COMMUTATIVE ALGEBRA PETE L. CLARK Contents Introduction 5 0.1. What is Commutative Algebra? 5 0.2. Why study Commutative Algebra? 5 0.3. Acknowledgments 7 1. Commutative rings 7 1.1. Fixing terminology 7 1.2. Adjoining elements 10 1.3. Ideals and quotient rings 11 1.4. The monoid of ideals of R 14 1.5. Pushing and pulling ideals 15 1.6. Maximal and prime ideals 16 1.7. Products of rings 17 1.8. A cheatsheet 19 2. Galois Connections 20 2.1. The basic formalism 20 2.2. Lattice Properties 22 2.3. Examples of Antitone Galois Connections 22 2.4. Antitone Galois Connections Decorticated: Relations 25 2.5. Isotone Galois Connections 26 2.6. Examples of Isotone Galois Connections 26 3. Modules 27 3.1. Basic definitions 27 3.2. Finitely presented modules 32 3.3. Torsion and torsionfree modules 34 3.4. Tensor and Hom 35 3.5. Projective modules 37 3.6. Injective modules 44 3.7. Flat modules 51 3.8. Nakayama's Lemma 52 3.9. Ordinal Filtrations and Applications 56 3.10. Tor and Ext 64 3.11. More on flat modules 72 3.12. Faithful flatness 78 4. First Properties of Ideals in a Commutative Ring 81 4.1. Introducing maximal and prime ideals 81 4.2. Radicals 84 Date: March 9, 2015. 1 2 PETE L. CLARK 4.3. Comaximal ideals 87 4.4. Local rings 89 4.5. The Prime Ideal Principle of Lam and Reyes 90 4.6. Minimal Primes 93 5. Examples of Rings 94 5.1. Rings of numbers 94 5.2. Rings of continuous functions 95 5.3. Rings of holomorphic functions 102 5.4. Polynomial rings 104 5.5. Semigroup algebras 106 6. Swan's Theorem 112 6.1. Introduction to (topological) vector bundles 113 6.2. Swan's Theorem 114 6.3. Proof of Swan's Theorem 115 6.4. Applications of Swan's Theorem 118 6.5. Stably Free Modules 119 6.6. The Theorem of Bkouche and Finney-Rotman 126 7. Localization 126 7.1. Definition and first properties 126 7.2. Pushing and pulling via a localization map 128 7.3. The fibers of a morphism 130 7.4. Commutativity of localization and passage to a quotient 131 7.5. Localization at a prime ideal 131 7.6. Localization of modules 131 7.7. Local properties 133 7.8. Local characterization of finitely generated projective modules 135 8. Noetherian rings 139 8.1. Chain conditions on partially ordered sets 139 8.2. Chain conditions on modules 140 8.3. Semisimple modules and rings 141 8.4. Normal Series 144 8.5. The Krull-Schmidt Theorem 145 8.6. Some important terminology 150 8.7. Introducing Noetherian rings 151 8.8. Theorems of Eakin-Nagata, Formanek and Jothilingam 152 8.9. The Bass-Papp Theorem 154 8.10. Artinian rings: structure theory 155 8.11. The Hilbert Basis Theorem 158 8.12. The Krull Intersection Theorem 160 8.13. Krull's Principal Ideal Theorem 162 8.14. The Dimension Theorem, following [BMRH] 165 8.15. The Artin-Tate Lemma 165 9. Boolean rings 166 9.1. First Properties 166 9.2. Boolean Algebras 166 9.3. Ideal Theory in Boolean Rings 169 9.4. The Stone Representation Theorem 171 9.5. Boolean Spaces 172 COMMUTATIVE ALGEBRA 3 9.6. Stone Duality 175 9.7. Topology of Boolean Rings 176 10. Associated Primes and Primary Decomposition 176 10.1. Associated Primes 176 10.2. The support of a module 179 10.3. Primary Ideals 180 10.4. Primary Decomposition, Lasker and Noether 182 10.5. Irredundant primary decompositions 184 10.6. Uniqueness properties of primary decomposition 185 10.7. Applications in dimension zero 187 10.8. Applications in dimension one 187 11. Nullstellens¨atze 188 11.1. Zariski's Lemma 188 11.2. Hilbert's Nullstellensatz 189 11.3. The Real Nullstellensatz 193 11.4. The Combinatorial Nullstellensatz 196 11.5. The Finite Field Nullstellensatz 198 11.6. Terjanian's Homogeneous p-Nullstellensatz 199 12. Goldman domains and Hilbert-Jacobson rings 203 12.1. Goldman domains 203 12.2. Hilbert rings 206 12.3. Jacobson Rings 207 12.4. Hilbert-Jacobson Rings 208 13. Spec R as a topological space 209 13.1. The Zariski spectrum 209 13.2. Properties of the spectrum: quasi-compactness 210 13.3. Properties of the spectrum: separation and specialization 211 13.4. Irreducible spaces 213 13.5. Noetherian spaces 214 13.6. Hochster's Theorem 216 13.7. Rank functions revisited 217 14. Integrality in Ring Extensions 219 14.1. First properties of integral extensions 219 14.2. Integral closure of domains 221 14.3. Spectral properties of integral extensions 224 14.4. Integrally closed domains 225 14.5. The Noether Normalization Theorem 227 14.6. Some Classical Invariant Theory 229 14.7. Galois extensions of integrally closed domains 233 14.8. Almost Integral Extensions 234 15. Factorization 235 15.1. Kaplansky's Theorem (II) 236 15.2. Atomic domains, (ACCP) 237 15.3. EL-domains 238 15.4. GCD-domains 239 15.5. GCDs versus LCMs 241 15.6. Polynomial rings over UFDs 243 15.7. Application: the Sch¨onemann-EisensteinCriterion 247 4 PETE L. CLARK 15.8. Application: Determination of Spec R[t] for a PID R 248 15.9. Power series rings over UFDs 249 15.10. Nagata's Criterion 250 16. Principal rings and B´ezoutdomains 253 16.1. Principal ideal domains 253 16.2. Some structure theory of principal rings 256 16.3. Euclidean functions and Euclidean rings 257 16.4. B´ezoutdomains 259 17. Valuation rings 260 17.1. Basic theory 260 17.2. Ordered abelian groups 263 17.3. Connections with integral closure 267 17.4. Another proof of Zariski's Lemma 268 17.5. Discrete valuation rings 269 18. Normalization theorems 272 18.1. The First Normalization Theorem 272 18.2. The Second Normalization Theorem 274 18.3. The Krull-Akizuki Theorem 274 19. The Picard Group and the Divisor Class Group 276 19.1. Fractional ideals 276 19.2. The Ideal Closure 278 19.3. Invertible fractional ideals and the Picard group 279 19.4. Divisorial ideals and the Divisor Class Group 283 20. Dedekind domains 285 20.1. Characterization in terms of invertibility of ideals 285 20.2. Ideal factorization in Dedekind domains 286 20.3. Local characterization of Dedekind domains 288 20.4. Factorization into primes implies Dedekind 288 20.5. Generation of ideals in Dedekind domains 289 20.6. Finitely generated modules over a Dedekind domain 290 20.7. Injective Modules 292 21. Pr¨uferdomains 294 21.1. Characterizations of Pr¨uferDomains 294 21.2. Butts's Criterion for a Dedekind Domain 297 21.3. Modules over a Pr¨uferdomain 298 22. One Dimensional Noetherian Domains 299 22.1. Finite Quotient Domains 299 23. Structure of overrings 302 23.1. Introducing overrings 302 23.2. Overrings of Dedekind domains 303 23.3. Elasticity in Replete Dedekind Domains 307 23.4. Overrings of Pr¨uferDomains 310 23.5. Kaplansky's Theorem (III) 311 23.6. Every commutative group is a class group 312 References 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 fic- tional 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 math- ematics? 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 re- sults 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 al- gebraic 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 seri- ous 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.