ Galois connection, p. 36

|C|, |P| length of the chain C, of the ordered set P, p. 37

 depth, height of x, p. 37

Groups (section 2.2)

 symmetric group, p. 41

xH, Hx left, right coset of x mod H, p. 41

G/H, G\H set of left cosets, right cosets mod H (H ⊆ G), p. 41

(G : 1) order of the group G, p. 41

(G : H) index of H in G, p. 41

ω(g) order of g ∈ G, p. 41

ε(G) exponent of the group G, p. 41

NG(H) normalizer of H, p. 41

i∈IGi free product of the family of groups (Gi), p. 42

F(I) free group on I, p. 42

 support of a family, p. 42

⊕i∈I Gi direct sum of abelian groups, p. 42

 normal subgroup of G, p. 42

H.K product of H, K ⊆ G , p. 43

Aut (G) automorphism group of G, p. 42

 center of G, p. 43

yx conjugate of y, p. 43

〈S〉 group generated by S, p. 46

ε(σ) signature of the permutation σ, p. 47

 alternating group, p. 47

 lattice of normal subgroups of G, p. 47

|G| order of the group G, p. 48

(h, k) commutator of the elements h, k, p. 48

(H, K) subgroup generated by the commutators (h,k), p. 48

G′,G(k) derived groups, pp. 48–49

Gab abelianization of G, p. 49

(Cn (G))n≥1 descending central series, p. 50

Gx stabilizer of x, p. 51

Rings and algebras (section 2.3)

Rop opposite ring, pp. 52–53

HomR (M, N) p. 53

0 module reduced to the 0 element, p. 53

[S]R R-module generated by S, p. 53

 left, right, two-sided ideal, p. 54

(S) two-sided ideal generated by S, p. 54

Lat (M) lattice of submodules of M , p. 55

AnnlR (m), AnnlR (M) annihilator, p. 55

 product of ideals, p. 56

Spm (R) maximal spectrum of R, p. 58

Spec (R) prime spectrum of R, p. 58

, p. 60

Char (K) characteristic of K, p. 61

L/K field extension, p. 62

d° (x) degree of the algebraic element x, p. 62

[L : K] dimension of a field extension, p. 62

 field of algebraic numbers, p. 62

End (V ) ring of endomorphisms of V , p. 63

 ring of square matrices, p. 63

rad (R) (Jacobson) radical, p. 65

 nilradical, p. 66

, p. 66

κR residue field of the local ring R, p. 67

 similarity, p. 68

 ring of entire functions, p. 71

θ Euclidean function, p. 72

K[X1,…, Xn] , K[(Xi)i∈I] ring of polynomials, p. 74

K [[X1, …, Xn]] , K[(Xi)i∈I] ring of formal power series, p. 75

ω(a) order of a formal power series, p. 76

 center of the algebra A, p. 77

K-Alg category of K-algebras, p. 77

K-Alga category of associative and unitary K-algebras, p. 77

A = K (xi)i∈I, p. 78

An (K) nth Weyl algebra over K, p. 78

K-Algc category of commutative K-algebras, p. 78

GLn (R) general linear group of degree n over R, p. 79

diag (a1, …, an) possibly rectangular diagonal matrix, p. 81

SLn (K) special linear group of degree n over K, p. 82

deg (x) degree of x in a graded algebra, p. 85

d derivation, antiderivation, p. 85

Additional concepts from linear algebra (section 3.1)

M* dual of M, p. 89

〈 − , − 〉 duality bracket, p. 89

tf transpose of f, p. 90

canM canonical homomorphism, p. 90

R ∈ AQ×(K) matrix of rows with finite support, p. 91

rkR (E) rank of the free R-module E, p. 92

R~R′ equivalent matrices, p. 92

~l, ~r left-equivalence, right-equivalence of matrices, p. 92

dimK (V) dimension of the K-vector space V, p. 93

A⊗R B tensor product, p. 97

s⊗t tensor product of linear mappings, p. 98

s⊗i∈I Ai skew tensor product, pp. 100–101

ρ*,ρ* extension, restriction of the ring of scalars, p. 102

-adic completion, pp. 107–108

S − 1A, AS − 1 ring of left fractions, right fractions, p. 110

 S-torsion submodule of M, p. 111

Q(A) field of fractions of A, p. 112

K ((Xi)i∈I) field of rational functions, p. 112

K ((X)) field of Laurent series, p. 112

 torsion submodule of M, p. 112

rkA (M) rank of an A-module M , p. 113

rk (f) rank of an A-homomorphism f , p. 113

K [X; σ,δ] , K[X;δ], K[X; σ] ring of skew polynomials, p. 119

K [Y, Y − 1; σ] ring of skew Laurent polynomials, p. 121

A1 (k), A1′ (k), B1 (k), p. 121

Notions of Commutative Algebra (section 3.2)

, p. 122

, p. 122

 zero set of A in K, p. 125

 Gelfand transform of x, p. 125

 ring of p-adic integers, p. 127

 field of p-adic numbers, p. 127

Γ, Γ∞, p. 127

Supp (M) support of the module M, p. 128

Ass (M) set of prime ideals associated with M, p. 129

dim (X) Krull dimension, p. 143

 affine space of dimension n over k, p. 146

 ideal of the algebraic set E, p. 146

Γ(E) algebra of regular functions over the algebraic set E, p. 148

AlSet category of algebraic sets, p. 148

Homological notions (section 3.3)

E (M) injective envelope of the module M, p. 153

C∞ (Ω) space of infinitely differentiable functions on Ω, p. 155

[x, y] interval with endpoints x, y, p. 163

 ring of integers of the number field K, p. 167

pd (M) , fd (M) , id (M) projective, flat, injective dimension, p. 170

gldA global (or homological) dimension, p. 171

coim (f) coimage, p. 176

dp codifferential, p. 184

Z (Cp), Bp (C•), Hp (C•) cycle, boundary, homology, p. 184

R-Comp, -Comp category of R-complexes, p. 185

dp differential, p. 188

Zp (C•), Bp (C•), Hp (C•) cocycle, coboundary, cohomology, p. 188

p E* space of p-forms over E, p. 194

Ωp (U) space of differential p-forms over U, p. 194

d exterior differential, p. 194

γ * δ juxtaposition of the two paths γ, δ, p. 195

γ~γ′ homotopic paths, p. 195

[γ] homotopy class of the path γ, p. 195

ϖ (X) Poincaré groupoid, p. 195

π1 (X, a), π1 (X) Poincaré group, p. 195

Toppc category of path-connected spaces, p. 195

EM left projective resolution, p. 198

 morphism of cochains over f, p. 198

 nth left derived , p. 199

TornR, p. 200

EM right projective resolution, p. 200

 nth right derived , p. 200

 nth right derived , p. 200

, p. 201

Modules over principal ideal domains and related notions (section 3.4)

Cf companion matrix, p. 215

 Jordan block, p. 220

1

Categories and Functors

Abstract

Category theory was introduced by S. Eilenberg and S. MacLane in an article published in 1945, which also axiomatized the notions of functor and natural transformation. MacLane wrote that the notion of category was defined in order to define the notion of functor, which was in turn defined in order to define the notion of natural transformation. The archetypal example of a natural (or canonical) isomorphism is the one that identifies finite-dimensional vector spaces with their biduals. The notion of structure gradually began to emerge at the end of the 19th Century, and was fully formalized in Éléments de mathématique by N. Bourbaki (vector space structures, topological spaces, etc.). A vector space, for example, is a set equipped with a vector space structure. This structuralist perspective adopted by Bourbaki is based on set theory. In the category theory, however, the objects of a category are not always sets, and consequently the morphisms are not always mappings. A functor that sends each object A in C and each morphism f : A → B .

Keywords

Bifunctor Hom; Categories; Concrete functors; Fibered products; Free objects, free functor; Functorial morphisms; Products and coproducts; Projective objects and injective objects; Structures; Universal arrows and Universal elements

Category theory was introduced by S. Eilenberg and S. MacLane in an article published in 1945 [EIL 45], which also axiomatized the notions of functor and natural transformation. MacLane [MCL 98, p. 18] wrote that the notion of category was defined in order to define the notion of functor, which was in turn defined in order to define the notion of natural transformation. The archetypal example of a natural (or canonical) isomorphism is the one that identifies finite-dimensional vector spaces with their biduals (see Theorem 3.12 and compare with Remark 3.13). The notion of structure gradually began to emerge at the end of the 19th Century, and was fully formalized in Éléments de mathématique by N. Bourbaki (vector space structures, topological spaces, etc.). A vector space, for example, is a set equipped with a vector space structure. This structuralist perspective adopted by Bourbaki is based on set theory. In the category theory, however, the objects of a category are not always sets, and consequently the morphisms are not always mappings. A functor that sends each object A in C and each morphism f : A → B .

1.1 Categories

1.1.1 General results about categories

(I) A category consists of:

;

whose elements are the morphisms (or arrows) from X to Y;

:

.

. The composition

   [1.1]

of two morphisms f and g is denoted by g f or g.f or gf. We call X the domain of f and Y the codomain. Composition is associative, and for each object X there exists a morphism idX : X → X such that, for each f : X → Y, f idX = f and idY f = f.

The opposite category and

is defined as

are the (f1, f2) such that f1, frespectively, with

.

(II) Examples. The objects of the category Set of sets are sets, and the morphisms of this category are mappings. As this example shows, it is always redundant to specify the objects of a category; however, it is necessary to state its morphisms. The morphisms of the category Mon of monoids (section 2.1.1(I)), the category Grp of groups (section 2.2), the category Rng of rings, the category KVec of left vector spaces over a division ring K, the category RMod of left R-modules over a ring R (section 2.3.1) and the category Top of topological spaces are respectively