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Unformatted text preview: COMMUTATIVE ALGEBRA PETE L. CLARK Contents Introduction 0.1. What is Commutative Algebra? 0.2. Why study Commutative Algebra? 0.3. Acknowledgements 1. Commutative rings 1.1. Fixing terminology 1.2. Adjoining elements 1.3. Ideals and quotient rings 1.4. The monoid of ideals of R 1.5. Pushing and pulling ideals 1.6. Maximal and prime ideals 1.7. Products of rings 1.8. A cheatsheet 2. Galois Connections 2.1. The basic formalism 2.2. Lattice Properties 2.3. Examples of Antitone Galois Connections 2.4. Antitone Galois Connections Decorticated: Relations 2.5. Isotone Galois Connections 2.6. Examples of Isotone Galois Connections 3. Modules 3.1. Basic definitions 3.2. Finitely presented modules 3.3. Torsion and torsionfree modules 3.4. Tensor and Hom 3.5. Projective modules 3.6. Injective modules 3.7. Flat modules 3.8. Nakayama’s Lemma 3.9. Ordinal Filtrations and Applications 3.10. Tor and Ext 3.11. More on flat modules 3.12. Faithful flatness 4. First Properties of Ideals in a Commutative Ring 4.1. Introducing maximal and prime ideals 4.2. Radicals Date: December 14, 2013. 1 4 4 5 7 7 7 10 11 14 15 16 16 19 19 19 21 22 24 25 26 26 26 31 34 34 36 43 50 52 56 64 72 77 80 80 83 2 PETE L. CLARK 4.3. Comaximal ideals 4.4. Local rings 4.5. The Prime Ideal Principle of Lam and Reyes 4.6. Minimal Primes 5. Examples of Rings 5.1. Rings of numbers 5.2. Rings of continuous functions 5.3. Rings of holomorphic functions 5.4. Polynomial rings 5.5. Semigroup algebras 6. Swan’s Theorem 6.1. Introduction to (topological) vector bundles 6.2. Swan’s Theorem 6.3. Proof of Swan’s Theorem 6.4. Applications of Swan’s Theorem 6.5. Stably Free Modules 6.6. The Theorem of Bkouche and Finney-Rotman 7. Localization 7.1. Definition and first properties 7.2. Pushing and pulling via a localization map 7.3. The fibers of a morphism 7.4. Commutativity of localization and passage to a quotient 7.5. Localization at a prime ideal 7.6. Localization of modules 7.7. Local properties 7.8. Local characterization of finitely generated projective modules 8. Noetherian rings 8.1. Chain conditions on partially ordered sets 8.2. Chain conditions on modules 8.3. Semisimple modules and rings 8.4. Normal Series 8.5. The Krull-Schmidt Theorem 8.6. Some important terminology 8.7. Introducing Noetherian rings 8.8. Theorems of Eakin-Nagata, Formanek and Jothilingam 8.9. The Bass-Papp Theorem 8.10. Artinian rings: structure theory 8.11. The Hilbert Basis Theorem 8.12. The Krull Intersection Theorem 8.13. Krull’s Principal Ideal Theorem 8.14. The Dimension Theorem, following [BMRH] 8.15. The Artin-Tate Lemma 9. Boolean rings 9.1. First Properties 9.2. Boolean Algebras 9.3. Ideal Theory in Boolean Rings 9.4. The Stone Representation Theorem 9.5. Boolean Spaces 86 89 90 93 94 94 95 101 104 106 112 112 113 115 118 119 126 126 126 128 130 130 131 131 133 135 138 139 140 140 143 145 149 150 151 154 155 158 159 162 164 165 165 165 166 169 170 172 COMMUTATIVE ALGEBRA 9.6. Stone Duality 9.7. Topology of Boolean Rings 10. Associated Primes and Primary Decomposition 10.1. Associated Primes 10.2. The support of a module 10.3. Primary Ideals 10.4. Primary Decomposition, Lasker and Noether 10.5. Irredundant primary decompositions 10.6. Uniqueness properties of primary decomposition 10.7. Applications in dimension zero 10.8. Applications in dimension one 11. Nullstellens¨ atze 11.1. Zariski’s Lemma 11.2. Hilbert’s Nullstellensatz 11.3. The Real Nullstellensatz 11.4. The Combinatorial Nullstellensatz 11.5. The Finite Field Nullstellensatz 11.6. Terjanian’s Homogeneous p-Nullstellensatz 12. Goldman domains and Hilbert-Jacobson rings 12.1. Goldman domains 12.2. Hilbert rings 12.3. Jacobson Rings 12.4. Hilbert-Jacobson Rings 13. Spec R as a topological space 13.1. The Zariski spectrum 13.2. Properties of the spectrum: quasi-compactness 13.3. Properties of the spectrum: separation and specialization 13.4. Irreducible spaces 13.5. Noetherian spaces 13.6. Hochster’s Theorem 13.7. Rank functions revisited 14. Integrality in Ring Extensions 14.1. First properties of integral extensions 14.2. Integral closure of domains 14.3. Spectral properties of integral extensions 14.4. Integrally closed domains 14.5. The Noether Normalization Theorem 14.6. Some Classical Invariant Theory 14.7. Galois extensions of integrally closed domains 14.8. Almost Integral Extensions 15. Factorization 15.1. Kaplansky’s Theorem (II) 15.2. Atomic domains, (ACCP) 15.3. EL-domains 15.4. GCD-domains 15.5. GCDs versus LCMs 15.6. Polynomial rings over UFDs 15.7. Application: the Sch¨onemann-Eisenstein Criterion 3 174 175 176 176 179 180 182 184 184 187 187 187 187 188 193 195 197 198 202 202 205 206 207 208 208 209 210 212 214 216 217 219 219 221 223 225 226 229 233 234 235 235 236 238 239 241 243 247 4 PETE L. CLARK 15.8. Application: Determination of Spec R[t] for a PID R 15.9. Power series rings over UFDs 15.10. Nagata’s Criterion 16. Principal rings and B´ezout domains 16.1. Principal ideal domains 16.2. Some structure theory of principal rings 16.3. Euclidean functions and Euclidean rings 16.4. B´ezout domains 17. Valuation rings 17.1. Basic theory 17.2. Ordered abelian groups 17.3. Connections with integral closure 17.4. Another proof of Zariski’s Lemma 17.5. Discrete valuation rings 18. Normalization theorems 18.1. The First Normalization Theorem 18.2. The Second Normalization Theorem 18.3. The Krull-Akizuki Theorem 19. The Picard Group and the Divisor Class Group 19.1. Fractional ideals 19.2. The Ideal Closure 19.3. Invertible fractional ideals and the Picard group 19.4. Divisorial ideals and the Divisor Class Group 20. Dedekind domains 20.1. Characterization in terms of invertibility of ideals 20.2. Ideal factorization in Dedekind domains 20.3. Local characterization of Dedekind domains 20.4. Factorization into primes implies Dedekind 20.5. Generation of ideals in Dedekind domains 20.6. Finitely generated modules over a Dedekind domain 20.7. Injective Modules 21. Pr¨ ufer domains 21.1. Characterizations of Pr¨ ufer Domains 21.2. Butts’s Criterion for a Dedekind Domain 21.3. Modules over a Pr¨ ufer domain 22. One Dimensional Noetherian Domains 22.1. Finite Quotient Domains 23. Structure of overrings 23.1. Introducing overrings 23.2. Overrings of Dedekind domains 23.3. Elasticity in Replete Dedekind Domains 23.4. Overrings of Pr¨ ufer Domains 23.5. Kaplansky’s Theorem (III) 23.6. Every commutative group is a class group References 248 249 250 253 253 256 257 259 260 260 263 267 268 269 272 272 274 274 276 276 278 279 283 285 285 286 288 288 289 290 292 294 294 297 299 299 299 302 302 303 307 310 311 312 316 COMMUTATIVE ALGEBRA 5 Introduction 0.1. What is Commutative Algebra? Commutative algebra is the study of commutative rings and attendant structures, especially ideals and modules. This is the only possible short answer I can think of, but it is not completely satisfying. We might as well say that Hamlet, Prince of Denmark is about a fictional royal family in late medieval Denmark and especially about the title (crown) prince, whose father (i.e., the King) has recently died and whose father’s brother has married his mother (i.e., the Queen). Informative, but not the whole story! 0.2. Why study Commutative Algebra? What are the purely mathematical reasons for studying any subject of pure mathematics? I can think of two: I. Commutative algebra is a necessary and/or useful prerequisite for the study of other fields of mathematics in which we are interested. II. We find commutative algebra to be intrinsically interesting and we want to learn more. Perhaps we even wish to discover new results in this area. Most beginning students of commutative algebra can relate to the first reason: they need, or are told they need, to learn some commutative algebra for their study of other subjects. Indeed, commutative algebra has come to occupy a remarkably central role in modern pure mathematics, perhaps second only to category theory in its ubiquitousness, but in a different way. Category theory provides a common language and builds bridges between different areas of mathematics: it is something like a circulatory system. Commutative algebra provides core results that other results draw upon in a foundational way: it is something like a skeleton. The branch of mathematics which most of all draws upon commutative algebra for its structural integrity is algebraic geometry, the study of geometric properties of manifolds and singular spaces which arise as the loci of solutions to systems of polynomial equations. In fact there is a hard lesson here: in the 19th century algebraic geometry split off from complex function theory and differential geometry as its own discipline and then burgeoned dramatically at the turn of the century and the years thereafter. But by 1920 or so the practitioners of the subject had found their way into territory in which “purely geometric” reasoning led to serious errors. In particular they had been making arguments about how algebraic varieties behave generically, but they lacked the technology to even give a precise meaning to the term. Thus the subject ultimately proved invertebrate and began to collapse under its own weight. Starting around 1930 there began a heroic shoring up process in which the foundations of the subject were recast with commutative algebraic methods at the core. This was done several times over, in different ways, by Zariski, Weil, Serre and Grothendieck, among others. For the last 60 years it 6 PETE L. CLARK has been impossible to deeply study algebraic geometry without knowing commutative algebra – a lot of commutative algebra. (More than is contained in these notes!) The other branch of mathematics which draws upon commutative algebra in an absolutely essential way is algebraic number theory. One sees this from the beginning in that the Fundamental Theorem of Arithmetic is the assertion that the ring Z is a unique factorization domain (UFD), a basic commutative algebraic concept. Moreover number theory was one of the historical sources of the subject. Notably the concept of Dedekind domain came from Richard Dedekind’s number-theoretic investigations. Knowledge of commutative algebra is not as indispensable for number theory (at least, not at the beginning) as it is for algebraic geometry, but such knowledge brings a great clarifying effect to the subject. In fact the interplay among number theory, algebraic geometry and commutative algebra flows in all directions. What Grothendieck did in the 1960s (with important contributions from Chevalley, Serre and others) was to create a single field of mathematics that encompassed commutative algebra, classical algebraic geometry and algebraic number theory: the theory of schemes. As a result, most contemporary number theorists are also partly commutative algebraists and partly algebraic geometers: we call this cosmopolitan take on the subject arithmetic geometry. There are other areas of mathematics that draw upon commutative algebra in important ways. To mention some which will show up in later in these notes: • • • • Differential topology. General topology. Invariant theory. Order theory. The task of providing a commutative algebraic foundation for algebraic geometry – or even the single, seminal text of R. Hartshorne – is a daunting one. Happily, this task has been completed by David Eisenbud (a leading contemporary expert on the interface of commutative algebra and algebraic geometry) in his text [Eis]. This work is highly recommended. It is also 797 pages long, so contains enough material for 3 − 5 courses in the subject. It would be folly to try to improve upon, or even successfully imitate, Eisenbud’s work here, and I certainly have not tried. I myself am an arithmetic geometer (which, as I tried to explain above, is a sort of uppity kind of number theorist), so it is not surprising that these notes are skewed more towards number theory than most introductory texts on commutative algebra. However for the most part a respectful distance is maintained: we rarely discuss number theory per se but rather classes of rings that a number theorist would like: Dedekind domains, valuation rings, B´ezout domains, and so forth. Just much as I have included some material of interest to number theorists I have included material making connections to other branches of mathematics, especially connections which are less traditionally made in commutative algebra texts. In fact at several points I have digressed to discuss topics and theorems which make COMMUTATIVE ALGEBRA 7 connections to other areas of mathematics: • • • • • • §2 on Galois connections. §5.2 on rings of continuous functions. §6 on vector bundles and Swan’s Theorem. §9 on Boolean rings, Boolean spaces and Stone Duality. §13 on the topology of prime spectra, including Hochster’s Theorem. §14.6 on invariant theory, including the Shephard-Todd-Chevalley Theorem. But I do find commutative algebra to be of interest unto itself, and I have tried to craft a sustained narrative rather than just a collection of results. 0.3. Acknowledgements. Thanks to Max Bender, Martin Brandenburg, John Doyle, Georges Elencwajg, Emil Jerabek, Keenan Kidwell, David Krumm, Allan Lacy, Casey LaRue, Stacy ´ Musgrave, Alon Regev, Jacob Schlather, Jack Schmidt, Mariano Su´arez-Alvarez and Matth´e van der Lee for catching errors1 and making other useful suggestions. Thanks to Hans Parshall for introducing me to the Stone-Tukey Theorem. 1. Commutative rings 1.1. Fixing terminology. We are interested in studying properties of commutative rings with unity. By a general algebra R, we mean a triple (R, +, ·) where R is a set endowed with a binary operation + : R × R → R – called addition – and a binary operation · : R × R → R – called multiplication – satisfying the following: (CG) (R, +) is a commutative group, (D) For all a, b, c ∈ R, (a + b) · c = a · c + b · c, a · (b + c) = a · b + a · c. For at least fifty years, there has been agreement that in order for an algebra to be a ring, it must satisfy the additional axiom of associativity of multiplication: (AM) For all a, b, c ∈ R, a · (b · c) = (a · b) · c. A general algebra which satisfies (AM) will be called simply an algebra. A similar convention that is prevalent in the literature is the use of the term nonassociative algebra to mean what we have called a general algebra: i.e., a not necessarily associative algebra. A ring R is said to be with unity if there exists a multiplicative identity, i.e., an element e of R such that for all a ∈ R we have e · a = a · e = a. If e and e0 are two such elements, then e = e · e0 = e0 . In other words, if a unity exists, it is unique, and we will denote it by 1. 1Of which many, many remain: your name could go here! 8 PETE L. CLARK A ring R is commutative if for all x, y ∈ R, x · y = y · x. In these notes we will be (almost) always working in the category of commutative rings with unity. In a sense which will shortly be made precise, this means that the identity 1 is regarded as part of the structure of a ring and must therefore be preserved by all homomorphisms. Probably it would be more natural to study the class of possibly non-commutative rings with unity, since, as we will see, many of the fundamental constructions of rings give rise, in general, to non-commutative rings. But if the restriction to commutative rings (with unity!) is an artifice, it is a very useful one, since two of the most fundamental notions in the theory, that of ideal and module, become significantly different and more complicated in the non-commutative case. It is nevertheless true that many individual results have simple analogues in the noncommutative case. But it does not seem necessary to carry along the extra generality of non-commutative rings; rather, when one is interested in the non-commutative case, one can simply remark “Proposition X.Y holds for (left) R-modules over a noncommutative ring R.” Notation: Generally we shall abbreviate x · y to xy. Moreover, we usually do not use different symbols to denote the operations of addition and multiplication in different rings: it will be seen that this leads to simplicity rather than confusion. Group of units: Let R be a ring with unity. An element x ∈ R is said to be a unit if there exists an element y such that xy = yx = 1. Exercise 1.1: a) Show that if x is a unit, the element y with xy = yx = 1 is unique, denoted x−1 . b) Show that if x is a unit, so is x−1 . c) Show that, for all x, y ∈ R, xy is a unit ⇐⇒ x and y are both units. d) Deduce that the units form a commutative group, denoted R× , under multiplication. Example (Zero ring): Our rings come with two distinguished elements, the additive identity 0 and the multiplicative identity 1. Suppose that 0 = 1. Then for x ∈ R, x = 1 · x = 0 · x, whereas in any rin g 0 · x = (0 + 0) · x = 0 · x + 0 · x, so 0 · x = 0. In other words, if 0 = 1, then this is the only element in the ring. It is clear that for any one element set R = {0}, 0 + 0 = 0 · 0 = 0 endows R with the structure of a ring. We call this ring the zero ring. The zero ring exhibits some strange behavior, such that it must be explicitly excluded in many results. For instance, the zero element is a unit in the zero ring, which is obviously not the case in any nonzero ring. A nonzero ring in which every nonzero element is a unit is called a division ring. A commutative division ring is called a field. Let R and S be rings (with unity). A homomorphism f : R → S is a map COMMUTATIVE ALGEBRA 9 of sets which satisfies (HOM1) For all x, y ∈ R, f (x + y) = f (x) + f (y). (HOM2) For all x, y ∈ R, f (xy) = f (x)f (y). (HOM3) f (1) = 1. Note that (HOM1) implies f (0) = f (0 + 0) = f (0) + f (0), so f (0) = 0. Thus we do not need to explcitly include f (0) = 0 in the definition of a group homomorphism. For the multiplicative identity however, this argument only shows that if f (1) is a unit, then f (1) = 1. Therefore, if we did not require (HOM3), then for instance the map f : R → R, f (x) = 0 for all x, would be a homomorphism, and we do not want this. Exercise 1.2: Suppose R and S are rings, and let f : R → S be a map satisfying (HOM1) and (HOM2). Show that f is a homomorphism of rings (i.e., satisfies also f (1) = 1) iff f (1) ∈ S × . A homomorphism f : R → S is an isomorphism if there exists a homomorphism g : S → R such that: for all x ∈ R, g(f (x)) = x; and for all y ∈ S, f (g(y)) = y. Exercise 1.3: Let f : R → S be a homomorphism of rings. Show TFAE: (i) f is a bijection. (ii) f is an isomorphism. Remark: In many algebra texts, an isomorphism of rings (or groups, etc.) is defined to be a bijective homomorphism, but this gives the wrong idea of what an isomorphism should be in other mathematical contexts (e.g. for topological spaces). Rather, having defined the notion of a morphism of any kind, one defines isomorphism in the way we have above. Exercise 1.4: a) Suppose R and S are both rings on a set containing exactly one element. Show that there is a unique ring isomorphism from R to S. (This is a triviality, but explains why are we able to speak of the zero ring, rather than simply the zero ring associated to one element set. We will therefore denote the zero ring just by 0.) b) Show that any ring R admits a unique homomorphism to the zero ring. One says that the zero ring is the final object in the category of rings. Exercise 1.5: Show that for a not-necessarily-commutative-ring S there exists a unique homomorphism from the ring Z of integers to S. (Thus Z is the initial object in the category of not-necessarily-commutative-rings. It follows immediately that it is also the initial object in the category of rings.) A subring R of a ring S is a subset R of S such that (SR1) 1 ∈ R. (SR2) For all r, s ∈ R, r + s ∈ R, r − s ∈ R, and rs ∈ R. 10 PETE L. CLARK Here (SR2) expresses that the subset R is an algebra under the operations of addition and multiplication defined on S. Working, as we are, with...
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