Algebra: Chapter 0 (Graduate Studies in Mathematics)

Algebra: Chapter 0 Is A Self-contained Introduction To The Main Topics Of Algebra, Suitable For A First Sequence On The Subject At The Beginning Graduate Or Upper Undergraduate Level. The Primary Distinguishing Feature Of The Book, Compared To Standard Textbooks In Algebra, Is The Early Introduction Of Categories, Used As A Unifying Theme In The Presentation Of The Main Topics. A Second Feature Consists Of An Emphasis On Homological Algebra: Basic Notions On Complexes Are Presented As Soon As Modules Have Been Introduced, And An Extensive Last Chapter On Homological Algebra Can Form The Basis For A Follow-up Introductory Course On The Subject. Approximately 1,000 Exercises Both Provide Adequate Practice To Consolidate The Understanding Of The Main Body Of The Text And Offer The Opportunity To Explore Many Other Topics, Including Applications To Number Theory And Algebraic Geometry. This Will Allow Instructors To Adapt The Textbook To Their Specific Choice Of Topics And Provide The Independent Reader With A Richer Exposure To Algebra. Many Exercises Include Substantial Hints, And Navigation Of The Topics Is Facilitated By An Extensive Index And By Hundreds Of Cross-references.--jacket. Preliminaries: Set Theory And Categories -- Groups, First Encounter -- Rings And Modules -- Groups, Second Encounter -- Irreducibility And Factorization In Integral Domains -- Linear Algebra -- Fields -- Linear Algebra, Reprise -- Homological Algebra. Paolo Aluffi. Includes Index. Cover Half-Title Title page Contents Preface to the second printing Introduction Chapter I. Preliminaries: Set theory and categories 1. Naive set theory 1.1. Sets 1.2. Inclusion of sets 1.3. Operations between sets 1.4. Disjoint unions, products 1.5. Equivalence relations, partitions, quotients Exercises 2. Functions between sets 2.1. Definition 2.2. Examples: Multisets, indexed sets 2.3. Composition of functions 2.4. Injections, surjections, bijections 2.5. Injections, surjections, bijections: Second viewpoint 2.6. Monomorphisms and epimorphisms 2.7. Basic examples 2.8. Canonical decomposition 2.9. Clarification Exercises 3. Categories 3.1. Definition 3.2. Examples Exercises 4. Morphisms 4.1. Isomorphisms 4.2. Monomorphisms and epimorphisms Exercises 5. Universal properties 5.1. Initial and final objects 5.2. Universal properties 5.3. Quotients 5.4. Products 5.5. Coproducts Exercises Chapter II. Groups, first encounter 1. Definition of group 1.1. Groups and groupoids 1.2. Definition 1.3. Basic properties 1.4. Cancellation 1.5. Commutative groups 1.6. Order Exercises 2. Examples of groups 2.1. Symmetric groups 2.2. Dihedral groups 2.3. Cyclic groups and modular arithmetic Exercises 3. The category Grp 3.1. Group homomorphisms 3.2. Grp: Definition 3.3. Pause for reflection 3.4. Products et al. 3.5. Abelian groups Exercises 4. Group homomorphisms 4.1. Examples 4.2. Homomorphisms and order 4.3. Isomorphisms 4.4. Homomorphisms of abelian groups Exercises 5. Free groups 5.1. Motivation 5.2. Universal property 5.3. Concrete construction 5.4. Free abelian groups Exercises 6. Subgroups 6.1. Definition 6.2. Examples: Kernel and image 6.3. Example: Subgroup generated by a subset 6.4. Example: Subgroups of cyclic groups 6.5. Monomorphisms Exercises 7. Quotient groups 7.1. Normal subgroups 7.2. Quotient group 7.3. Cosets 7.4. Quotient by normal subgroups 7.6. kernel ⟺ normal Exercises 8. Canonical decomposition and Lagrange’s theorem 8.1. Canonical decomposition 8.2. Presentations 8.3. Subgroups of quotients 8.4. HK/H vs. K/(H∩K) 8.5. The index and Lagrange’s theorem 8.6. Epimorphisms and cokernels Exercises 9. Group actions 9.1. Actions 9.2. Actions on sets 9.3. Transitive actions and the category G-Set Exercises 10. Group objects in categories 10.1. Categorical viewpoint Exercises Chapter III. Rings and modules 1. Definition of ring 1.1. Definition 1.2. First examples and special classes of rings 1.3. Polynomial rings 1.4. Monoid rings Exercises 2. The category Ring 2.1. Ring homomorphisms 2.2. Universal property of polynomial rings 2.3. Monomorphisms and epimorphisms 2.4. Products 2.5. End_{Ab}(G) Exercises 3. Ideals and quotient rings 3.1. Ideals 3.2. Quotients 3.3. Canonical decomposition and consequences Exercises 4. Ideals and quotients: Remarks and examples. Prime and maximal ideals 4.1. Basic operations 4.2. Quotients of polynomial rings 4.3. Prime and maximal ideals Exercises 5. Modules over a ring 5.1. Definition of (left-)R-module 5.2. The category R-Mod 5.3. Submodules and quotients 5.4. Canonical decomposition and isomorphism theorems Exercises 6. Products, coproducts, etc., in R-Mod 6.1. Products and coproducts 6.2. Kernels and cokernels 6.3. Free modules and free algebras 6.4. Submodule generated by a subset; Noetherian modules 6.5. Finitely generated vs. finite type Exercises 7. Complexes and homology 7.1. Complexes and exact sequences 7.2. Split exact sequences 7.3. Homology and the snake lemma Exercises Chapter IV. Groups, second encounter 1. The conjugation action 1.1. Actions of groups on sets, reminder 1.2. Center, centralizer, conjugacy classes 1.3. The Class Formula 1.4. Conjugation of subsets and subgroups Exercises 2. The Sylow theorems 2.1. Cauchy’s theorem 2.2. Sylow I 2.3. Sylow II 2.4. Sylow III 2.5. Applications Exercises 3. Composition series and solvability 3.1. The Jordan-Hölder theorem 3.2. Composition factors; Schreier’s theorem 3.3. The commutator subgroup, derived series, and solvability Exercises 4. The symmetric group 4.1. Cycle notation 4.2. Type and conjugacy classes in Sn 4.3. Transpositions, parity, and the alternating group 4.4. Conjugacy in An; simplicity of An and solvability of Sn Exercises 5. Products of groups 5.1. The direct product 5.2. Exact sequences of groups; extension problem 5.3. Internal/semidirect products Exercises 6. Finite abelian groups 6.1. Classification of finite abelian groups 6.2. Invariant factors and elementary divisors 6.3. Application: Finite subgroups of multiplicative groups of fields Exercises Chapter V. Irreducibility and factorization in integral domains 1. Chain conditions and existence of factorizations 1.1. Noetherian rings revisited 1.2. Prime and irreducible elements 1.3. Factorization into irreducibles; domains with factorizations Exercises 2. UFDs, PIDs, Euclidean domains 2.1. Irreducible factors and greatest common divisor 2.2. Characterization of UFDs 2.3. PID ⟹ UFD 2.4. Euclidean domain ⟹ PID Exercises 3. Intermezzo: Zorn’s lemma 3.1. Set theory, reprise 3.2. Application: Existence of maximal ideals Exercises 4. Unique factorization in polynomial rings 4.1. Primitivity and content; Gauss’s lemma 4.2. The field of fractions of an integral domain 4.3. R UFD ⟹ R[x] UFD Exercises 5. Irreducibility of polynomials 5.1. Roots and reducibility 5.2. Adding roots; algebraically closed fields 5.3. Irreducibility in C[x], R[x], Q[x] 5.4. Eisenstein’s criterion Exercises 6. Further remarks and examples 6.1. Chinese remainder theorem 6.2. Gaussian integers 6.3. Fermat’s theorem on sums of squares Exercises Chapter VI. Linear algebra 1. Free modules revisited 1.1. R-Mod 1.2. Linear independence and bases 1.3. Vector spaces 1.4. Recovering B from FR(B) Exercises 2. Homomorphisms of free modules, I 2.1. Matrices 2.2. Change of basis 2.3. Elementary operations and Gaussian elimination 2.4. Gaussian elimination over Euclidean domains Exercises 3. Homomorphisms of free modules, II 3.1. Solving systems of linear equations 3.2. The determinant 3.3. Rank and nullity 3.4. Euler characteristic and the Grothendieck group Exercises 4. Presentations and resolutions 4.1. Torsion 4.2. Finitely presented modules and free resolutions 4.3. Reading a presentation Exercises 5. Classification of finitely generated modules over PIDs 5.1. Submodules of free modules 5.2. PIDs and resolutions 5.3. The classification theorem Exercises 6. Linear transformations of a free module 6.1. Endomorphisms and similarity 6.2. The characteristic and minimal polynomials of an endomorphism 6.3. Eigenvalues, eigenvectors, eigenspaces Exercises 7. Canonical forms 7.1. Linear transformations of free modules; actions of polynomial rings 7.2. k[t]-modules and the rational canonical form 7.3. Jordan canonical form 7.4. Diagonalizability Exercises Chapter VII. Fields 1. Field extensions, I 1.1. Basic definitions 1.2. Simple extensions 1.3. Finite and algebraic extensions Exercises 2. Algebraic closure, Nullstellensatz, and a little algebraic geometry 2.1. Algebraic closure 2.2. The Nullstellensatz 2.3. A little affine algebraic geometry Exercises 3. Geometric impossibilities 3.1. Constructions by straightedge and compass 3.2. Constructible numbers and quadratic extensions 3.3. Famous impossibilities Exercises 4. Field extensions, II 4.1. Splitting fields and normal extensions 4.2. Separable polynomials 4.3. Separable extensions and embeddings in algebraic closures Exercises 5. Field extensions, III 5.1. Finite fields 5.2. Cyclotomic polynomials and fields 5.3. Separability and simple extensions Exercises 6. A little Galois theory 6.1. The Galois correspondence and Galois extensions 6.2. The fundamental theorem of Galois theory, I 6.3. The fundamental theorem of Galois theory, II 6.4. Further remarks and examples Exercises 7. Short march through applications of Galois theory 7.1. Fundamental theorem of algebra 7.2. Constructibility of regular n-gons 7.3. Fundamental theorem on symmetric functions 7.4. Solvability of polynomial equations by radicals 7.5. Galois groups of polynomials 7.6. Abelian groups as Galois groups over Q Exercises Chapter VIII. Linear algebra, reprise 1. Preliminaries, reprise 1.1. Functors 1.2. Examples of functors 1.3. When are two categories ‘equivalent’? 1.4. Limits and colimits 1.5. Comparing functors Exercises 2. Tensor products and the Tor functors 2.1. Bilinear maps and the definition of tensor product 2.2. Adjunction with Hom and explicit computations 2.3. Exactness properties of tensor; flatness 2.4. The Tor functors Exercises 3. Base change 3.1. Balanced maps 3.2. Bimodules; adjunction again 3.3. Restriction and extension of scalars Exercises 4. Multilinear algebra 4.1. Multilinear, symmetric, alternating maps 4.2. Symmetric and exterior powers 4.3. Very small detour: Graded algebra 4.4. Tensor algebras Exercises 5. Hom and duals 5.1. Adjunction again 5.2. Dual modules 5.3. Duals of free modules 5.4. Duality and exactness 5.5. Duals and matrices; biduality 5.6. Duality on vector spaces Exercises 6. Projective and injective modules and the Ext functors 6.1. Projectives and injectives 6.2. Projective modules 6.3. Injective modules 6.4. The Ext functors 6.5. Ext_Z*(G, Z) Exercises Chapter IX. Homological algebra 1. (Un)necessary categorical preliminaries 1.1. Undesirable features of otherwise reasonable categories 1.2. Additive categories 1.3. Abelian categories 1.4. Products, coproducts, and direct sums 1.5. Images; canonical decomposition of morphisms Exercises 2. Working in abelian categories 2.1. Exactness in abelian categories 2.2. The snake lemma, again 2.3. Working with ‘elements’ in a small abelian category 2.4. What is missing? Exercises 3. Complexes and homology, again 3.1. Reminder of basic definitions; general strategy 3.2. The category of complexes 3.3. The long exact cohomology sequence 3.4. Triangles Exercises 4. Cones and homotopies 4.1. The mapping cone of a morphism 4.2. Quasi-isomorphisms and derived categories 4.3. Homotopy Exercises 5. The homotopic category. Complexes of projectives and injectives 5.1. Homotopic maps are identified in the derived category 5.2. Definition of the homotopic category of complexes 5.3. Complexes of projective and injective objects 5.4. Homotopy equivalences vs. quasi-isomorphisms in K(A) 5.5. Proof of Theorem 5.9 Exercises 6. Projective and injective resolutions and the derived category 6.1. Recovering A 6.2. From objects to complexes 6.3. Poor man’s derived category Exercises 7. Derived functors 7.1. Viewpoint shift 7.2. Universal property of the derived functor 7.3. Taking cohomology 7.4. Long exact sequence of derived functors 7.5. Relating F, LiF, RiF 7.6. Example: A little group cohomology Exercises 8. Double complexes 8.1. Resolution by acyclic objects 8.2. Complexes of complexes 8.3. Exactness of the total complex 8.4. Total complexes and resolutions 8.5. Acyclic resolutions again and balancing Tor and Ext Exercises 9. Further topics 9.1. Derived categories 9.2. Triangulated categories 9.3. Spectral sequences Exercises Index Back Cover