A term of commutative algebra

434 Pages·2017·4.618 MB·English

by  Allen Altman, Steven Kleiman

A Term of
Commutative Algebra
By Allen B. ALTMAN
and Steven L. KLEIMAN
Version of August 6, 2017: tca.tex
⃝c2013, Worldwide Center of Mathematics, LLC
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike
3.0 Unported License.
v. edition number for publishing purposes
ISBN 978-0-9885572-1-5
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . vi
Part I Subject Matter
1. Rings and Ideals . . . . . . . . . . . . . . . . . . . 2
A. Text . . . . . . . . . . . . . . . . . . . . . 2
B. Exercises . . . . . . . . . . . . . . . . . . . . 7
2. Prime Ideals . . . . . . . . . . . . . . . . . . . . 11
A. Text . . . . . . . . . . . . . . . . . . . . . 11
B. Exercises . . . . . . . . . . . . . . . . . . . . 15
3. Radicals . . . . . . . . . . . . . . . . . . . . . 18
A. Text . . . . . . . . . . . . . . . . . . . . . 18
B. Exercises . . . . . . . . . . . . . . . . . . . . 22
4. Modules . . . . . . . . . . . . . . . . . . . . . 24
A. Text . . . . . . . . . . . . . . . . . . . . . 24
B. Exercises . . . . . . . . . . . . . . . . . . . . 30
5. Exact Sequences . . . . . . . . . . . . . . . . . . . 32
A. Text . . . . . . . . . . . . . . . . . . . . . 32
B. Exercises . . . . . . . . . . . . . . . . . . . . 37
C. Appendix: Fitting Ideals . . . . . . . . . . . . . . . 38
D. Appendix: Exercises . . . . . . . . . . . . . . . . 42
6. Direct Limits . . . . . . . . . . . . . . . . . . . . 44
A. Text . . . . . . . . . . . . . . . . . . . . . 44
B. Exercises . . . . . . . . . . . . . . . . . . . . 49
7. Filtered Direct Limits . . . . . . . . . . . . . . . . . 52
A. Text . . . . . . . . . . . . . . . . . . . . . 52
B. Exercises . . . . . . . . . . . . . . . . . . . . 57
8. Tensor Products . . . . . . . . . . . . . . . . . . . 59
A. Text . . . . . . . . . . . . . . . . . . . . . 59
B. Exercises . . . . . . . . . . . . . . . . . . . . 65
9. Flatness . . . . . . . . . . . . . . . . . . . . . 66
A. Text . . . . . . . . . . . . . . . . . . . . . 66
B. Exercises . . . . . . . . . . . . . . . . . . . . 70
10. Cayley{Hamilton Theorem . . . . . . . . . . . . . . . 73
A. Text . . . . . . . . . . . . . . . . . . . . . 73
B. Exercises . . . . . . . . . . . . . . . . . . . . 78
11. Localization of Rings . . . . . . . . . . . . . . . . . 81
A. Text . . . . . . . . . . . . . . . . . . . . . 81
B. Exercises . . . . . . . . . . . . . . . . . . . . 85
12. Localization of Modules . . . . . . . . . . . . . . . . 87
A. Text . . . . . . . . . . . . . . . . . . . . . 87
B. Exercises . . . . . . . . . . . . . . . . . . . . 91
13. Support . . . . . . . . . . . . . . . . . . . . . 94
A. Text . . . . . . . . . . . . . . . . . . . . . 94
iii
Contents
B. Exercises . . . . . . . . . . . . . . . . . . . . 98
14. Cohen{Seidenberg Theory . . . . . . . . . . . . . . . 103
A. Text . . . . . . . . . . . . . . . . . . . . . 103
B. Exercises . . . . . . . . . . . . . . . . . . . . 106
15. Noether Normalization . . . . . . . . . . . . . . . . . 108
A. Text . . . . . . . . . . . . . . . . . . . . . 108
B. Exercises . . . . . . . . . . . . . . . . . . . . 113
C. Appendix: Jacobson Rings . . . . . . . . . . . . . . 114
D. Appendix: Exercises . . . . . . . . . . . . . . . . 116
16. Chain Conditions . . . . . . . . . . . . . . . . . . 118
A. Text . . . . . . . . . . . . . . . . . . . . . 118
B. Exercises . . . . . . . . . . . . . . . . . . . . 122
C. Appendix: Noetherian Spaces . . . . . . . . . . . . . 124
D. Appendix: Exercises . . . . . . . . . . . . . . . . 129
17. Associated Primes . . . . . . . . . . . . . . . . . . 130
A. Text . . . . . . . . . . . . . . . . . . . . . 130
B. Exercises . . . . . . . . . . . . . . . . . . . . 135
18. Primary Decomposition . . . . . . . . . . . . . . . . 138
A. Text . . . . . . . . . . . . . . . . . . . . . 138
B. Exercises . . . . . . . . . . . . . . . . . . . . 144
C. Appendix: Old-primary Submodules . . . . . . . . . . . 145
D. Appendix: Exercises . . . . . . . . . . . . . . . . 150
19. Length . . . . . . . . . . . . . . . . . . . . . . 153
A. Text . . . . . . . . . . . . . . . . . . . . . 153
B. Exercises . . . . . . . . . . . . . . . . . . . . 157
20. Hilbert Functions . . . . . . . . . . . . . . . . . . 159
A. Text . . . . . . . . . . . . . . . . . . . . . 159
B. Exercises . . . . . . . . . . . . . . . . . . . . 165
C. Appendix: Homogeneity . . . . . . . . . . . . . . . 165
D. Appendix: Exercises . . . . . . . . . . . . . . . . 167
21. Dimension . . . . . . . . . . . . . . . . . . . . . 168
A. Text . . . . . . . . . . . . . . . . . . . . . 168
B. Exercises . . . . . . . . . . . . . . . . . . . . 173
22. Completion . . . . . . . . . . . . . . . . . . . . 176
A. Text . . . . . . . . . . . . . . . . . . . . . 176
B. Exercises . . . . . . . . . . . . . . . . . . . . 184
C. Appendix: Henselian Rings . . . . . . . . . . . . . . 188
D. Appendix: Exercises . . . . . . . . . . . . . . . . 196
23. Discrete Valuation Rings . . . . . . . . . . . . . . . . 198
A. Text . . . . . . . . . . . . . . . . . . . . . 198
B. Exercises . . . . . . . . . . . . . . . . . . . . 203
C. Appendix: M-sequences . . . . . . . . . . . . . . . 203
D. Appendix: Exercises . . . . . . . . . . . . . . . . 211
24. Dedekind Domains . . . . . . . . . . . . . . . . . . 213
A. Text . . . . . . . . . . . . . . . . . . . . . 213
B. Exercises . . . . . . . . . . . . . . . . . . . . 216
iv
Contents
25. Fractional Ideals . . . . . . . . . . . . . . . . . . . 218
A. Text . . . . . . . . . . . . . . . . . . . . . 218
B. Exercises . . . . . . . . . . . . . . . . . . . . 222
26. Arbitrary Valuation Rings . . . . . . . . . . . . . . . . 223
A. Text . . . . . . . . . . . . . . . . . . . . . 223
B. Exercises . . . . . . . . . . . . . . . . . . . . 227
Part II Solutions
1. Rings and Ideals . . . . . . . . . . . . . . . . . . . 230
2. Prime Ideals . . . . . . . . . . . . . . . . . . . . 238
3. Radicals . . . . . . . . . . . . . . . . . . . . . 244
4. Modules . . . . . . . . . . . . . . . . . . . . . 250
5. Exact Sequences . . . . . . . . . . . . . . . . . . . 255
Appendix: Fitting Ideals . . . . . . . . . . . . . . . . 258
6. Direct Limits . . . . . . . . . . . . . . . . . . . . 260
7. Filtered direct limits . . . . . . . . . . . . . . . . . 264
8. Tensor Products . . . . . . . . . . . . . . . . . . . 268
9. Flatness . . . . . . . . . . . . . . . . . . . . . 272
10. Cayley{Hamilton Theorem . . . . . . . . . . . . . . . 277
11. Localization of Rings . . . . . . . . . . . . . . . . . 283
12. Localization of Modules . . . . . . . . . . . . . . . . 288
13. Support . . . . . . . . . . . . . . . . . . . . . 293
14. Cohen{Seidenberg Theory . . . . . . . . . . . . . . . 307
15. Noether Normalization . . . . . . . . . . . . . . . . . 314
Appendix: Jacobson Rings . . . . . . . . . . . . . . . 318
16. Chain Conditions . . . . . . . . . . . . . . . . . . 321
Appendix: Noetherian Spaces . . . . . . . . . . . . . . 327
17. Associated Primes . . . . . . . . . . . . . . . . . . 330
18. Primary Decomposition . . . . . . . . . . . . . . . . 335
Appendix: Old-primary Submodules . . . . . . . . . . . . 339
19. Length . . . . . . . . . . . . . . . . . . . . . . 344
20. Hilbert Functions . . . . . . . . . . . . . . . . . . 349
Appendix: Homogeneity . . . . . . . . . . . . . . . . 351
21. Dimension . . . . . . . . . . . . . . . . . . . . . 353
22. Completion . . . . . . . . . . . . . . . . . . . . 361
Appendix: Hensel’s Lemma . . . . . . . . . . . . . . . 381
23. Discrete Valuation Rings . . . . . . . . . . . . . . . . 385
Appendix: Cohen{Macaulay Modules . . . . . . . . . . . . 389
24. Dedekind Domains . . . . . . . . . . . . . . . . . . 396
25. Fractional Ideals . . . . . . . . . . . . . . . . . . . 399
26. Arbitrary Valuation Rings . . . . . . . . . . . . . . . . 401
References . . . . . . . . . . . . . . . . . . . . . . . 405
Disposition of the Exercises in [4] . . . . . . . . . . . . . . . 406
Use of the Exercises in this Book . . . . . . . . . . . . . . . 409
Notation . . . . . . . . . . . . . . . . . . . . . . . . 416
Index . . . . . . . . . . . . . . . . . . . . . . . . . 418
v
Preface
There is no shortage of books on Commutative Algebra, but the present book is
different. Most books are monographs, with extensive coverage. But there is one
notable exception: Atiyah and Macdonald’s 1969 classic [4]. 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 (cid:13)aws 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 (cid:12)rst elementary course in
general algebra. On the other hand, it is not intended as a substitute for the more
voluminoustractsonCommutativeAlgebra....Thelecture-noteoriginofthisbook
accountsfortherathertersestyle,withlittlegeneralpadding,andforthecondensed
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
ofwhatisbynowanelegantandattractivetheory." Theyendeavor\tobuildupto
themaintheoremsinasuccessionofsimplestepsandtoomitroutineveri(cid:12)cations."
AtiyahandMacdonald’ssuccessfulphilosophyiswholeheartedlyembracedbelow
(itisafeature,nota(cid:13)aw!),andalsore(cid:12)nedabit. Thepresentbookalso\grewout
of a course of lectures." That course was based primarily on their book, but has
beenofferedanumberoftimes,andhasevolvedovertheyears,in(cid:13)uencedbyother
publications, especially [16], and the reactions of the students. That course had
as prerequisite a \(cid:12)rst elementary course in general algebra" based on [3]. Below,
to further clarify and streamline the \mathematical structure" of the theory, the
theory is usually developed in its natural generality, where the settings are just
what is appropriate for the arguments.
Atiyah and Macdonald’s book comprises eleven chapters, split into forty-two
sections. The present book comprises twenty-six chapters; each chapter represents
a single lecture, and is self-contained. Lecturers are encouraged to emphasize the
meaning of statements and the ideas of proofs, especially those in the longer and
richer chapters, \waving their hands" and leaving the details for students to read
on their own and to discuss with others.
AtiyahandMacdonald\provided...exercisesattheendofeachchapter,"aswell
as some exercises within the text. They \provided hints, and sometimes complete
solutions,tothehard"exercises. Furthermore,theydevelopedasigni(cid:12)cantamount
ofnewmaterialintheexercises. Bycontrast,inthepresentbook, theexercisesare
morecloselytiedintothetext,andcompletesolutionsaregiveninthesecondpart
ofthebook. Doingsolengthenedthebookconsiderably. Thesolutions(cid:12)llnearlyas
much space as the text. Moreover, seven chapters have appendices; they elaborate
on important issues, most stemming from Atiyah and Macdonald’s exercises.
There are 585 exercises below, including all of Atiyah and Macdonald’s. The
disposition of the latter is indicated in a special index. The 578 also include 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, to solidify, and
to expand their understanding. The 578 are intentionally not difficult, tricky, or
vi
Preface
involved. Rarely do they introduce new techniques, although some introduce new
concepts, and many are used later. All the exercises within the text are used right
away. Another special index indicates all the exercises that are used, and where.
Students are encouraged to try to solve lots of exercises, without (cid:12)rst reading
the solutions. If they become stuck on an exercise, then they should review the
relevant material; if they remain stuck, then they should change tack by studying
the solution, possibly discussing it with others, but always making sure they can,
eventually, solve the whole exercise entirely on their own. In any event, students
should always read the given solutions, just to make sure they haven’t missed any
details; also, some solutions provide enlightening alternative arguments.
As to prioritizing the exercises, here is one reasonable order: (cid:12)rst, those that
appear within the text; second, those that are used more often, as indicated in the
index, \Use of the Exercises..."; third, those whose solutions are less involved, as
indicated by their length; fourth, those whose statements sound interesting; (cid:12)fth,
thosestemmingfromtheexercisesinAtiyahandMacdonald’sbook,asindicatedin
the index, \Disposition...." Of course, no one should exhaust all the exercises of
one level of priority before considering exercises of lower level; rather, if there’s no
other good reason to choose one exercise over another, then the order of priorities
could serve as the deciding factor.
Instructors are encouraged to assign six exercises with short solutions, say a
paragraph or two long, per lecture, and to ask students to write up solutions in
their own words. Instructors are encouraged to examine students, possibly orally
at a blackboard, possibly via written tests, on a small, randomly chosen subset of
the assigned exercises. For use during each exam, instructors are urged to provide
each student with a copy of the book that omits the solutions. A reasonable way
to grade is to count the exerecises as 30%, a midterm as 30%, and a (cid:12)nal as 40%.
AtiyahandMacdonaldexplainthat\apropertreatmentofHomologicalAlgebra
is impossible within the con(cid:12)nes 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
studyofhomology." Again,theirphilosophyisembracedandre(cid:12)nedinthepresent
book. Notably, below, elementarymethodsareused, notTor’sastheydo, toprove
theIdealCriterionfor(cid:13)atness, andtoprovethat, overlocalrings, (cid:13)atmodulesare
free. Also, projective modules are treated below, but not in their book.
Inthepresentbook,CategoryTheoryisabasictool;inAtiyahandMacdonald’s,
itseemslikeaforeignlanguage. Thustheydiscusstheuniversal(mapping)property
(UMP) of localization of a ring, but provide an ad hoc characterization. They also
provetheUMPoftensorproductofmodules, butdonotnameitthistime. Below,
theUMPisfundamental: therearemanystandardconstructions; eachhasaUMP,
which serves to characterize the resulting object up to unique isomorphism owing
to one general observation of Category Theory. For example, the Left Exactness of
Homisviewedsimplyasexpressinginotherwordsthatthekernelandthecokernel
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
saysthatTensorProductistheleftadjointofHom. FromitandtheLeftExactness
of Hom, they deduce the Right Exactness of Tensor Product. They note that this
derivationshowsthatany\leftadjointisrightexact." Moregenerally,asexplained
vii
Preface
below, this derivation shows that any left adjoint preserves arbitrary direct limits,
ones indexed by any small category. Atiyah and Macdonald consider only direct
limitsindexedbyadirectedset,andsketchanadhocargumentshowingthattensor
productpreservesdirectlimit. Also,arbitrarydirectsumsaredirectlimitsindexed
byadiscretecategory(itisnotadirectedset); hence,thegeneralresultyieldsthat
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, de(cid:12)ned on the appropriate
category of functors. As such, Direct Limit is a left adjoint. Hence, direct limits
preserve other direct limits. Here the theory brie(cid:13)y climbs to a higher level of
abstraction. The discussion is completely elementary, but by far the most abstract
in the book. The extra abstraction can be difficult, especially for beginners.
Below, (cid:12)ltered direct limits are treated too. They are closer to the kind of limits
treated by Atiyah and Macdonald. In particular, (cid:12)ltered direct limits preserve
exactnessand(cid:13)atness. Further,theyappearinthefollowinglovelyformofLazard’s
Theorem: in a canonical way, every module is the direct limit of free modules of
(cid:12)nite rank; moreover, the module is (cid:13)at if and only if that direct limit is (cid:12)ltered.
Atiyah and Macdonald treat primary decomposition in a somewhat dated way.
First, they study primary decompositions of ideals. Then, in the exercises, they
indicatehowtotranslatethetheorytomodules. Associatedprimesplayasecondary
role: theyarede(cid:12)nedastheradicalsoftheprimarycomponents,thencharacterized
as the primes that are the radicals of annihilators of elements. Finally, when the
ringsandmodulesareNoetherian,primarydecompositionsareprovedtoexist,and
associated primes to be annihilators themselves.
Below, as is standard nowadays, associated primes of modules are studied right
from the start; they are de(cid:12)ned as the primes that are annihilators of elements.
Submodules are called primary if the quotient modules have only one associated
prime. Below,AtiyahandMacdonald’sprimarysubmodulesarecalledold-primary
submodules, and they are studied too, mostly in an appendix. In the Noetherian
case, the two notions agree; so the two studies provide alternative proofs.
Below, general dimension theory is developed for Noetherian modules; whereas,
Atiyah and Macdonald treat only Noetherian rings. Moreover, the modules below
are often assumed to be semilocal|that is, their annihilator lies in only (cid:12)nitely
many maximal ideals|correspondingly, Atiyah and Macdonald’s rings are local.
There are several other signi(cid:12)cant differences between Atiyah and Macdonald’s
treatment and the one below. First, the Noether Normalization Lemma is proved
below in a stronger form for nested sequences of ideals; consequently, for algebras
that are (cid:12)nitely generated over a (cid:12)eld, dimension theory can be developed directly
viii
Preface
andmoreextensively, withouttreatingNoetherianlocalrings(cid:12)rst(see(21.24)for
thelatterapproach). Second,inanumberofresultsbelow,themodulesareassumed
tobe(cid:12)nitelypresentedoveranarbitraryring, ratherthan(cid:12)nitelygeneratedovera
Noetherianring. Third,thereisanelementarytreatmentofregularsequencesbelow
and a proof of Serre’s Criterion for Normality; this important topic is developed
furtherinanappendix. Fourth,below,theAdjoint-AssociativityFormulaisproved
overapairofbaserings;hence,ityieldsbothaleftandarightadjointtothefunctor
of restriction of scalars.
Many people have contributed to the quality of the present book. Pavel Etingof
and Bjorn Poonen lectured from an earlier edition, and Dan Grayson and Amnon
Yekutieli read parts of it; all four have made a number of good comments and
suggestions, which were incorporated. Many people have pointed out typos, which
were corrected. For this service to the community, the authors are grateful, and
they welcome any future such remarks from anyone.
It is rarely easy to learn anything new of substance, value, and beauty, like
Commutative Algebra, but it is always satisfying, enjoyable, and worthwhile to do
so. The authors bid their readers much success in learning Commutative Algebra.
Allen B. Altman and Steven L. Kleiman
July 2017
ix