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Commutative Algebra Version of September 3, 2012 A Term of Commutative Algebra By Allen ALTMAN and Steven KLEIMAN Contents Preface . iii 1. Rings and Ideals ................... 1 2. Prime Ideals .................... 6 3. Radicals ...................... 10 4. Modules ...................... 14 5. Exact Sequences ................... 20 6. Direct Limits .................... 26 7. Filtered Direct Limits . 33 8. Tensor Products ................... 37 9. Flatness ...................... 43 10. Cayley{Hamilton Theorem . 49 11. Localization of Rings . 55 12. Localization of Modules . 61 13. Support ..................... 66 14. Krull{Cohen{Seidenberg Theory . 71 15. Noether Normalization . 75 Appendix: Jacobson Rings . 80 16. Chain Conditions . 82 17. Associated Primes . 87 18. Primary Decomposition . 91 19. Length ...................... 97 20. Hilbert Functions . 101 Appendix: Homogeneity . 107 21. Dimension . 109 22. Completion . 115 23. Discrete Valuation Rings . 122 24. Dedekind Domains . 127 25. Fractional Ideals . 131 26. Arbitrary Valuation Rings . 136 Solutions . 141 1. Rings and Ideals . 141 2. Prime Ideals . 143 3. Radicals . 145 4. Modules . 148 5. Exact Sequences . 149 i 6. Direct Limits . 153 7. Filtered direct limits . 156 8. Tensor Products . 158 9. Flatness . 159 10. Cayley{Hamilton Theorem . 161 11. Localization of Rings . 164 12. Localization of Modules . 167 13. Support . 168 14. Krull{Cohen{Seidenberg Theory . 171 15. Noether Normalization . 174 16. Chain Conditions . 177 17. Associated Primes . 179 18. Primary Decomposition . 180 19. Length . 183 20. Hilbert Functions . 185 21. Dimension . 188 22. Completion . 190 23. Discrete Valuation Rings . 193 24. Dedekind Domains . 197 25. Fractional Ideals . 199 26. Arbitrary Valuation Rings . 200 References . 202 Index . 203 ii Preface There is no shortage of books on Commutative Algebra, but the present book is different. Most books are monographs, with extensive coverage. There is one notable exception: Atiyah and Macdonald's 1969 classic [2]. It is a clear, concise, and efficient textbook, aimed at beginners, with a good selection of topics. So it has remained popular. However, its age and flaws do show. So there is need for an updated and improved version, which the present book aims to be. Atiyah and Macdonald explain their philosophy in their introduction. They say their book \has the modest aim of providing a rapid introduction to the subject. It is designed to be read by students who have had a first elementary course in general algebra. On the other hand, it is not intended as a substitute for the more voluminous tracts on Commutative Algebra . The lecture-note origin of this book accounts for the rather terse style, with little general padding, and for the condensed account of many proofs." They \resisted the temptation to expand it in the hope that the brevity of [the] presentation will make clearer the mathematical structure of what is by now an elegant and attractive theory." They endeavor \to build up to the main theorems in a succession of simple steps and to omit routine verifications.” Their successful philosophy is wholeheartedly embraced below (it is a feature, not a flaw!), and also refined a bit. The present book also \grew out of a course of lectures." That course was based primarily on their book, but has been offered a number of times, and has evolved over the years, influenced by other publications and the reactions of the students. Their book comprises eleven chapters, split into forty-two sections. The present book comprises twenty-six sections; each represents a single lecture, and is self-contained. Atiyah and Macdonald \provided . exercises at the end of each chapter." They \provided hints, and sometimes complete solutions, to the hard" exercises. More- over, they developed a significant amount of the main content in the exercises. By contrast, in the present book, the exercises are integrated into the development, and complete solutions are given at the end of the book. There are well over two hundred exercises below. Included are nearly all the exercises in Atiyah and Macdonald's book. Included also are many exercises that come from other publications and many that originate here. Here the exercises are tailored to provide a means for students to check, solidify, and expand their understanding of the material. The exercises are intentionally not difficult, tricky, or involved. Rarely do they introduce new techniques, although some introduce new concepts and many statements are used later. Students are encouraged to try to solve each and every exercise, and to do so before looking up its solution. If they become stuck, then they should review the relevant material; if they remain stuck, then they should study the given solution, possibly discussing it with others, but always making sure they can eventually solve the whole exercise completely on their own. In any event, students should read the given solution, even if they think they already know it, just to make sure; also, some exercises provide enlightening alternative solutions. Finally, instructors are encouraged to examine their students, possibly orally at a blackboard, on a small iii iv Preface randomly chosen subset of exercises that have been assigned for the students to write up in their own words over the course of the term. Atiyah and Macdonald explain that \a proper treatment of Homological Algebra is impossible within the confines of a small book; on the other hand, it is hardly sensible to ignore it completely." So they \use elementary homological methods | exact sequence, diagrams, etc. | but . stop short of any results requiring a deep study of homology." Again, their philosophy is embraced and refined in the present book. Notably, below, elementary methods are used, not Tor's as they do, to prove the Ideal Criterion for flatness, and to relate flat modules and free modules over local rings. Also, projective modules are treated below, but not in their book. In the present book, Category Theory is a basic tool; in Atiyah and Macdonald's, it seems like a foreign language. Thus they discuss the universal (mapping) property (UMP) of localization of a ring, but provide an ad hoc characterization. They also prove the UMP of tensor product of modules, but do not use the term this time. Below, the UMP is fundamental: there are many canonical constructions; each has a UMP, which serves to characterize the construction up to unique isomorphism owing to one general observation of Category Theory. For example, the Left Exactness of Hom is viewed simply as expressing in other words that the kernel and the cokernel of a map are characterized by their UMPs; by contrast, Atiyah and Macdonald prove the Left Exactness via a tedious elementary argument. Atiyah and Macdonald prove the Adjoint-Associativity Formula. They note it says that Tensor Product is the left adjoint of Hom. From it and the Left Exactness of Hom, they deduce the Right Exactness of Tensor Product. They note that this derivation shows that any \left adjoint is right exact." More generally, as explained below, this derivation shows that any left adjoint preserves arbitrary direct limits, ones indexed by any small category. Atiyah and Macdonald consider only direct limits indexed by a directed set, and sketch an ad hoc argument showing that tensor product preserves direct limit. Also, arbitrary direct sums are direct limits indexed by a discrete category (it is not a directed set); hence, the general result yields that Tensor Product and other left adjoints preserve arbitrary Direct Sum. Below, left adjoints are proved unique up to unique isomorphism. Therefore, the functor of localization of a module is canonically isomorphic to the functor of tensor product with the localized base ring, as both are left adjoints of the same functor, Restriction of Scalars from the localized ring to the base ring. There is an alternative argument. Since Localization is a left adjoint, it preserves Direct Sum and Cokernel; whence, it is isomorphic to that tensor-product functor by Watts Theorem, which characterizes all tensor-product functors as those linear functors that preserve Direct Sum and Cokernel. Atiyah and Macdonald's treatment is ad hoc. However, they do use the proof of Watts Theorem directly to show that, under the appropriate conditions, Completion of a module is Tensor Product with the completed base ring. Below, Direct Limit is also considered as a functor, defined on the appropriate category of functors. As such, Direct Limit is a left adjoint. Hence, direct limits preserve other direct limits. Here the theory briefly reaches a higher level of ab- straction. This discussion is completely elementary, but by far the most abstract part of the book. The extra abstraction can be difficult, especially for beginners. Below, filtered direct limits are treated too. They are closer to the kind of limits treated by Atiyah and Macdonald. In particular, filtered direct limits preserve Preface v exactness and flatness. Further, they appear in the following lovely form of Lazard's Theorem: in a canonical way, every module is the direct limit of free modules of finite rank; moreover, the module is flat if and only if that direct limit is filtered. Atiyah and Macdonald handle primary decomposition in a somewhat personal and dated fashion. First, they study primary decompositions of ideals in rings. Then, in the exercises, they indicate how to translate the theory to modules. The decompositions need not exist, as the rings and modules need not be Noetherian. Associated primes play a secondary role: they are defined as the radicals of the primary components, and then characterized as the primes that are the radicals of annihilators of elements. Finally, they prove that, when the rings and modules are Noetherian, decompositions exist and the associated primes are annihilators. To prove existence, they study irreducible modules. Nowadays, associated primes are normally defined as prime annihilators of elements, and studied on their own at first; sometimes, as below, irreducible modules are not considered.
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