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Separable Algebras [Hardback]

  • Formāts: Hardback, 664 pages, height x width: 254x178 mm, weight: 1280 g
  • Sērija : Graduate Studies in Mathematics
  • Izdošanas datums: 30-Sep-2017
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470437708
  • ISBN-13: 9781470437701
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Separable AlgebrasPalielināt
  • Formāts: Hardback, 664 pages, height x width: 254x178 mm, weight: 1280 g
  • Sērija : Graduate Studies in Mathematics
  • Izdošanas datums: 30-Sep-2017
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470437708
  • ISBN-13: 9781470437701
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781470437701.html
Keywords:
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 etale 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.
Preface xv
Chapter 1 Background Material on Rings and Modules
1(52)
§1 Rings and Modules
1(13)
1.1 Categories and Functors
2(3)
1.2 Progenerator Modules
5(3)
1.3 Exercises
8(1)
1.4 Nakayama's Lemma
9(2)
1.5 Exercise
11(1)
1.6 Module Direct Summands of Rings
11(2)
1.7 Exercises
13(1)
§2 Polynomial Functions
14(9)
2.1 The Ring of Polynomial Functions on a Module
14(1)
2.2 Resultant of Two Polynomials
15(4)
2.3 Polynomial Functions on an Algebraic Curve
19(2)
2.4 Exercises
21(2)
§3 Horn and Tensor
23(14)
3.1 Tensor Product
23(4)
3.2 Exercises
27(2)
3.3 Horn Groups
29(4)
3.4 Horn Tensor Relations
33(3)
3.5 Exercises
36(1)
§4 Direct Limit and Inverse Limit
37(10)
4.1 The Direct Limit
38(2)
4.2 The Inverse Limit
40(2)
4.3 Inverse Systems Indexed by Nonnegative Integers
42(3)
4.4 Exercises
45(2)
§5 The Morita Theorems
47(6)
5.1 Exercises
52(1)
Chapter 2 Modules over Commutative Rings
53(38)
§1 Localization of Modules and Rings
53(5)
1.1 Local to Global Lemmas
54(3)
1.2 Exercises
57(1)
§2 The Prime Spectrum of a Commutative Ring
58(7)
2.1 Exercises
63(2)
§3 Finitely Generated Projective Modules
65(5)
3.1 Exercises
68(2)
§4 Faithfully Flat Modules and Algebras
70(5)
4.1 Exercises
74(1)
§5 Chain Conditions
75(5)
5.1 Exercises
78(2)
§6 Faithfully Flat Base Change
80(11)
6.1 Fundamental Theorem on Faithfully Flat Base Change
80(3)
6.2 Locally Free Finite Rank is Finitely Generated Projective
83(2)
6.3 Invertible Modules and the Picard Group
85(3)
6.4 Exercises
88(3)
Chapter 3 The Wedderburn-Artin Theorem
91(24)
§1 The Jacobson Radical and Nakayama's Lemma
91(3)
1.1 Exercises
94(1)
§2 Semisimple Modules and Semisimple Rings
94(9)
2.1 Simple Rings and the Wedderburn-Artin Theorem
97(3)
2.2 Commutative Artinian Rings
100(2)
2.3 Exercises
102(1)
§3 Integral Extensions
103(3)
§4 Completion of a Linear Topological Module
106(9)
4.1 Graded Rings and Graded Modules
110(2)
4.2 Lifting of Idempotents
112(3)
Chapter 4 Separable Algebras, Definition and First Properties
115(44)
§1 Separable Algebra, the Definition
115(5)
1.1 Exercises
119(1)
§2 Examples of Separable Algebras
120(3)
§3 Separable Algebras Under a Change of Base Ring
123(4)
§4 Homomorphisms of Separable Algebras
127(10)
4.1 Exercises
133(4)
§5 Separable Algebras over a Field
137(9)
5.1 Central Simple Equals Central Separable
137(3)
5.2 Unique Decomposition Theorems
140(3)
5.3 The Skolem-Noether Theorem
143(1)
5.4 Exercises
144(2)
§6 Commutative Separable Algebras
146(9)
6.1 Separable Extensions of Commutative Rings
146(2)
6.2 Separability and the Trace
148(4)
6.3 Twisted Form of the Trivial Extension
152(1)
6.4 Exercises
153(2)
§7 Formally Unramified, Smooth and Etale Algebras
155(4)
Chapter 5 Background Material on Homological Algebra
159(40)
§1 Group Cohomology
159(14)
1.1 Cocycle and Coboundary Groups in Low Degree
161(2)
1.2 Applications and Computations
163(7)
1.3 Exercises
170(3)
§2 The Tensor Algebra of a Module
173(4)
2.1 Exercises
176(1)
§3 Theory of Faithfully Flat Descent
177(11)
3.1 The Amitsur Complex
177(1)
3.2 The Descent of Elements
178(2)
3.3 Descent of Homomorphisms
180(1)
3.4 Descent of Modules
181(5)
3.5 Descent of Algebras
186(2)
§4 Hochschild Cohomology
188(3)
§5 Amitsur Cohomology
191(8)
5.1 The Definition and First Properties
191(4)
5.2 Twisted Forms
195(4)
Chapter 6 The Divisor Class Group
199(44)
§1 Background Results from Commutative Algebra
200(6)
1.1 Krull Dimension
200(1)
1.2 The Serre Criteria for Normality
201(1)
1.3 The Hilbert-Serre Criterion for Regularity
202(2)
1.4 Discrete Valuation Rings
204(2)
§2 The Class Group of Weil Divisors
206(7)
2.1 Exercises
210(3)
§3 Lattices
213(13)
3.1 Definition and First Properties
213(3)
3.2 Reflexive Lattices
216(6)
3.3 A Local to Global Theorem for Reflexive Lattices
222(2)
3.4 Exercises
224(2)
§4 The Ideal Class Group
226(7)
4.1 Exercises
232(1)
§5 Functorial Properties of the Class Group
233(10)
5.1 Flat Extensions
233(2)
5.2 Finite Extensions
235(1)
5.3 Galois Descent of Divisor Classes
236(2)
5.4 The Class Group of a Regular Domain
238(4)
5.5 Exercises
242(1)
Chapter 7 Azumaya Algebras, I
243(44)
§1 First Properties of Azumaya Algebras
243(6)
§2 The Commutator Theorems
249(3)
§3 The Brauer Group
252(2)
§4 Splitting Rings
254(4)
4.1 Exercises
258(1)
§5 Azumaya Algebras over a Field
258(5)
§6 Azumaya Algebras up to Brauer Equivalence
263(4)
6.1 Exercises
266(1)
§7 Noetherian Reduction of Azumaya Algebras
267(7)
7.1 Exercises
273(1)
§8 The Picard Group of Invertible Bimodules
274(7)
8.1 Definition of the Picard Group
274(5)
8.2 The Skolem-Noether Theorem
279(1)
8.3 Exercise
280(1)
§9 The Brauer Group Modulo an Ideal
281(6)
9.1 Lifting Azumaya Algebras
284(2)
9.2 The Brauer Group of a Commutative Artinian Ring
286(1)
Chapter 8 Derivations, Differentials and Separability
287(42)
§1 Derivations and Separability
287(16)
1.1 The Definition and First Results
287(4)
1.2 A Noncommutative Binomial Theorem in Characteristic p
291(1)
1.3 Extensions of Derivations
292(2)
1.4 Exercises
294(2)
1.5 More Tests for Separability
296(5)
1.6 Locally of Finite Type is Finitely Generated as an Algebra
301(1)
1.7 Exercises
301(2)
§2 Differential Crossed Product Algebras
303(5)
2.1 Elementary p-Algebras
305(3)
§3 Differentials and Separability
308(9)
3.1 The Definition and Fundamental Exact Sequences
308(4)
3.2 More Tests for Separability
312(4)
3.3 Exercises
316(1)
§4 Separably Generated Extension Fields
317(8)
4.1 Emmy Noether's Normalization Lemma
320(3)
4.2 Algebraic Curves
323(2)
§5 Tests for Regularity
325(4)
5.1 A Differential Criterion for Regularity
325(1)
5.2 A Jacobian Criterion for Regularity
326(3)
Chapter 9 Etale Algebras
329(38)
§1 Complete Noetherian Rings
329(7)
§2 Etale and Smooth Algebras
336(12)
2.1 Etale Algebras
336(3)
2.2 Formally Smooth Algebras
339(7)
2.3 Formally Etale is Etale
346(1)
2.4 An Example of Raynaud
346(2)
§3 More Properties of Etale Algebras
348(13)
3.1 Quasi-finite Algebras
348(2)
3.2 Exercises
350(1)
3.3 Standard Etale Algebras
350(3)
3.4 Theorems of Permanence
353(2)
3.5 Etale Algebras over a Normal Ring
355(2)
3.6 Topological Invariance of Etale Coverings
357(2)
3.7 Etale Neighborhood of a Local Ring
359(2)
§4 Ramified Radical Extensions
361(6)
4.1 Exercises
364(3)
Chapter 10 Henselization and Splitting Rings
367(40)
§1 Henselian Local Rings
368(12)
1.1 The Definition
368(8)
1.2 Henselian Noetherian Local Rings
376(3)
1.3 Exercises
379(1)
§2 Henselization of a Local Ring
380(7)
2.1 Henselization of a Noetherian Local Ring
381(3)
2.2 Henselization of an Arbitrary Local Ring
384(1)
2.3 Strict Henselization of a Noetherian Local Ring
385(2)
2.4 Exercises
387(1)
§3 Splitting Rings for Azumaya Algebras
387(8)
3.1 Existence of Splitting Rings (Local Version)
387(4)
3.2 Local to Global Lemmas
391(3)
3.3 Splitting Rings for Azumaya Algebras
394(1)
§4 Cech Cohomology
395(12)
4.1 The Definition
396(2)
4.2 The Brauer group and Amitsur Cohomology
398(9)
Chapter 11 Azumaya Algebras, II
407(38)
§1 Invariants Attached to Elements in Azumaya Algebras
407(7)
1.1 The Characteristic Polynomial
408(4)
1.2 Exercises
412(1)
1.3 The Rank of an Element
412(2)
§2 The Brauer Group is Torsion
414(5)
2.1 Applications to Division Algebras
417(2)
§3 Maximal Orders
419(17)
3.1 Definition, First Properties
419(3)
3.2 Localization and Completion of Maximal Orders
422(2)
3.3 When is a Maximal Order an Azumaya Algebra?
424(2)
3.4 Azumaya Algebras at the Generic Point
426(2)
3.5 Azumaya Algebras over a DVR
428(2)
3.6 Locally Trivial Azumaya Algebras
430(1)
3.7 An Example of Ojanguren
431(3)
3.8 Exercises
434(2)
§4 Brauer Groups in Characteristic p
436(9)
4.1 The Brauer Group is p-divisible
437(2)
4.2 Generators for the Subgroup Annihilated by p
439(3)
4.3 Exercises
442(3)
Chapter 12 Galois Extensions of Commutative Rings
445(52)
§1 Crossed Product Algebras, the Definition
445(2)
§2 Galois Extension, the Definition
447(9)
2.1 Noetherian Reduction of a Galois Extension
456(1)
§3 Induced Galois Extensions and Galois Extensions of Fields
456(3)
§4 Galois Descent of Modules and Algebras
459(3)
§5 The Fundamental Theorem of Galois Theory
462(6)
5.1 Fundamental Theorem for a Connected Galois Extension
463(3)
5.2 Exercises
466(2)
§6 The Embedding Theorem
468(5)
6.1 Embedding a Separable Algebra
468(2)
6.2 Embedding a Connected Separable Algebra
470(3)
§7 Separable Polynomials
473(5)
7.1 Exercise
478(1)
§8 Separable Closure and Infinite Galois Theory
478(8)
8.1 The Separable Closure
478(5)
8.2 The Fundamental Theorem of Infinite Galois Theory
483(1)
8.3 Exercises
484(2)
§9 Cyclic Extensions
486(11)
9.1 Kummer Theory
486(5)
9.2 Artin-Schreier Extensions
491(1)
9.3 Exercises
492(5)
Chapter 13 Crossed Products and Galois Cohomology
497(60)
§1 Crossed Product Algebras
498(3)
§2 Generalized Crossed Product Algebras
501(12)
2.1 Exercises
512(1)
§3 The Seven Term Exact Sequence of Galois Cohomology
513(12)
3.1 The Theorem and Its Corollaries
513(7)
3.2 Exercises
520(1)
3.3 Galois Cohomology Agrees with Amitsur Cohomology
521(2)
3.4 Galois Cohomology and the Brauer Group
523(2)
3.5 Exercise
525(1)
§4 Cyclic Crossed Product Algebras
525(7)
4.1 Symbol Algebras
528(1)
4.2 Cyclic Algebras in Characteristic p
528(2)
4.3 The Brauer Group of a Henselian Local Ring
530(1)
4.4 Exercises
531(1)
§5 Generalized Cyclic Crossed Product Algebras
532(9)
§6 The Brauer Group of a Polynomial Ring
541(16)
6.1 The Brauer Group of a Graded Ring
544(1)
6.2 The Brauer Group of a Laurent Polynomial Ring
545(1)
6.3 Examples of Brauer Groups
546(6)
6.4 Exercises
552(5)
Chapter 14 Further Topics
557(58)
§1 Corestriction
557(27)
1.1 Norms of Modules and Algebras
561(5)
1.2 Applications of Corestriction
566(2)
1.3 Corestriction and Galois Descent
568(3)
1.4 Corestriction and Amitsur Cohomology
571(6)
1.5 Corestriction and Galois Cohomology
577(4)
1.6 Corestriction and Generalized Crossed Products
581(2)
1.7 Exercises
583(1)
§2 A Mayer-Vietoris Sequence for the Brauer Group
584(15)
2.1 Milnor's Theorem
585(6)
2.2 Mayer-Vietoris Sequences
591(7)
2.3 Exercises
598(1)
§3 Brauer Groups of Some Nonnormal Domains
599(16)
3.1 The Brauer Group of an Algebraic Curve
600(1)
3.2 Every Finite Abelian Group is a Brauer Group
601(1)
3.3 A Family of Nonnormal Subrings of k[ x, y]
602(3)
3.4 The Brauer Group of a Subring of a Global Field
605(7)
3.5 Exercises
612(3)
Acronyms 615(2)
Glossary of Notations 617(4)
Bibliography 621(10)
Index 631
Timothy J. Ford, Florida Atlantic University, Boca Raton, FL.