Homological Algebra Notes Sean Sather-Wagstaff Department of Mathematics, 300 Minard Hall, North Dakota State University, Fargo, North Dakota 58105-5075, USA E-mail address: [email protected] URL: http://math.ndsu.nodak.edu/faculty/ssatherw/ September 8, 2009. Contents Introduction September 8, 2009 v Format of these notes vii Acknowledgements viii Notation and conventions viii Chapter I. Universal Constructions September 8, 2009 1 I.1. Finitely Generated Free Modules 1 I.2. Products of Modules 3 I.3. Coproducts of Modules 5 I.4. Localization 7 I.5. Hom: Functoriality and Localization 10 Chapter II. Tensor Products September 8, 2009 21 II.1. Existence and Uniqueness 21 II.2. Functoriality and Base-Change 25 II.3. Commutativity and Associativity 31 II.4. Right-Exactness 35 II.5. Hom-Tensor Adjointness 40 Chapter III. Injective, Projective, and Flat Modules September 8, 2009 43 III.1. Injective and Projective Modules 43 III.2. Flat Modules 51 III.3. Faithfully Flat Modules 58 III.4. Power Series Rings 60 III.5. Flat Ring Homomorphisms 65 III.6. Completions: A Survey 69 III.7. Completions: Some Details 69 Chapter IV. Homology, Resolutions, Ext and Tor September 8, 2009 71 IV.1. Chain Complexes and Homology 71 IV.2. Resolutions 76 IV.3. Ext-Modules 79 IV.4. Tor-Modules 84 IV.5. Epilogue 86 Chapter V. Depth September 8, 2009 89 V.1. Assumptions 89 V.2. Associated Primes and Supports of Modules 91 V.3. Prime Filtrations 96 V.4. Prime Avoidance and Nakayama's Lemma 100 V.5. Regular Sequences and Ext 105 iii iv CONTENTS V.6. Four Lemmas 112 Chapter VI. Chain Maps and Induced Maps on Ext and Tor September 8, 2009 115 VI.1. Chain Maps 115 VI.2. Isomorphisms for Ext and Tor 117 VI.3. Liftings of Resolutions 121 VI.4. Induced Chain Maps 125 VI.5. Ext-maps via projective resolutions 131 VI.6. Ext-maps via injective resolutions 135 VI.7. Tor-maps 137 Chapter VII. Ext, Tor, and Homological Dimensions September 8, 2009 143 VII.1. Assumptions 143 VII.2. Depth and Dimension 143 VII.3. Ext and Projective Dimension 146 VII.4. Tor and Projective Dimension 152 VII.5. Ext and Injective Dimension 154 VII.6. Tor and Flat Dimension 157 Chapter VIII. Long Exact Sequences September 8, 2009 161 VIII.1. General Long Exact Sequences 161 VIII.2. Long Exact Sequences in Ext and Tor 167 VIII.3. Horseshoe Lemmas 170 VIII.4. Mapping Cones 174 VIII.5. Ext, Tor, and Resolutions 179 VIII.6. Koszul Complexes 183 VIII.7. Epilogue: Tor and Torsion 192 Chapter IX. Depth and Homological Dimensions September 8, 2009 195 IX.1. Projective Dimension and Regular Sequences 195 IX.2. The Auslander-Buchsbaum Formula 197 IX.3. Depth and Flat Ring Homomorphisms 200 IX.4. Injective Dimension and Regular Sequences 205 Chapter X. Regular Local Rings September 8, 2009 207 X.1. Background from Dimension Theory 207 X.2. Definitions and Basic Properties 209 X.3. Theorems of Auslander, Buchsbaum and Serre 212 X.4. Regularity and Flat Local Homomorphisms 216 Index 221 Bibliography 225 Introduction September 8, 2009 An important question to ask (and re-ask) when one is learning a new subject is, \What does this subject do for me?" A complete answer to this question is usually hard to give, especially because the answer almost certainly depends on the interests of the person asking it. Here are a couple of motivating answers for the (commutative) algebraist who is thinking about learning some homological algebra. Let R be a commutative ring (with identity). Ext and Tor. Given an R-module N and an exact sequence of R-modules 0 ! M 0 ! M ! M 00 ! 0 (&) the operators HomR(−; −) and − ⊗R − give rise to three exact sequences 0 00 0 ! HomR(N; M ) ! HomR(N; M) ! HomR(N; M )(∗) 00 0 0 ! HomR(M ;N) ! HomR(M; N) ! HomR(M ;N)(y) 0 00 M ⊗R N ! M ⊗R N ! M ⊗R N ! 0: (#) One may be tempted to feel cheated by the loss of zeroes. When N is projective, we get to add \! 0" onto the first sequence, and we get to add \0 !" onto the last sequence. And when N is injective, we get to add \! 0" onto the second sequence. But why are the last maps in the first two sequences not surjective in general? And why is the first map in the last sequence not injective? The answers to these questions are given in terms of Ext and Tor. There are two sequences of operators n R fExtR(−; −) j n = 1; 2;:::g and fTorn (−; −) j n = 1; 2;:::g that satisfy the following properties. n (a) An R-moduleN is projective if and only if ExtR(N; −) = 0 for all n > 1. (b) Given an R-module N and an exact sequence of R-modules (&) there is a \long exact sequence" 0 00 0 ! HomR(N; M ) ! HomR(N; M) ! HomR(N; M ) 1 0 1 1 00 −! ExtR(N; M ) −! ExtR(N; M) −! ExtR(N; M ) !··· n 0 n n 00 · · · −! ExtR(N; M ) −! ExtR(N; M) −! ExtR(N; M ) !··· : n (c) An R-module N is injective if and only if ExtR(−;N) = 0 for all n > 1. v vi INTRODUCTION September 8, 2009 (d) Given an R-module N and an exact sequence of R-modules (&) there is a \long exact sequence" 00 0 0 ! HomR(M ;N) ! HomR(M; N) ! HomR(M ;N) 1 00 1 1 0 −! ExtR(M ;N) −! ExtR(M; N) −! ExtR(M ;N) !··· n 00 n n 0 · · · −! ExtR(M ;N) −! ExtR(M; N) −! ExtR(M ;N) !··· : R (e) An R-module N is flat if and only if Torn (−;N) = 0 for all n > 1. (f) Given an R-module N and an exact sequence of R-modules (&) there is a \long exact sequence" R 0 R R 00 ···! Torn (M ;N) ! Torn (M; N) ! Torn (M ;N) !··· R 0 R R 00 ···! Tor1 (M ;N) ! Tor1 (M; N) ! Tor1 (M ;N) 0 00 −−! M ⊗R N −−−! M ⊗R N −−−! M ⊗R N ! 0: The sequence in (b) shows exactly what is missing from (∗). Furthermore, when N is projective, item (a) explains exactly why we can add \! 0" onto the sequence (∗). Similar comments hold for the sequence (y); also for (#) once one knows that every projective R-module is flat. The constructions of Ext and Tor are homological in nature. So, the first answer to the question of what homological algebra gives you is: it shows you what has been missing and gives a full explanation for some special-case behaviors. Another thing homological algebra gives you is invariants for studying rings and modules. Consider the following example. How do you distinguish between the vector spaces R2 and R3? Answer: by looking at the dimensions. The first one has dimension 2 and the second one has dimension 3. Therfore, they are not isomorphic. In the study of modules over a commutative ring, even when there is a rea- sonable vector space dimension, it may not be enough to distinguish between non-isomorphic R-modules. Take for example the ring R[X; Y ] and the modules R[X; Y ]=(X; Y 2) and R[X; Y ]=(X2;Y ). Each has vector space dimension 2 (over R) but they are not isomorphic as R-modules. Homological algebra gives you new invariants (numbers, functors, categories, etc.) to attach to an R-module that give you the power to detect (sometimes) when two modules are non-isomorphic. Of course, in the last example, one doesn't need to work very hard to see why the modules are not isomorphic. But in other situations, these homological invariants can be extremely powerful tools for the study of rings and modules. And these tools are so useful that many of them have become indispensable, almost unavoidable, items for the ring theorists' toolbox. Regular sequences. Assume that R is noetherian and local with maximal ideal m. A sequence x1; : : : ; xn 2 m is R-regular if (1) the element x1 is a non-zero- divisor on R, and (2) for i = 1; : : : ; n − 1 the element xi+1 is a non-zero-divisor on the quotient R=(x1; : : : ; xi). The fact that R is noetherian implies that every R-regular sequence can be extended to a maximal one, that is, to one that cannot be further extended. It is not obvious, though, whether two maximal R-regular sequences have the same length. The fact that this works is a consequence of the following Ext-characterization: For each integer n > 1, the following conditions are equivalent: FORMAT OF THESE NOTES vii i (i) We have ExtR(R=m;M) = 0 for all i < n; (ii) Every R-regular sequence in m of length 6 n can be extended to an R-regular sequence in m of length n; and (iii) There exists an R-regular sequence of length n in m. The depth of a R is the length of a maximal R-regular sequence in m. It is the subject of Chapter V. This is a handy invariant for induction arguments because when x 2 m is an R-regular element, the rings R and R=xR are homologically very similar, but we have depth(R=xR) = depth(R) − 1. Hence, if one is proving a result by induction on depth(R), one can often apply the induction hypothesis to R=xR and then show that the desired conclusion for R=xR implies the desired conclusion for R.