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Separable Algebras GRADUATE STUDIES IN MATHEMATICS 183 Separable Algebras Timothy J. Ford American Mathematical Society 10.1090/gsm/183 Separable Algebras GRADUATE STUDIES IN MATHEMATICS 183 Separable Algebras Timothy J. Ford American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed (Chair) Gigliola Staffilani Jeff A. Viaclovsky 2010 Mathematics Subject Classification. Primary 16H05, 15B05, 13A15, 13C20, 14F20, 14B25, 16-01, 13-01. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-183 Library of Congress Cataloging-in-Publication Data Names: Ford, Timothy J., 1954- Title: Separable algebras / Timothy J. Ford. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Gradu- ate studies in mathematics ; volume 183 | Includes bibliographical references and index. Identifiers: LCCN 2017013677 | ISBN 9781470437701 (alk. paper) Subjects: LCSH: Separable algebras–Textbooks. | Associative rings–Textbooks. | AMS: Associa- tive rings and algebras – Algebras and orders – Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.). msc | Commutative algebra – Ring extensions and related topics – Etale´ 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 – Etale´ and other Grothendieck topologies and (co)homologies. msc | Algebraic geometry – Local theory – Local structure of morphisms: Etale,´ flat, etc. msc | Associative rings and alge- bras – Instructional exposition (textbooks, tutorial papers, etc.). msc | Commutative algebra – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA251.5 .F67 2017 | DDC 512/.46–dc23 LC record available at https://lccn. loc.gov/2017013677 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2017 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 222120191817 Dedicated to Frank DeMeyer 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. Hom and Tensor 23 3.1. Tensor Product 23 3.2. Exercises 27 3.3. Hom Groups 29 3.4. Hom 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 viii Contents 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 243 §1. First Properties of Azumaya Algebras 243 §2. The Commutator Theorems 249 §3. The Brauer Group 252 §4. Splitting Rings 254 4.1. Exercises 258 §5. Azumaya Algebras over a Field 258 §6. Azumaya Algebras up to Brauer Equivalence 263 6.1. Exercises 266 §7. Noetherian Reduction of Azumaya Algebras 267 7.1. Exercises 273 §8. The Picard Group of Invertible Bimodules 274 8.1. Definition of the Picard Group 274 8.2. The Skolem-Noether Theorem 279 8.3. Exercise 280 §9. The Brauer Group Modulo an Ideal 281 9.1. Lifting Azumaya Algebras 284 9.2. The Brauer Group of a Commutative Artinian Ring 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. Etale´ Algebras 329 §1. Complete Noetherian Rings 329 §2. Etale´ and Smooth Algebras 336 2.1. Etale´ Algebras 336 2.2. Formally Smooth Algebras 339 2.3. Formally Etale´ is Etale´ 346 2.4. An Example of Raynaud 346 §3. More Properties of Etale´ Algebras 348 3.1. Quasi-finite Algebras 348 3.2. Exercises 350 3.3. Standard Etale´ Algebras 350 3.4. Theorems of Permanence 353 3.5. Etale´ Algebras over a Normal Ring 355 3.6. Topological Invariance of Etale´ Coverings 357 3.7. Etale´ 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.
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