Kooperativer Bibliotheksverbund

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Format: 1 online resource (528 pages)

Edition: First edition.

ISBN: 3-031-50062-8

Note: Intro -- Preface -- Contents -- Chapter 1 Prerequisites and Preliminaries -- 1.1 Prerequisites -- 1.1.1 Groups: -components of elements -- 1.1.2 Nilpotent groups -- 1.1.3 Complements on Sylow subgroups -- 1.1.4 Solvability and the Schur-Zassenhaus Theorem -- 1.2 Preliminary: the Language of Categories -- 1.2.1 What is a category? -- 1.2.2 First examples -- 1.2.3 Monomorphisms, epimorphisms -- 1.2.4 Functors -- 1.2.5 Examples -- 1.2.6 Yet another example: presheaves -- 1.2.7 Faithful and full functors -- 1.2.8 Morphisms of functors -- 1.2.9 Isomorphisms of functors -- 1.2.10 Morphisms of functors: Yoneda's Lemma -- 1.2.11 Universals -- 1.2.12 Adjoint functors and adjunctions -- 1.2.13 Equivalences of categories -- 1.2.14 Complements: Going on and on -- 1.2.15 Horizontal composition -- Part I Rings and Modules -- Chapter 2 Rings, Polynomials, Divisibility -- 2.1 Rings, Morphisms, Modules -- 2.1.1 Morphisms -- 2.1.2 Subrings -- 2.1.3 Endomorphisms of abelian groups and modules -- 2.2 Polynomials and Power Series -- 2.2.1 Generalities -- 2.2.2 Infinite sums and products -- 2.2.3 Derivatives -- 2.2.4 Logarithm and exponential -- Definition 2.2.14 -- Proposition 2.2.15 -- Proposition 2.2.16 -- Remark 2.2.17 -- 2.2.5 Euclidean division -- Proposition 2.2.18 (Euclidean division by a monic polynomial) -- Remarks 2.2.19 -- Corollary 2.2.20 -- Corollary 2.2.21 -- Exercise 2.2.22 -- Remark 2.2.23 -- Theorem 2.2.24 (Cayley-Hamilton) -- Lemma 2.2.25 -- Exercise 2.2.26 -- 2.3 Canonical Morphisms -- 2.3.1 Prime ring and characteristic -- 2.3.2 Universal property of the polynomial ring -- 2.4 Ideals -- 2.4.1 Left, right, two-sided ideals -- 2.4.2 Ideals and morphisms -- 2.4.3 Chinese Remainder Theorem -- 2.5 Factorial Domains, Principal Ideal Domains, Euclidean Domains -- 2.5.1 Divisors and irreducible elements -- 2.5.2 Factorial domains -- Definition 2.5.4. , Remark 2.5.5 -- Lemma 2.5.6 (Gauß' Lemma) -- Example 2.5.7 -- Proposition-Definition 2.5.8 -- Lemma 2.5.9 -- 2.5.3 Principal ideal domains -- Definition 2.5.10 -- Proposition 2.5.11 -- Exercise 2.5.12 -- Lemma 2.5.13 -- Remark 2.5.14 -- 2.5.4 Euclidean rings -- Definition and first properties Definition 2.5.15 -- Examples 2.5.16 -- Remark 2.5.17 -- Exercise 2.5.18 -- Proposition 2.5.19 -- Lemma 2.5.20 -- Remark 2.5.21 -- Complement without proofs: quadratic extensions of -- Theorem 2.5.22 -- Remark 2.5.23 -- Theorem 2.5.24 -- About Euclidean rings: Euclid's algorithm -- 2.5.5 Case of -- and application -- Remark 2.5.25 -- Theorem 2.5.26 -- Example 2.5.27 -- Exercise 2.5.28 -- 2.6 Roots of Unity, Cyclotomic Polynomials -- 2.7 More Exercises -- Chapter 3 Polynomial Rings in Several Indeterminates -- 3.1 Universal Property, Substitutions -- 3.1.1 First particular case -- 3.1.2 Second particular case: evaluation function -- 3.1.3 Third particular case: substitution -- 3.1.4 Fourth particular case: specialization -- 3.2 Symmetric Polynomials -- 3.2.1 Definition and fundamental theorem -- 3.2.2 Newton formulae -- 3.2.3 Symmetric fractions -- 3.2.4 Antisymmetric polynomials -- 3.3 Resultant and Discriminant -- 3.3.1 Resultant of two polynomials -- 3.3.2 First properties -- 3.3.3 Resultant and roots -- 3.3.4 A geometric application -- 3.3.5 Discriminant -- 3.4 More Exercises -- Chapter 4 More on Modules -- 4.1 Several Equivalent Definitions -- 4.1.1 Two definitions of "module" -- 4.1.2 Morphisms -- 4.2 Submodules -- 4.2.1 Generalities -- 4.2.2 Direct sums -- 4.2.3 Quotients -- 4.2.4 Kernels, images, cokernels, coimages -- 4.2.5 Exact sequences -- 4.2.6 Ideals and modules -- 4.3 Torsion Elements, Torsion Submodule -- 4.3.1 Cyclic modules -- 4.3.2 Torsion and torsion free elements -- 4.4 Free and Generating Systems, Free Modules. , 4.4.1 Free systems, generating systems, bases -- 4.4.2 A property of free modules -- 4.4.3 Projective modules -- 4.5 Sums and Products -- 4.5.1 Direct sums (coproducts) and products -- 4.5.2 Split exact sequences -- 4.6 -Linear and Abelian -Linear Categories -- 4.6.1 Initial, terminal, null objects -- 4.6.2 -linear categories -- 4.6.3 -linear functors -- 4.6.4 An example: stable category -- 4.6.5 Kernels and cokernels -- 4.6.6 Canonical decomposition of a morphism -- 4.6.7 A bunch of definitions for abelian categories -- 4.6.8 Grothendieck group -- 4.6.9 Functors between abelian -linear categories -- 4.7 More Exercises -- 4.8 Tensor Products -- 4.8.1 Definition of the tensor product -- 4.8.2 Functoriality and other properties of the tensor product -- 4.8.3 Exact sequences, Hom and ⊗ -- 4.8.4 Tensor product and duality -- 4.8.5 Extension of scalars -- 4.8.6 Extending scalars for an algebra -- 4.8.7 Trace and restriction of scalars -- 4.8.8 Complement: Kronecker product of matrices -- 4.9 Tensor, Symmetric and Exterior Algebras -- 4.9.1 Symmetric and alternating squares -- 4.9.2 Tensor algebra -- 4.9.3 Symmetric algebra -- 4.9.4 Exterior algebra -- 4.10 More on Algebras -- 4.10.1 Generalities about algebras -- 4.10.2 Left and right modules -- 4.10.3 Tensor product of left with right modules -- 4.10.4 Tensor product and bimodules -- Exercise 4.10.14 -- Proposition 4.10.15 -- 4.10.5 A famous adjunction -- Proposition 4.10.16 -- mod -- Exercises 4.10.17 -- mod -- 4.11 Modules Over a Matrix Algebra -- 4.11.1 An equivalence of categories -- 4.11.2 An application: the Skolem-Noether theorem -- 4.12 More Exercises -- Chapter 5 On Representations of Finite Groups -- 5.1 Generalities on Representations -- 5.1.1 Introduction -- 5.1.2 Representations on a category -- 5.2 Set-Representations -- 5.2.1 Union and product -- 5.2.2 Transitive representations. , 5.2.3 Classification of transitive representations -- 5.2.4 Burnside's marks -- 5.2.5 Induction and restriction -- 5.2.6 Generalized transfer -- 5.3 Linear Representations -- 5.3.1 Generalities -- 5.3.2 The group algebra -- 5.3.3 Induction and restriction for finite group algebras -- 5.3.3.1 Restriction -- 5.3.3.2 Induction Definition -- 5.3.3.3 Universal property of induction -- 5.3.3.4 Induction and tensor product -- 5.3.3.5 Another definition of induction -- 5.3.4 Mackey's formula -- 5.3.5 Trace on induced modules -- 5.3.6 Generalized tensor induction -- 5.3.7 Complement: fixed and cofixed points -- 5.4 Projective Representations, Twisted Group Algebras -- 5.4.1 Preliminary: fragments on cohomology -- 5.4.2 Projective representations, ×-groups, twisted group algebras -- 5.4.3 Above a stable module for a normal subgroup -- 5.5 More Exercises -- Part II Integral Domains, Polynomials, Fields -- Chapter 6 Prime and Maximal Ideals, Integral Domains -- 6.1 Definition and First Examples -- 6.2 Examples in Polynomial Rings -- 6.2.1 Generalities -- 6.2.2 Example of maximal ideals of Z[ ] -- 6.3 Nilradical and Radical -- 6.3.1 Characterizations -- 6.3.2 Local rings -- 6.3.3 Finite-dimensional algebras over a field -- 6.4 Integral Domains, Fields of Fractions -- 6.4.1 Construction of field of fractions -- 6.4.2 Universal property of the field of fractions -- 6.5 Localizations -- 6.5.1 Localizations on rings -- 6.5.2 Localizations on modules -- 6.5.3 Local properties of modules -- 6.5.4 On localization and projectivity -- 6.6 Irreducibility Criteria in [ ] -- 6.6.1 Primitive and irreducible polynomials -- 6.6.2 Reduction modulo a prime ideal -- 6.6.3 Case of [ ] for factorial -- 6.6.4 Content and primitive part -- 6.6.5 Example-Exercise: the decimal numbers -- 6.6.6 An application: automorphisms of ( ) -- 6.6.7 Eisenstein criterion. , 6.6.8 More on irreducible elements in ( -- 6.7 Transfer Properties -- 6.7.1 Transfer of some properties to polynomial rings -- 6.7.2 Yet another proof of the Cayley-Hamilton theorem -- 6.8 More Exercises -- Chapter 7 Fields, Division Rings -- 7.1 Finite Subgroups of the Multiplicative Group of a Field -- 7.2 Algebraic Extensions -- 7.2.1 First properties -- 7.2.2 Algebraic closure -- 7.3 Splitting Polynomials, Normal Extensions -- 7.4 Separable Polynomials, Separable Extensions -- 7.5 Norm and Traces for Normal Separable Extensions -- 7.6 Short Introduction to Galois Theory -- 7.6.1 Quick overview -- 7.6.2 The Galois group as a permutation group -- 7.6.3 The generic equation -- 7.7 Finite Fields -- 7.8 Quaternions -- 7.8.1 Rings of quaternions -- 7.8.2 The quaternion group of order 8 -- 7.9 More Exercises -- Part III Finitely Generated Modules -- Chapter 8 Integrality, Noetherianity -- 8.1 Integrality Over a Ring -- 8.1.1 Definition and characterization -- 8.1.2 Integral extensions -- 8.1.3 Integrality and localization -- 8.1.4 Integral closure and field extensions -- 8.2 Complement: Jacobson Rings, Hilbert's Nullstellensatz -- 8.2.1 On maximal ideals of polynomial algebras -- 8.2.2 Application to algebraic varieties -- 8.3 Noetherian Rings and Modules -- 8.3.1 Noetherian modules -- 8.3.2 Noetherian rings -- 8.3.3 Hilbert's Basis Theorem -- 8.3.4 Localization over Noetherian rings -- 8.3.5 More exercises -- Chapter 9 Finitely Generated Projective Modules -- 9.1 Rank and Basis of a Finitely Generated Free Module -- 9.1.1 Rank: another proof -- 9.1.2 The dual of a free module of finite rank -- 9.1.3 About finitely generated torsion-free modules -- 9.2 Finitely Generated Projective Modules -- 9.2.1 Characterization, dual -- 9.2.2 Projective morphisms -- 9.2.3 A series of characterizations -- 9.2.4 The case of local rings -- 9.3 More Exercises. , Chapter 10 Finitely Generated Modules Over Dedekind Domains.

Additional Edition: Print version: Broué, Michel From Rings and Modules to Hopf Algebras Cham : Springer International Publishing AG,c2024 ISBN 9783031500619

Language: English