Graduate Studies in Mathematics
Volume: 183;
2017;
637 pp;
Hardcover
MSC: Primary 16; 15; 13; 14;
Print ISBN: 978-1-4704-3770-1
Product Code: GSM/183
List Price: $94.00
AMS Member Price: $75.20
MAA Member Price: $84.60
Electronic ISBN: 978-1-4704-4230-9
Product Code: GSM/183.E
List Price: $94.00
AMS Member Price: $75.20
MAA Member Price: $84.60
You may also like
Supplemental Materials
Separable Algebras
Share this pageTimothy J. Ford
This book presents a comprehensive introduction to the theory of
separable algebras over commutative rings. After a thorough
introduction to the general theory, the fundamental roles played by
separable algebras are explored. For example, Azumaya algebras, the
henselization of local rings, and Galois theory are rigorously
introduced and treated. Interwoven throughout these applications is
the important notion of étale algebras. Essential connections are
drawn between the theory of separable algebras and Morita theory, the
theory of faithfully flat descent, cohomology, derivations,
differentials, reflexive lattices, maximal orders, and class groups.
The text is accessible to graduate students who have finished a first
course in algebra, and it includes necessary foundational material,
useful exercises, and many nontrivial examples.
Readership
Graduate students and researchers interested in algebra.
Reviews & Endorsements
The thorough and comprehensive treatment of separable, Azumaya, and tale algebras, Hensel rings, the Galois theory of rings, and Galois cohomology of rings makes the book under review an indispensable reference for the graduate student interested in these topics. As an added bonus, the book comes with a rich, 155 item, bibliography, well-chosen examples, calculations, and sets of exercises in each chapter, which makes this book an excellent textbook for self-study or for a topics course on separable algebras.
-- Felipe Zaldivar, MAA Reviews
Table of Contents
Table of Contents
Separable Algebras
Table of Contents pages: 1 2
- Cover Cover11
- Title page iii4
- Contents vii8
- Preface xv16
- Chapter 1. Background Material on Rings and Modules 124
- Chapter 2. Modules over Commutative Rings 5376
- Chapter 3. The Wedderburn-Artin Theorem 91114
- Chapter 4. Separable Algebras, Definition and First Properties 115138
- 1. Separable Algebra, the Definition 115138
- 2. Examples of Separable Algebras 120143
- 3. Separable Algebras Under a Change of Base Ring 123146
- 4. Homomorphisms of Separable Algebras 127150
- 5. Separable Algebras over a Field 137160
- 6. Commutative Separable Algebras 146169
- 7. Formally Unramified, Smooth and Étale Algebras 155178
- Chapter 5. Background Material on Homological Algebra 159182
- Chapter 6. The Divisor Class Group 199222
- Chapter 7. Azumaya Algebras, I 243266
- 1. First Properties of Azumaya Algebras 243266
- 2. The Commutator Theorems 249272
- 3. The Brauer Group 252275
- 4. Splitting Rings 254277
- 5. Azumaya Algebras over a Field 258281
- 6. Azumaya Algebras up to Brauer Equivalence 263286
- 7. Noetherian Reduction of Azumaya Algebras 267290
- 8. The Picard Group of Invertible Bimodules 274297
- 9. The Brauer Group Modulo an Ideal 281304
- Chapter 8. Derivations, Differentials and Separability 287310
- 1. Derivations and Separability 287310
- 1.1. The Definition and First Results 287310
- 1.2. A Noncommutative Binomial Theorem in Characteristic 𝑝 291314
- 1.3. Extensions of Derivations 292315
- 1.4. Exercises 294317
- 1.5. More Tests for Separability 296319
- 1.6. Locally of Finite Type is Finitely Generated as an Algebra 301324
- 1.7. Exercises 301324
- 2. Differential Crossed Product Algebras 303326
- 3. Differentials and Separability 308331
- 4. Separably Generated Extension Fields 317340
- 5. Tests for Regularity 325348
- Chapter 9. Étale Algebras 329352
- Chapter 10. Henselization and Splitting Rings 367390
- Chapter 11. Azumaya Algebras, II 407430
Table of Contents pages: 1 2