Separable algebras:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2017]
|
| Schriftenreihe: | Graduate studies in mathematics
183 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | xxi, 637 Seiten Diagramme |
| ISBN: | 9781470437701 |
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| 490 | 1 | |a Graduate studies in mathematics |v 183 | |
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| 650 | 4 | |a Associative rings and algebras ... Algebras and orders ... Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) / msc | |
| 650 | 4 | |a Commutative algebra ... Ring extensions and related topics ... Étale and flat extensions; Henselization; Artin approximation / msc | |
| 650 | 4 | |a Commutative algebra ... Ring extensions and related topics ... Galois theory / msc | |
| 650 | 4 | |a Commutative algebra ... General commutative ring theory ... Ideals; multiplicative ideal theory / msc | |
| 650 | 4 | |a Commutative algebra ... Theory of modules and ideals ... Class groups / msc | |
| 650 | 4 | |a Algebraic geometry ... (Co)homology theory ... Étale and other Grothendieck topologies and (co)homologies / msc | |
| 650 | 4 | |a Algebraic geometry ... Local theory ... Local structure of morphisms: étale, flat, etc. / msc | |
| 650 | 4 | |a Associative rings and algebras ... Instructional exposition (textbooks, tutorial papers, etc.) / msc | |
| 650 | 4 | |a Commutative algebra ... Instructional exposition (textbooks, tutorial papers, etc.) / msc | |
| 650 | 4 | |a Separable algebras |v Textbooks | |
| 650 | 4 | |a Associative rings |v Textbooks | |
| 650 | 4 | |a Associative rings and algebras ... Algebras and orders ... Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) | |
| 650 | 4 | |a Commutative algebra ... Ring extensions and related topics ... Étale and flat extensions; Henselization; Artin approximation | |
| 650 | 4 | |a Commutative algebra ... Ring extensions and related topics ... Galois theory | |
| 650 | 4 | |a Commutative algebra ... General commutative ring theory ... Ideals; multiplicative ideal theory | |
| 650 | 4 | |a Commutative algebra ... Theory of modules and ideals ... Class groups | |
| 650 | 4 | |a Algebraic geometry ... (Co)homology theory ... Étale and other Grothendieck topologies and (co)homologies | |
| 650 | 4 | |a Algebraic geometry ... Local theory ... Local structure of morphisms: étale, flat, etc. | |
| 650 | 4 | |a Associative rings and algebras ... Instructional exposition (textbooks, tutorial papers, etc.) | |
| 650 | 4 | |a Commutative algebra ... Instructional exposition (textbooks, tutorial papers, etc.) | |
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Datensatz im Suchindex
| _version_ | 1804177924256956416 |
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| adam_text | Titel: Separable algebras
Autor: Ford, Timothy J
Jahr: 2017
Contents Preface xv Chapter 1. Background Material on Rings and Modules 1 §1. Rings and Modules 1 1.1. Categories and Functors 2 1.2. Progenerator Modules 5 1.3. Exercises 8 1.4. Nakayama’s Lemma 9 1.5. Exercise 11 1.6. Module Direct Summands of Rings 11 1.7. Exercises 13 §2. Polynomial Functions 14 2.1. The Ring of Polynomial Functions on a Module 14 2.2. Resultant of Two Polynomials 15 2.3. Polynomial Functions on an Algebraic Curve 19 2.4. Exercises 21 §3. Horn and Tensor 23 3.1. Tensor Product 23 3.2. Exercises 27 3.3. Horn Groups 29 3.4. Horn Tensor Relations 33 3.5. Exercises 36 §4. Direct Limit and Inverse Limit 37 4.1. The Direct Limit 38 4.2. The Inverse Limit 40 4.3. Inverse Systems Indexed by Nonnegative Integers 42 vii
Contents viii 4.4. Exercises 45 §5. The Morita Theorems 47 5.1. Exercises 52 Chapter 2. Modules over Commutative Rings 53 §1. Localization of Modules and Rings 53 1.1. Local to Global Lemmas 54 1.2. Exercises 57 §2. The Prime Spectrum of a Commutative Ring 58 2.1. Exercises 63 §3. Finitely Generated Projective Modules 65 3.1. Exercises 68 §4. Faithfully Flat Modules and Algebras 70 4.1. Exercises 74 §5. Chain Conditions 75 5.1. Exercises 78 §6. Faithfully Flat Base Change 80 6.1. Fundamental Theorem on Faithfully Flat Base Change 80 6.2. Locally Free Finite Rank is Finitely Generated Projective 83 6.3. Invertible Modules and the Picard Group 85 6.4. Exercises 88 Chapter 3. The Wedderburn-Artin Theorem 91 §1. The Jacobson Radical and Nakayama’s Lemma 91 1.1. Exercises 94 §2. Semisimple Modules and Semisimple Rings 94 2.1. Simple Rings and the Wedderburn-Artin Theorem 97 2.2. Commutative Artinian Rings 100 2.3. Exercises 102 §3. Integral Extensions 103 §4. Completion of a Linear Topological Module 106 4.1. Graded Rings and Graded Modules 110 4.2. Lifting of Idempotents 112 Chapter 4. Separable Algebras, Definition and First Properties 115 §1. Separable Algebra, the Definition 115 1.1. Exercises 119 §2. Examples of Separable Algebras 120 §3. Separable Algebras Under a Change of Base Ring 123 §4. Homomorphisms of Separable Algebras 127
Contents IX 4.1. Exercises 133 §5. Separable Algebras over a Field 137 5.1. Central Simple Equals Central Separable 137 5.2. Unique Decomposition Theorems 140 5.3. The Skolem-Noether Theorem 143 5.4. Exercises 144 §6. Commutative Separable Algebras 146 6.1. Separable Extensions of Commutative Rings 146 6.2. Separability and the Trace 148 6.3. Twisted Form of the Trivial Extension 152 6.4. Exercises 153 §7. Formally Unramified, Smooth and Etale Algebras 155 Chapter 5. Background Material on Homological Algebra 159 §1. Group Cohomology 159 1.1. Cocycle and Coboundary Groups in Low Degree 161 1.2. Applications and Computations 163 1.3. Exercises 170 §2. The Tensor Algebra of a Module 173 2.1. Exercises 176 §3. Theory of Faithfully Flat Descent 177 3.1. The Amitsur Complex 177 3.2. The Descent of Elements 178 3.3. Descent of Homomorphisms 180 3.4. Descent of Modules 181 3.5. Descent of Algebras 186 §4. Hochschild Cohomology 188 §5. Amitsur Cohomology 191 5.1. The Definition and First Properties 191 5.2. Twisted Forms 195 Chapter 6. The Divisor Class Group 199 §1. Background Results from Commutative Algebra 200 1.1. Krull Dimension 200 1.2. The Serre Criteria for Normality 201 1.3. The Hilbert-Serre Criterion for Regularity 202 1.4. Discrete Valuation Rings 204 §2. The Class Group of Weil Divisors 206 2.1. Exercises 210 §3. Lattices 213 3.1. Definition and First Properties 213
X Contents 3.2. Reflexive Lattices 216 3.3. A Local to Global Theorem for Reflexive Lattices 222 3.4. Exercises 224 §4. The Ideal Class Group 226 4.1. Exercises 232 §5. Functorial Properties of the Class Group 233 5.1. Flat Extensions 233 5.2. Finite Extensions 235 5.3. Galois Descent of Divisor Classes 236 5.4. The Class Group of a Regular Domain 238 5.5. Exercises 242 Chapter 7. Azumaya Algebras, I §1. First Properties of Azumaya Algebras §2. The Commutator Theorems §3. The Brauer Group §4. Splitting Rings 4.1. Exercises §5. Azumaya Algebras over a Field §6. Azumaya Algebras up to Brauer Equivalence 6.1. Exercises §7. Noetherian Reduction of Azumaya Algebras 7.1. Exercises §8. The Picard Group of Invertible Bimodules 8.1. Definition of the Picard Group 8.2. The Skolem-Noether Theorem 8.3. Exercise §9. The Brauer Group Modulo an Ideal 9.1. Lifting Azumaya Algebras 9.2. The Brauer Group of a Commutative Artinian Ring 243 243 249 252 254 258 258 263 266 267 273 274 274 279 280 281 284 286 Chapter 8. Derivations, Differentials and Separability 287 §1. Derivations and Separability 287 1.1. The Definition and First Results 287 1.2. A Noncommutative Binomial Theorem in Characteristic p 291 1.3. Extensions of Derivations 292 1.4. Exercises 294 1.5. More Tests for Separability 296 1.6. Locally of Finite Type is Finitely Generated as an Algebra 301 1.7. Exercises 301
Contents xi §2. Differential Crossed Product Algebras 303 2.1. Elementary p -Algebras 305 §3. Differentials and Separability 308 3.1. The Definition and Fundamental Exact Sequences 308 3.2. More Tests for Separability 312 3.3. Exercises 316 §4. Separably Generated Extension Fields 317 4.1. Emmy Noether’s Normalization Lemma 320 4.2. Algebraic Curves 323 §5. Tests for Regularity 325 5.1. A Differential Criterion for Regularity 325 5.2. A Jacobian Criterion for Regularity 326 Chapter 9. Étale Algebras 329 §1. Complete Noetherian Rings 329 §2. Étale and Smooth Algebras 336 2.1. Étale Algebras 336 2.2. Formally Smooth Algebras 339 2.3. Formally Étale is Étale 346 2.4. An Example of Raynaud 346 §3. More Properties of Étale Algebras 348 3.1. Quasi-finite Algebras 348 3.2. Exercises 350 3.3. Standard Étale Algebras 350 3.4. Theorems of Permanence 353 3.5. Étale Algebras over a Normal Ring 355 3.6. Topological Invariance of Étale Coverings 357 3.7. Étale Neighborhood of a Local Ring 359 §4. Ramified Radical Extensions 361 4.1. Exercises 364 Chapter 10. Henselization and Splitting Rings 367 §1. Henselian Local Rings 368 1.1. The Definition 368 1.2. Henselian Noetherian Local Rings 376 1.3. Exercises 379 §2. Henselization of a Local Ring 380 2.1. Henselization of a Noetherian Local Ring 381 2.2. Henselization of an Arbitrary Local Ring 384 2.3. Strict Henselization of a Noetherian Local Ring 385 2.4. Exercises 387
Contents xii §3. Splitting Rings for Azumaya Algebras 387 3.1. Existence of Splitting Rings (Local Version) 387 3.2. Local to Global Lemmas 391 3.3. Splitting Rings for Azumaya Algebras 394 §4. Cech Cohomology 395 4.1. The Definition 396 4.2. The Brauer group and Amitsur Cohomology 398 Chapte r 11. Azumaya Algebras, II 407 §1. Invariants Attached to Elements in Azumaya Algebras 407 1.1. The Characteristic Polynomial 408 1.2. Exercises 412 1.3. The Rank of an Element 412 §2. The Brauer Group is Torsion 414 2.1. Applications to Division Algebras 417 §3. Maximal Orders 419 3.1. Definition, First Properties 419 3.2. Localization and Completion of Maximal Orders 422 3.3. When is a Maximal Order an Azumaya Algebra? 424 3.4. Azumaya Algebras at the Generic Point 426 3.5. Azumaya Algebras over a DVR 428 3.6. Locally Trivial Azumaya Algebras 430 3.7. An Example of Ojanguren 431 3.8. Exercises 434 §4- Brauer Groups in Characteristic p 436 4.1. The Brauer Group is p-divisible 437 4.2. Generators for the Subgroup Annihilated by p 439 4.3. Exercises 442 Chapter 12. Galois Extensions of Commutative Rings 445 §1. Crossed Product Algebras, the Definition 445 §2. Galois Extension, the Definition 447 2.1. Noetherian Reduction of a Galois Extension 456 §3. Induced Galois Extensions and Galois Extensions of Fields 456 §4. Galois Descent of Modules and Algebras 459 §5. The Fundamental Theorem of Galois Theory 462 5.1. Fundamental Theorem for a Connected Galois Extension 463 5.2. Exercises 466 §6. The Embedding Theorem 468 6.1. Embedding a Separable Algebra 468 6.2. Embedding a Connected Separable Algebra 470
Contents xiii §7. Separable Polynomials 473 7.1. Exercise 478 §8. Separable Closure and Infinite Galois Theory 478 8.1. The Separable Closure 478 8.2. The Fundamental Theorem of Infinite Galois Theory 483 8.3. Exercises 484 §9. Cyclic Extensions 486 9.1. Kummer Theory 486 9.2. Artin-Schreier Extensions 491 9.3. Exercises 492 Chapter 13. Crossed Products and Galois Cohomology 497 §1. Crossed Product Algebras 498 §2. Generalized Crossed Product Algebras 501 2.1. Exercises 512 §3. The Seven Term Exact Sequence of Galois Cohomology 513 3.1. The Theorem and Its Corollaries 513 3.2. Exercises 520 3.3. Galois Cohomology Agrees with Amitsur Cohomology 521 3.4. Galois Cohomology and the Brauer Group 523 3.5. Exercise 525 §4. Cyclic Crossed Product Algebras 525 4.1. Symbol Algebras 528 4.2. Cyclic Algebras in Characteristic p 528 4.3. The Brauer Group of a Henselian Local Ring 530 4.4. Exercises 531 §5. Generalized Cyclic Crossed Product Algebras 532 §6. The Brauer Group of a Polynomial Ring 541 6.1. The Brauer Group of a Graded Ring 544 6.2. The Brauer Group of a Laurent Polynomial Ring 545 6.3. Examples of Brauer Groups 546 6.4. Exercises 552 Chapter 14. Further Topics 557 §1. Corestriction 557 1.1. Norms of Modules and Algebras 561 1.2. Applications of Corestriction 566 1.3. Corestriction and Galois Descent 568 1.4. Corestriction and Amitsur Cohomology 571 1.5. Corestriction and Galois Cohomology 577 1.6. Corestriction and Generalized Crossed Products 581
XIV Contents 1.7. Exercises 583 §2. A Mayer-Vietoris Sequence for the Brauer Group 584 2.1. Milnor’s Theorem 585 2.2. Mayer-Vietoris Sequences 591 2.3. Exercises 598 §3. Brauer Groups of Some Nonnormal Domains 599 3.1. The Brauer Group of an Algebraic Curve 600 3.2. Every Finite Abelian Group is a Brauer Group 601 3.3. A Family of Nonnormal Subrings of k[x, y] 602 3.4. The Brauer Group of a Subring of a Global Field 605 3.5. Exercises 612 Acronyms 615 Glossary of Notations 617 Bibliography 621 Index 631
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| series2 | Graduate studies in mathematics |
| spelling | Ford, Timothy J. 1954- Verfasser (DE-588)114532942X aut Separable algebras Timothy J. Ford Providence, Rhode Island American Mathematical Society [2017] xxi, 637 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 183 Includes bibliographical references and index Associative rings and algebras ... Algebras and orders ... Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) / msc Commutative algebra ... Ring extensions and related topics ... Étale and flat extensions; Henselization; Artin approximation / msc Commutative algebra ... Ring extensions and related topics ... Galois theory / msc Commutative algebra ... General commutative ring theory ... Ideals; multiplicative ideal theory / msc Commutative algebra ... Theory of modules and ideals ... Class groups / msc Algebraic geometry ... (Co)homology theory ... Étale and other Grothendieck topologies and (co)homologies / msc Algebraic geometry ... Local theory ... Local structure of morphisms: étale, flat, etc. / msc Associative rings and algebras ... Instructional exposition (textbooks, tutorial papers, etc.) / msc Commutative algebra ... Instructional exposition (textbooks, tutorial papers, etc.) / msc Separable algebras Textbooks Associative rings Textbooks Associative rings and algebras ... Algebras and orders ... Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) Commutative algebra ... Ring extensions and related topics ... Étale and flat extensions; Henselization; Artin approximation Commutative algebra ... Ring extensions and related topics ... Galois theory Commutative algebra ... General commutative ring theory ... Ideals; multiplicative ideal theory Commutative algebra ... Theory of modules and ideals ... Class groups Algebraic geometry ... (Co)homology theory ... Étale and other Grothendieck topologies and (co)homologies Algebraic geometry ... Local theory ... Local structure of morphisms: étale, flat, etc. Associative rings and algebras ... Instructional exposition (textbooks, tutorial papers, etc.) Commutative algebra ... Instructional exposition (textbooks, tutorial papers, etc.) Graduate studies in mathematics 183 (DE-604)BV009739289 183 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029955058&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ford, Timothy J. 1954- Separable algebras Graduate studies in mathematics Associative rings and algebras ... Algebras and orders ... Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) / msc Commutative algebra ... Ring extensions and related topics ... Étale and flat extensions; Henselization; Artin approximation / msc Commutative algebra ... Ring extensions and related topics ... Galois theory / msc Commutative algebra ... General commutative ring theory ... Ideals; multiplicative ideal theory / msc Commutative algebra ... Theory of modules and ideals ... Class groups / msc Algebraic geometry ... (Co)homology theory ... Étale and other Grothendieck topologies and (co)homologies / msc Algebraic geometry ... Local theory ... Local structure of morphisms: étale, flat, etc. / msc Associative rings and algebras ... Instructional exposition (textbooks, tutorial papers, etc.) / msc Commutative algebra ... Instructional exposition (textbooks, tutorial papers, etc.) / msc Separable algebras Textbooks Associative rings Textbooks Associative rings and algebras ... Algebras and orders ... Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) Commutative algebra ... Ring extensions and related topics ... Étale and flat extensions; Henselization; Artin approximation Commutative algebra ... Ring extensions and related topics ... Galois theory Commutative algebra ... General commutative ring theory ... Ideals; multiplicative ideal theory Commutative algebra ... Theory of modules and ideals ... Class groups Algebraic geometry ... (Co)homology theory ... Étale and other Grothendieck topologies and (co)homologies Algebraic geometry ... Local theory ... Local structure of morphisms: étale, flat, etc. Associative rings and algebras ... Instructional exposition (textbooks, tutorial papers, etc.) Commutative algebra ... Instructional exposition (textbooks, tutorial papers, etc.) |
| title | Separable algebras |
| title_auth | Separable algebras |
| title_exact_search | Separable algebras |
| title_full | Separable algebras Timothy J. Ford |
| title_fullStr | Separable algebras Timothy J. Ford |
| title_full_unstemmed | Separable algebras Timothy J. Ford |
| title_short | Separable algebras |
| title_sort | separable algebras |
| topic | Associative rings and algebras ... Algebras and orders ... Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) / msc Commutative algebra ... Ring extensions and related topics ... Étale and flat extensions; Henselization; Artin approximation / msc Commutative algebra ... Ring extensions and related topics ... Galois theory / msc Commutative algebra ... General commutative ring theory ... Ideals; multiplicative ideal theory / msc Commutative algebra ... Theory of modules and ideals ... Class groups / msc Algebraic geometry ... (Co)homology theory ... Étale and other Grothendieck topologies and (co)homologies / msc Algebraic geometry ... Local theory ... Local structure of morphisms: étale, flat, etc. / msc Associative rings and algebras ... Instructional exposition (textbooks, tutorial papers, etc.) / msc Commutative algebra ... Instructional exposition (textbooks, tutorial papers, etc.) / msc Separable algebras Textbooks Associative rings Textbooks Associative rings and algebras ... Algebras and orders ... Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) Commutative algebra ... Ring extensions and related topics ... Étale and flat extensions; Henselization; Artin approximation Commutative algebra ... Ring extensions and related topics ... Galois theory Commutative algebra ... General commutative ring theory ... Ideals; multiplicative ideal theory Commutative algebra ... Theory of modules and ideals ... Class groups Algebraic geometry ... (Co)homology theory ... Étale and other Grothendieck topologies and (co)homologies Algebraic geometry ... Local theory ... Local structure of morphisms: étale, flat, etc. Associative rings and algebras ... Instructional exposition (textbooks, tutorial papers, etc.) Commutative algebra ... Instructional exposition (textbooks, tutorial papers, etc.) |
| topic_facet | Associative rings and algebras ... Algebras and orders ... Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) / msc Commutative algebra ... Ring extensions and related topics ... Étale and flat extensions; Henselization; Artin approximation / msc Commutative algebra ... Ring extensions and related topics ... Galois theory / msc Commutative algebra ... General commutative ring theory ... Ideals; multiplicative ideal theory / msc Commutative algebra ... Theory of modules and ideals ... Class groups / msc Algebraic geometry ... (Co)homology theory ... Étale and other Grothendieck topologies and (co)homologies / msc Algebraic geometry ... Local theory ... Local structure of morphisms: étale, flat, etc. / msc Associative rings and algebras ... Instructional exposition (textbooks, tutorial papers, etc.) / msc Commutative algebra ... Instructional exposition (textbooks, tutorial papers, etc.) / msc Separable algebras Textbooks Associative rings Textbooks Associative rings and algebras ... Algebras and orders ... Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) Commutative algebra ... Ring extensions and related topics ... Étale and flat extensions; Henselization; Artin approximation Commutative algebra ... Ring extensions and related topics ... Galois theory Commutative algebra ... General commutative ring theory ... Ideals; multiplicative ideal theory Commutative algebra ... Theory of modules and ideals ... Class groups Algebraic geometry ... (Co)homology theory ... Étale and other Grothendieck topologies and (co)homologies Algebraic geometry ... Local theory ... Local structure of morphisms: étale, flat, etc. Associative rings and algebras ... Instructional exposition (textbooks, tutorial papers, etc.) Commutative algebra ... Instructional exposition (textbooks, tutorial papers, etc.) |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029955058&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT fordtimothyj separablealgebras |