Universitext For further volumes: http://www.springer.com/series/223 Joseph J. Rotman An Introduction to Homological Algebra Second Edition 123 Joseph J. Rotman Department of Mathematics University of Illinois at Urbana-Champaign Urbana IL 61801 USA [email protected] Editorial board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Universita` degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley Claude Sabbah, CNRS, Ecole´ Polytechnique Endre Suli,¨ University of Oxford Wojbor Woyczynski, Case Western Reserve University ISBN: 978-0-387-24527-0 e-ISBN: 978-0-387-68324-9 DOI 10.1007/978-0-387-68324-9 Library of Congress Control Number: 2008936123 Mathematics Subject Classification (2000): 18-01 c Springer Science+Business Media, LLC 2009 All rights reserved. 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Printed on acid-free paper springer.com To the memory of my mother Rose Wolf Rotman Contents Preface to the Second Edition ................... x How to Read This Book .......................xiii Chapter 1 Introduction 1.1 Simplicial Homology . ..................... 1 1.2 Categories and Functors ..................... 7 1.3 Singular Homology . ..................... 29 Chapter 2 Hom and Tensor 2.1 Modules . ............................ 37 2.2 Tensor Products . ..................... 69 2.2.1 Adjoint Isomorphisms . ..................... 91 Chapter 3 Special Modules 3.1 Projective Modules . ..................... 98 3.2 Injective Modules . .....................115 3.3 Flat Modules . .....................131 3.3.1Purity..............................146 Chapter 4 Specific Rings 4.1 Semisimple Rings . .....................154 4.2 von Neumann Regular Rings . ..............159 4.3 Hereditary and Dedekind Rings . ..............160 vii viii Contents 4.4 Semihereditary and Prufer¨ Rings . ..............169 4.5 Quasi-Frobenius Rings .....................173 4.6 Semiperfect Rings . .....................179 4.7 Localization . .....................188 4.8 Polynomial Rings . .....................203 Chapter 5 Setting the Stage 5.1 Categorical Constructions . ..............213 5.2 Limits..............................229 5.3 Adjoint Functor Theorem for Modules . ..........256 5.4 Sheaves.............................273 5.4.1 Manifolds ............................288 5.4.2 Sheaf Constructions . .....................294 5.5 Abelian Categories . .....................303 5.5.1Complexes............................317 Chapter 6 Homology 6.1 Homology Functors . .....................323 6.2 Derived Functors . .....................340 6.2.1 Left Derived Functors . .....................343 6.2.2 Axioms . ............................357 6.2.3 Covariant Right Derived Functors . ..............364 6.2.4 Contravariant Right Derived Functors . ..........369 6.3 Sheaf Cohomology . .....................377 6.3.1 Cechˇ Cohomology . .....................384 6.3.2 Riemann–Roch Theorem . ..............392 Chapter 7 Tor and Ext 7.1 Tor................................404 7.1.1 Domains . ............................412 7.1.2 Localization . .....................415 7.2 Ext................................418 7.2.1BaerSum............................428 7.3 Cotorsion Groups . .....................438 7.4 Universal Coefficients . .....................448 Chapter 8 Homology and Rings 8.1 Dimensions of Rings . .....................453 8.2 Hilbert’s Syzygy Theorem . ..............467 8.3 Stably Free Modules . .....................476 8.4 Commutative Noetherian Local Rings . ..........484 Contents ix Chapter 9 Homology and Groups 9.1 Group Extensions . .....................495 9.1.1 Semidirect Products . .....................500 9.1.2 General Extensions and Cohomology ..............504 9.1.3 Stabilizing Automorphisms . ..............514 9.2 Group Cohomology . .....................519 9.3 Bar Resolutions . .....................525 9.4 Group Homology . .....................535 9.4.1 Schur Multiplier . .....................541 9.5 Change of Groups . .....................559 9.5.1 Restriction and Inflation .....................564 9.6 Transfer . ............................572 9.7 Tate Groups . .....................580 9.8 Outer Automorphisms of p-Groups . ..............587 9.9 Cohomological Dimension . ..............591 9.10 Division Rings and Brauer Groups . ..............595 Chapter 10 Spectral Sequences 10.1 Bicomplexes . .....................609 10.2 Filtrations and Exact Couples . ..............616 10.3 Convergence . .....................624 10.4 Homology of the Total Complex . ..............628 10.5 Cartan–Eilenberg Resolutions . ..............647 10.6 Grothendieck Spectral Sequences . ..............656 10.7 Groups . ............................660 10.8 Rings . ............................666 10.9Sheaves.............................675 10.10 Kunneth¨ Theorems . .....................678 References ...............................689 Special Notation ............................695 Index ...................................697 Preface to the Second Edition Homological Algebra has grown in the nearly three decades since the first edi- tion of this book appeared in 1979. Two books discussing more recent results are Weibel, An Introduction to Homological Algebra, 1994, and Gelfand– Manin, Methods of Homological Algebra, 2003. In their Foreword, Gelfand and Manin divide the history of Homological Algebra into three periods: the first period ended in the early 1960s, culminating in applications of Homo- logical Algebra to regular local rings. The second period, greatly influenced by the work of A. Grothendieck and J.-P. Serre, continued through the 1980s; it involves abelian categories and sheaf cohomology. The third period, in- volving derived categories and triangulated categories, is still ongoing. Both of these newer books discuss all three periods (see also Kashiwara–Schapira, Categories and Sheaves). The original version of this book discussed the first period only; this new edition remains at the same introductory level, but it now introduces the second period as well. This change makes sense peda- gogically, for there has been a change in the mathematics population since 1979; today, virtually all mathematics graduate students have learned some- thing about functors and categories, and so I can now take the categorical viewpoint more seriously. When I was a graduate student, Homological Algebra was an unpopular subject. The general attitude was that it was a grotesque formalism, boring to learn, and not very useful once one had learned it. Perhaps an algebraic topologist was forced to know this stuff, but surely no one else should waste time on it. The few true believers were viewed as workers at the fringe of mathematics who kept tinkering with their elaborate machine, smoothing out rough patches here and there. x Preface to the Second Edition xi This attitude changed dramatically when J.-P. Serre characterized regular local rings using Homological Algebra (they are the commutative noetherian local rings of “finite global dimension”), for this enabled him to prove that any localization of a regular local ring is itself regular (until then, only spe- cial cases of this were known). At the same time, M. Auslander and D. A. Buchsbaum also characterized regular local rings, and they went on to com- plete work of M. Nagata by using global dimension to prove that every regular local ring is a unique factorization domain. As Grothendieck and Serre revolu- tionized Algebraic Geometry by introducing schemes and sheaves, resistance to Homological Algebra waned. Today, it is just another standard tool in a mathematician’s kit. For more details, we recommend C. A. Weibel’s chapter, “History of Homological Algebra,” in the book of James, History of Topology. Homological Algebra presents a great pedagogical challenge for authors and for readers. At first glance, its flood of elementary definitions (which often originate in other disciplines) and its space-filling diagrams appear for- bidding. To counter this first impression, S. Lang set the following exercise on page 105 of his book, Algebra: Take any book on homological algebra and prove all the theorems without looking at the proofs given in that book. Taken literally, the statement of the exercise is absurd. But its spirit is ab- solutely accurate; the subject only appears difficult. However, having rec- ognized the elementary character of much of the early material, one is often tempted to “wave one’s hands”: to pretend that minutiae always behave well. It should come as no surprise that danger lurks in this attitude. For this rea- son, I include many details in the beginning, at the risk of boring some readers by so doing (of course, such readers are free to turn the page). My intent is twofold: to allow readers to see that complete proofs can, in fact, be written