This book is an English translation of an entirely revised version of the 1958 edition of the eighth chapter of the book Algebra, the second Book of the Elements of Mathematics. It is devoted to the study of certain classes of rings and of modules, in particular to the notions of Noetherian or Artinian modules and rings, as well as that of radical. This chapter studies Morita equivalence of module and algebras, it describes the structure of semisimple rings. Various Grothendieck groups are defined that play a universal role for module invariants.The chapter also presents two particular cases of algebras over a field. The theory of central simple algebras is discussed in detail; their classification involves the Brauer group, of which severaldescriptions are given. Finally, the chapter considers group algebras and applies the general theory to representations of finite groups. At the end of the volume, a historical note taken from the previous edition recounts the evolution of many of the developed notions.
Author(s): N. Bourbaki
Publisher: Springer Nature
Year: 2023
Language: English
Pages: 505
To the Reader
CONTENTS
INTRODUCTION
CHAPTER VIII. Semisimple Modules and Rings
§ 1. ARTINIAN MODULES AND NOETHERIAN MODULES
1. Artinian Modules and Noetherian Modules
2. Artinian Rings and Noetherian Rings
3. Countermodule
4. Polynomials with Coefficients in a Noetherian Ring
Exercises
§ 2. THE STRUCTURE OF MODULES OF FINITE LENGTH
1. Local Rings
2. Weyr–Fitting Decomposition
3. Indecomposable Modules and Primordial Modules
4. Semiprimordial Modules
5. The Structure of Modules of Finite Length
Exercises
§ 3. SIMPLE MODULES
1. Simple Modules
2. Schur’s Lemma
3. Maximal Submodules
4. Simple Modules over an Artinian Ring
5. Classes of Simple Modules
Exercises
§ 4. SEMISIMPLE MODULES
1. Semisimple Modules
2. The homomorphism sum of homomorphisms
3. Some Operations on Modules
4. Isotypical Modules
5. Description of an Isotypical Module
6. Isotypical Components of a Module
7. Description of a Semisimple Module
8. Multiplicities and Lengths in Semisimple Modules
Exercises
§ 5. COMMUTATION
1. The Commutant and Bicommutant of a Module
2. Generating Modules
3. The Bicommutant of a Generating Module
4. The Countermodule of a Semisimple Module
5. Density Theorem
6. Application to Field Theory
Exercises
§ 6. MORITA EQUIVALENCE OF MODULES AND ALGEBRAS
1. Commutant and Duality
2. Generating Modules and Finitely Generated Projective Modules
3. Invertible Bimodules and Morita Equivalence
4. The Morita Correspondence of Modules
5. Ordered Sets of Submodules
6. Other Properties Preserved by the Morita Correspondence
7. Morita Equivalence of Algebras
Exercises
§ 7. SIMPLE RINGS
1. Simple Rings
2. Modules over a Simple Ring
3. Degrees
4. Ideals of Simple Rings
Exercises
§ 8. SEMISIMPLE RINGS
1. Semisimple Rings
2. Modules over a Semisimple Ring
3. Factors of a Semisimple Ring
4. Idempotents and Semisimple Rings
Exercises
§ 9. RADICAL
1. The Radical of a Module
2. The Radical of a Ring
3. Nakayama’s Lemma
4. Lifts of Idempotents
5. Projective Cover of a Module
Exercises
§ 10. MODULES OVER AN ARTINIAN RING
1. The Radical of an Artinian Ring
2. Modules over an Artinian Ring
3. Projective Modules over an Artinian Ring
Exercises
§ 11. GROTHENDIECK GROUPS Modules
1. Additive Functions of Modules
2. The Grothendieck Group of an Additive Set of Modules
3. Using Composition Series
4. The Grothendieck Group R(A)
5. Change of Rings
6. The Grothendieck Group R(A)
7. Multiplicative Structure on K(C)
8. The Grothendieck Group K(A)
9. The Grothendieck Group K(A) of an Artinian Ring
10. Change of Rings for K(A)
11. Frobenius Reciprocity
12. The Case of Simple Rings
Exercises
§ 12. TENSOR PRODUCTS OF SEMISIMPLE MODULES
1. Semisimple Modules over Tensor Products of Algebras
2. Tensor Products of Simple Modules
3. Tensor Products of Semisimple Commutative Algebras
4. The Radical of a Tensor Product of Algebras
5. Tensor Products of Semisimple Modules
6. Tensor Products of Semisimple Algebras
7. Extension of Scalars in Semisimple Modules
Exercises
§ 13. ABSOLUTELY SEMISIMPLE ALGEBRAS
1. Absolutely Semisimple Modules
2. Algebras over Separably Closed Fields
3. Absolutely Semisimple Algebras
4. Characterization of Absolutely Semisimple Modules
5. Derivations on Semisimple Algebras
6. Cohomology of Algebras
7. Cohomology of Absolutely Semisimple Algebras
8. The Splitting of Artinian Algebras
Exercises
§ 14. CENTRAL SIMPLE ALGEBRAS
1. Central Simple Algebras
2. Two Lemmas on Bimodules
3. Conjugacy Theorems
4. Automorphisms of Semisimple Algebras
5. Simple Subalgebras of Simple Algebras
6. Maximal Commutative Subalgebras
7. Maximal Étale Subalgebras
8. Diagonalizable Subalgebras of Simple Algebras
Exercises
§ 15. BRAUER GROUPS
1. Classes of Algebras
2. Definition of the Brauer Group
3. Change of Base Field
4. Examples of Brauer Groups
Exercises
§ 16. OTHER DESCRIPTIONS OF THE BRAUER GROUP
1. τ-Extensions of Groups
2. Inverse Image of a τ-Extension
3. Direct Image of a τ-Extension
4. Group Law on the Classes of τ-Extensions
5. Cohomological Description
6. Restriction and Corestriction
7. Galois Algebras
8. Actions on Galois Algebras
9. Cross Products
10. Application to the Brauer Group
11. Index and Exponent
Exercises
§ 17. REDUCED NORMS AND TRACES
1. Complements on Characteristic Polynomials
2. Reduced Norms and Traces
3. Properties of Reduced Norms and Traces
4. The Reduced Norm is a Polynomial Function
5. Transitivity of Reduced Norms and Traces
6. Reduced Norms and Determinants
Exercises
§ 18. SIMPLE ALGEBRAS OVER A FINITE FIELD
1. Polynomials over a Finite Field
2. Simple Algebras over Finite Fields
Exercises
§ 19. QUATERNION ALGEBRAS
1. General Properties of Quaternion Algebras
2. The Center of a Quaternion Algebra
3. Simplicity of Quaternion Algebras
4. Criteria for a Quaternion Algebra to Be a Field
5. Algebras over Maximal Ordered Fields
Exercises
§ 20. LINEAR REPRESENTATIONS OF ALGEBRAS
1. Linear Representations of Algebras
2. Restricted Dual of an Algebra
3. Coefficients of a Module
4. Restricted Dual and Matrix Coefficients
5. Dual of a Semisimple Algebra
6. Character of a Representation
7. Coefficients of a Set of Classes of Modules
8. Cogebra Structure on the Restricted Dual
Exercises
§ 21. LINEAR REPRESENTATIONS OF FINITE GROUPS
1. Linear Representations
2. Maschke’s Theorem
3. Induced and Coinduced Representations
4. Representations and the Grothendieck Group
5. Fourier Inversion Formula
6. Schur Orthogonality Relations
7. Orthogonality Relation for Characters
8. Central Functions on a Finite Group
9. The Case of Abelian Groups
10. Characters and Grothendieck Groups
11. Dimension of Simple Representations
12. Change of Base Field
13. Complex Linear Representations
Exercises
APPENDIX 1 ALGEBRAS WITHOUT UNIT ELEMENT
1. Regular Ideals
2. Adjunction of a Unit Element
3. The Radical of an Algebra
4. Density Theorem
Exercises
APPENDIX 2 DETERMINANTS OVER A NONCOMMUTATIVE FIELD
1. A Generalization of Alternating Multilinear Forms
2. A Uniqueness Theorem
3. Determinant of an Automorphism
4. Determinant of a Square Matrix
5. The Unimodular Group
Exercises
APPENDIX 3 HILBERT’S NULLSTELLENSATZ
APPENDIX 4 TRACE OF AN ENDOMORPHISM OF FINITE RANK
1. Linear Mappings of Finite Rank
2. Trace of an Endomorphism of Finite Rank
Exercises
HISTORICAL NOTE
BIBLIOGRAPHY
NOTATION INDEX
TERMINOLOGY INDEX
