Commutative Algebra [Lecture notes]

363 Pages·2016·1.937 MB·English

by  Pete L. Clark

Commutative Algebra
Pete L. Clark

Contents
Introduction 9
1. What is Commutative Algebra? 9
2. Why study Commutative Algebra? 9
3. What distinguishes this text 10
4. More on the contents 11
5. Acknowledgments 12
Chapter 1. Commutative rings 13
1. Fixing terminology 13
2. Adjoining elements 16
3. Ideals and quotient rings 17
4. The monoid of ideals of R 20
5. Pushing and pulling ideals 21
6. Maximal and prime ideals 22
7. Products of rings 22
8. A cheatsheet 24
Chapter 2. Galois Connections 27
1. The basic formalism 27
2. Lattice properties 29
3. Examples of Antitone Galois Connections 29
4. Antitone Galois Connections Decorticated: Relations 31
5. Isotone Galois Connections 32
6. Examples of Isotone Galois Connections 33
Chapter 3. Modules 35
1. Basic definitions 35
2. Finitely presented modules 40
3. Torsion and torsionfree modules 42
4. Tensor and Hom 43
5. Projective modules 44
6. Injective modules 51
7. Flat modules 58
8. Nakayama’s Lemma 60
9. Ordinal Filtrations and Applications 64
10. Tor and Ext 72
11. More on flat modules 80
12. Faithful flatness 85
Chapter 4. First Properties of Ideals in a Commutative Ring 89
3
4 CONTENTS
1. Introducing maximal and prime ideals 89
2. Radicals 91
3. Comaximal ideals 95
4. Local rings 98
5. The Prime Ideal Principle of Lam and Reyes 99
6. Minimal Primes 101
7. An application to unit groups 102
Chapter 5. Examples of Rings 103
1. Rings of numbers 103
2. Rings of continuous functions 104
3. Rings of holomorphic functions 110
4. Kapovich’s Theorem 112
5. Polynomial rings 116
6. Semigroup algebras 118
Chapter 6. Swan’s Theorem 125
1. Introduction to (topological) vector bundles 125
2. Swan’s Theorem 126
3. Proof of Swan’s Theorem 127
4. Applications of Swan’s Theorem 131
5. Stably free modules 131
Chapter 7. Localization 139
1. Definition and first properties 139
2. Pushing and pulling via a localization map 141
3. The fibers of a morphism 143
4. Commutativity of localization and passage to a quotient 143
5. Localization at a prime ideal 144
6. Localization of modules 144
7. Local properties 145
8. Local characterization of finitely generated projective modules 150
Chapter 8. Noetherian rings 155
1. Chain conditions on partially ordered sets 155
2. Chain conditions on modules 156
3. Semisimple modules and rings 157
4. Normal Series 160
5. The Krull-Schmidt Theorem 161
6. Some important terminology 166
7. Introducing Noetherian rings 167
8. Theorems of Eakin-Nagata, Formanek and Jothilingam 168
9. The Bass-Papp Theorem 170
10. Artinian rings: structure theory 171
11. The Hilbert Basis Theorem 174
12. The Krull Intersection Theorem 176
13. Krull’s Principal Ideal Theorem 178
14. The Dimension Theorem, following [BMRH] 181
15. The Artin-Tate Lemma 181
CONTENTS 5
Chapter 9. Boolean rings 183
1. First Properties 183
2. Boolean Algebras 183
3. Ideal Theory in Boolean Rings 186
4. The Stone Representation Theorem 188
5. Boolean Spaces 189
6. Stone Duality 191
Chapter 10. Associated Primes and Primary Decomposition 193
1. Associated Primes 193
2. The support of a module 196
3. Primary Ideals 197
4. Primary Decomposition, Lasker and Noether 199
5. Irredundant primary decompositions 200
6. Uniqueness properties of primary decomposition 201
7. Applications in dimension zero 204
8. Applications in dimension one 204
Chapter 11. Nullstellens¨atze 205
1. Zariski’s Lemma 205
2. Hilbert’s Nullstellensatz 206
3. The Real Nullstellensatz 210
4. The Combinatorial Nullstellensatz 213
5. The Finite Field Nullstellensatz 214
6. Terjanian’s Homogeneous p-Nullstellensatz 215
Chapter 12. Goldman domains and Hilbert-Jacobson rings 221
1. Goldman domains 221
2. Hilbert rings 224
3. Jacobson Rings 225
4. Hilbert-Jacobson Rings 226
Chapter 13. SpecR as a topological space 227
1. The Prime Spectrum 227
2. Properties of the spectrum: quasi-compactness 228
3. Properties of the spectrum: connectedness 229
4. Properties of the spectrum: separation and specialization 229
5. Irreducible spaces 232
6. Noetherian spaces 234
7. Krull Dimension of Topological Spaces 236
8. Jacobson spaces 236
9. Hochster’s Theorem 240
10. Rank functions revisited 241
11. The Forster-Swan Theorem 242
Chapter 14. Integral Extensions 245
1. First properties of integral extensions 245
2. Integral closure of domains 247
3. Spectral properties of integral extensions 249
4. Integrally closed domains 250
6 CONTENTS
5. The Noether Normalization Theorem 252
6. Some Classical Invariant Theory 255
7. Galois extensions of integrally closed domains 259
8. Almost Integral Extensions 260
Chapter 15. Factorization 261
1. Kaplansky’s Theorem (II) 261
2. Atomic domains, ACCP 262
3. EL-domains 264
4. GCD-domains 265
5. GCDs versus LCMs 267
6. Polynomial rings over UFDs 269
7. Application: the Sch¨onemann-Eisenstein Criterion 273
8. Application: Determination of SpecR[t] for a PID R 274
9. Power series rings over UFDs 275
10. Nagata’s Criterion 276
Chapter 16. Principal rings and B´ezout domains 281
1. Principal ideal domains 281
2. Some structure theory of principal rings 283
3. Euclidean functions and Euclidean rings 285
4. B´ezout domains 287
Chapter 17. Valuation rings 289
1. Basic theory 289
2. Ordered abelian groups 291
3. Connections with integral closure 295
4. Another proof of Zariski’s Lemma 297
5. Discrete valuation rings 298
Chapter 18. Normalization theorems 301
1. The First Normalization Theorem 301
2. The Second Normalization Theorem 302
3. The Krull-Akizuki Theorem 303
Chapter 19. The Picard Group and the Divisor Class Group 305
1. Fractional ideals 305
2. The Ideal Closure 307
3. Invertible fractional ideals and the Picard group 307
4. Divisorial ideals and the Divisor Class Group 312
Chapter 20. Dedekind domains 315
1. Characterization in terms of invertibility of ideals 315
2. Ideal factorization in Dedekind domains 316
3. Local characterization of Dedekind domains 317
4. Factorization into primes implies Dedekind 318
5. Generation of ideals in Dedekind domains 319
6. Finitely generated modules over a Dedekind domain 320
7. Injective Modules 322
Chapter 21. Pru¨fer domains 325
CONTENTS 7
1. Characterizations of Pru¨fer Domains 325
2. Butts’s Criterion for a Dedekind Domain 328
3. Modules over a Pru¨fer domain 330
Chapter 22. One Dimensional Noetherian Domains 331
1. Residually Finite Domains 331
2. Cohen-Kaplansky domains 334
3. Rings of Finite Rank 338
Chapter 23. Structure of overrings 343
1. Introducing overrings 343
2. Overrings of Dedekind domains 343
3. Elasticity in Replete Dedekind Domains 347
4. Overrings of Pru¨fer Domains 350
5. Kaplansky’s Theorem (III) 352
6. Every commutative group is a class group 353
Bibliography 357

Introduction
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-
tionalroyalfamilyinlatemedievalDenmarkandespeciallyaboutthetitle(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!
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:
theyneed,oraretoldtheyneed,tolearnsomecommutativealgebrafortheirstudy
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
languageandbuildsbridgesbetweendifferentareasofmathematics: itissomething
likeacirculatorysystem. Commutativealgebraprovidescoreresultsandstructures
that other results and structures draw upon are overlayed upon: 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 solution sets to systems of polyno-
mialequations. Thereisahardlessonhere: inthe19thcenturyalgebraicgeometry
splitofffromcomplexfunctiontheoryanddifferentialgeometryasitsowndiscipline
and then burgeoned dramatically at the turn of the century and the years there-
after. 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
9
10 INTRODUCTION
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 eventually became invertebrate and began to collapse under its own
weight. Then (starting in about 1930) came a heroic shoring up process in which
the foundations of the subject were recast with commutative algebraic methods at
thecore. Thiswasdoneseveraltimesover,indifferentways,byZariski,Weil,Serre
and Grothendieck, among others. For the last 60 years it 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
essential way is algebraic number theory. One sees this from the beginning in that
theFundamentalTheoremofArithmeticistheassertionthattheringZisaunique
factorization domain (UFD), a basic commutative algebraic concept. Moreover
number theory was one of the historical sources of the subject. Notably the con-
cept of Dedekind domain came from Dedekind’s number-theoretic investigations.
At the student level, algebraic number theory does not embrace commutative al-
gebra as early or as thoroughly as algebraic geometry. This seems to me to be
a pedagogical mistake: although one can do a good amount of algebraic number
theorywithoutexplicitrelianceoncommutativealgebra, thisseemstocomeatthe
expense of not properly explaining what is going on. A modicum of commutative
algebra greatly enriches the study of algebraic number theory: it clarifies it, gener-
alizes it and (I believe) makes it more interesting.
The interplay among number theory, algebraic geometry and commutative alge-
bra flows in all directions. What Grothendieck did in the 1960s (with important
contributionsfromChevalley,Serreandothers)wastocreateasinglefieldofmath-
ematics 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 ge-
ometers: 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.
3. What distinguishes this text
Themoststraightforwardraison d’ˆetre foracommutativealgebratextwouldbeto
provide a foundation for the subjects of algebraic geometry, arithmetic geometry
and algebraic number theory. The bad news is that this task – even, restricted to
providingfoundationsforthesingle,seminaltextofHartshorne[Ha]–isdauntingly
large. The good news is that this has nevertheless been achieved some time ago by
David Eisenbud (a leading contemporary expert on the interface of commutative
algebraandalgebraicgeometry)inhistext[Ei]. Thisworkishighlyrecommended.