Yanqi Lake Lectures on Algebra

99 Pages·2019·1.57 MB·English

by  Wen-Wei Li

Y L
ANQI AKE
L
ECTURES ON
A
LGEBRA
/ Wen-Wei Li
Part 3
Commutative rings
Completions
Dimension theory
Yanqi Lake Lectures on Algebra
Part 3
Wen-Wei Li
Chinese Academy of Sciences
Version: 2019-06-14
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Contents
Introduction 1
1 Warmingup 3
1.1 Reviewonringtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Localizationofringsandmodules . . . . . . . . . . . . . . . . . . . . . . 5
1.3 RadicalsandNakayama’slemma . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 NoetherianandArtinianrings . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Whatiscommutativealgebra? . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Primarydecompositions 15
2.1 Thesupportofamodule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Associatedprimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Primaryandcoprimarymodules . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Primarydecomposition: themaintheorem . . . . . . . . . . . . . . . . . 19
2.5 Examplesandremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Integraldependence,Nullstellensatzandflatness 25
3.1 Integralextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Flatness: thefirstglance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Structureofflatmodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Faithfulflatnessandsurjectivity . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Going-up,going-down,gradingsandfiltrations 39
4.1 Going-upandgoing-down . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Subsetsinthespectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Gradedringsandmodules . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5 TheoremsofArtin–ReesandKrull . . . . . . . . . . . . . . . . . . . . . . 48
5 Fromcompletionstodimensions 51
5.1 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 Furtherpropertiesofcompletion . . . . . . . . . . . . . . . . . . . . . . . 55
5.3 Hilbert–Samuelpolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.4 DefinitionofKrulldimension . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.5 Krull’stheoremsandregularity . . . . . . . . . . . . . . . . . . . . . . . . 63
⋅ iv ⋅ CONTENTS
6 Dimensionoffinitelygeneratedalgebras 65
6.1 Dimensionsinfibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.2 Calculationforpolynomialalgebras . . . . . . . . . . . . . . . . . . . . . 67
6.3 Noethernormalizationanditsconsequences . . . . . . . . . . . . . . . . 68
7 Serre’scriterionfornormalityanddepth 73
7.1 Reviewofdiscretevaluationrings . . . . . . . . . . . . . . . . . . . . . . 73
7.2 Auxiliaryresultsonthetotalfractionring . . . . . . . . . . . . . . . . . . 74
7.3 Onnormality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.4 Serre’scriterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.5 Introductiontodepth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8 SomeaspectsofKoszulcomplexes 83
8.1 Preparationsinhomologicalalgebra . . . . . . . . . . . . . . . . . . . . . 83
8.2 Auxiliaryresultsondepth . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8.3 Koszulcomplexesanddepth . . . . . . . . . . . . . . . . . . . . . . . . . 86
Bibliography 91
Index 93
Introduction
Inthebeginning,theselecturenoteswerepreparedforthegraduatecourseAlgebraIII
(ID: 011M4002Y) in Spring 2016, University of the Chinese Academy of Sciences. For
somereasons,ittookplaceintheYuquanlucampusinsteadoftheYanqiLakecampus
asitshouldbe.
ThecourseisasequeltoAlgebraI(fields,modulesandrepresentations)andII(ho-
mologicalalgebra). ThetopicofthisPartIIIiscommutativealgebra,ormoreprecisely
commutative ring theory. Each “Lecture” in these notes took roughly one week, say ap-
proximately four hours of lecture, but the materials were only partially covered. My
initial intention was to give a traditional course on commutative algebra as proposed
by the syllabus prescribed by UCAS. For various reasons, my plan failed. For exam-
ple, there are too few discussions on depth, regular sequences and Cohen–Macaulay
modules, too few applications of completions, and the computational aspects haven’t
beentouched. Moreover,thehomologicalaspectofcommutativealgebraisalmostnon-
existent in these notes, namely the Auslander–Buchsbaum Formula, the properties of
regularlocalrings,etc. Lastbutnotleast,theexerciseshereinarescarcelysufficient.
These notes were also used for the Enhanced Program for Graduate Study held at
the Beijing International Center of Mathematical Research, Peking University, during
Spring2019(courseID:00102057).
As the title suggests, some backgrounds from the Part I are presumed, namely the
basic notions of rings, modules and their chain conditions, as well as familiarity with
tensor products and some Galois theory. Weoccasionallypresume some basic knowl-
𝑖
edgeofhomologicalalgebra,suchasthefunctorsTor andExt .
𝑖
Sometimes I made free use of the language of derived categories. This was indeed
coveredintheprecedingcourseAlgebraII,anditshouldbecommonsenseforthefuture
generations.
As the reader might have observed, these notes were prepared in a rush; certain
paragraphs have not been proofread yet and many proofs are silly. I am very grateful
tothestudentsforvariouscorrectionsandimprovements,andIwilltrytopolishthese
notesinthefuture.
Conventions
Throughouttheselectures,weconsideronlyassociativeringswithunit1,andtherings
andalgebrasareassumedtobecommutativeandnonzerounlessotherwisespecified.
Theidealgeneratedbyelements𝑥 ,𝑥 ,…inaring𝑅isdenotedby(𝑥 ,𝑥 ,…)orsome-
1 2 1 2
times ⟨𝑥 ,𝑥 ,…⟩; the 𝑅-algebra of polynomials in variables 𝑋,𝑌,… with coefficients in
1 2
⋅ 2 ⋅ Introduction
𝑅isdenotedby𝑅[𝑋,𝑌,…]. Wewrite𝑅×forthegroupofinvertibleelementsinaring𝑅.
Aringwithoutzero-divisorsexcept0iscalledanintegraldomain,orsimplyadomain.
The localization of a ring 𝑅 with respect to a multiplicative subset 𝑆 will be written as
𝑅[𝑆−1].
Foranysets𝐸,𝐹,let𝐸∖𝐹 ∶= {𝑥 ∈ 𝐸 ∶ 𝑥 ∉ 𝐹}. Thecardinalityof𝐸isdenotedby|𝐸|.
The usual logical connectives such as ∃, ∀, ∧, ∨ and so forth will occasionally be
used. Writing𝐴 ∶= 𝐵meansthattheexpression𝐴isdefinedtobe𝐵.
We will use the standard notations ℤ,ℚ,ℝ,ℂ to denote the set of integers, of ra-
tional numbers, etc. Sans serif fonts are reserved for categories, such as Ab (abelian
groups)and Ring.
When denoting morphisms in a category by arrows, monomorphisms (resp. epi-
∼
morphisms,isomorphisms)willbeindicated↪(resp.↠,→).
Possible references
The reader is expected to have basic familiarity with groups, rings and modules, as
coveredinmylecturenotesonAlgebraI.Wewillmakeuseofsomereallyelementary
homologicalalgebraasourcourseproceeds—sokeepcalm.
Ourmainreferenceswillbe[11]and[8]. TheBourbakivolumes[5,3]serveasour
ultimate source. The readers are also encouraged to consult the relevant materials in
StacksProject.
《教授生涯》,李桦,木刻版画,(cid:62295)(cid:62303)(cid:62298)(cid:62302)年.
Lecture 1
Warming up
Thereadermightbefamiliarwithmostofthematerialsinthislecture. Ourgoalistofix
notation and present the basic structural results on Noetherian and Artinian rings or
modules,includingthecelebratedNakayama’sLemmawhichwillbeusedrepeatedly.
1.1 Review on ring theory
Let𝑅bearing,supposedtobecommutativewithunit1 ≠ 0ascustomary. Recallthat
anideal 𝐼 ⊊ 𝑅iscalled
⋄ prime,if𝑎𝑏 ∈ 𝐼 ⟺ (𝑎 ∈ 𝐼)∨(𝑏 ∈ 𝐼);
⋄ maximal,if 𝐼 ismaximalamongtheproperidealsof𝑅withrespecttoinclusion.
Recallthefollowingstandardfacts
⋄ 𝐼 isprimeifandonlyif𝑅/𝐼 isanintegraldomain,i.e. hasnozerodivisorsexcept
0;
⋄ 𝐼 ismaximalifandonlyif𝑅/𝐼 isafield;inparticular,maximalidealsareprime;
⋄ every proper ideal 𝐼 is contained in a maximal ideal (an application of Zorn’s
Lemma).
Definition1.1.1(Localrings). Thering𝑅iscalledlocalifithasauniquemaximalideal,
semi-localifithasonlyfinitelymanymaximalideals.
Let 𝔪 be the maximal ideal of a local ring 𝑅. We call 𝑅/𝔪 the residue field of 𝑅. A
local homomorphism between local rings 𝜑 ∶ 𝑅 → 𝑅 is a ring homomorphism such
1 2
that 𝜑(𝔪 ) ⊂ 𝔪 . Consequently, local homomorphisms induce embeddings on the
1 2
levelofresiduefields.
Sometimeswedenotealocalringbythepair(𝑅,𝔪).
Remark1.1.2. Let𝑅bealocalringwithmaximalideal𝔪,then𝑅× = 𝑅∖𝔪. Theiseasily
seenasfollows. Anelement𝑥 ∈ 𝑅isinvertibleifandonlyif𝑅𝑥 = 𝑅. Notethat𝑅𝑥 = 𝑅
is equivalent to that 𝑥 is not contained in any maximal ideal, and the only maximal
idealis 𝔪.
⋅ 4 ⋅ Warmingup
Throughouttheselectures,weshallwrite
Spec(𝑅) ∶= {primeidealsof𝑅},
MaxSpec(𝑅) ∶= {maximalidealsof𝑅}.
Theyarecalledthespectrumandthemaximalspectrumof𝑅,respectively. Theupshotis
that Spec(𝑅)comeswithanaturaltopology.
Definition 1.1.3 (Zariski topology). For any ideal 𝔞 ⊂ 𝑅, set 𝑉(𝔞) ∶= {𝔭 ∈ Spec(𝑅) ∶
𝔭 ⊃ 𝔞}. Then there is a topology on Spec(𝑅), called the Zariski topology, whose closed
subsetareprecisely𝑉(𝔞),forvariousideals 𝔞.
Indeed, we only have to prove the family of subsets {𝑉(𝔞) ∶ 𝔞 ⊂ 𝑅} is closed un-
der finite union and arbitrary intersections. It boils down to the easy observation that
𝑉(𝔞) ∪ 𝑉(𝔟) = 𝑉(𝔞𝔟) (check this!) and ⋂ 𝑉(𝔞) = 𝑉(∑ 𝔞), where 𝒜 is any
𝔞∈𝒜 𝔞∈𝒜
familyofideals.
Givenaringhomomorphism𝜑 ∶ 𝑅 → 𝑅 ,if𝐼 ⊂ 𝑅 isanideal,then𝜑−1(𝐼) ⊂ 𝑅 is
1 2 2 1
alsoanideal.
Proposition1.1.4. Given𝜑asabove,itinducesacontinuousmap
𝜑♯ ∶ Spec(𝑅 ) ⟶ Spec(𝑅 )
2 1
𝔭 ⟼ 𝜑−1(𝔭)
withrespecttotheZariskitopologiesonspectra.
Proof. Clearly, 𝑎𝑏 ∈ 𝜑−1(𝔭) is equivalent to 𝜑(𝑎)𝜑(𝑏) ∈ 𝔭, which is in turn equivalent
to(𝜑(𝑎) ∈ 𝔭)∨(𝜑(𝑏) ∈ 𝔭)when𝔭isprime.
To show the continuity of 𝜑♯, observe that for any ideal 𝔞 ⊂ 𝑅 and 𝔭 ∈ Spec(𝑅 ),
1 2
wehave𝜑−1(𝔭) ⊃ 𝔞 ifandonlyif𝔭 ⊃ 𝜑(𝔞),i.e. 𝔭 ∈ 𝑉(𝜑(𝔞)𝑅 ). Hencethepreimageof
2
closedsubsetsarestillclosed.
Moreoperationsonspectra:
⋄ Take𝑅 tobeasubringof𝑅 and𝜑betheinclusionmap,themapabovebecomes
1 2
𝔭 ↦ 𝔭∩𝑅 .
1
⋄ Take𝜑 ∶ 𝑅 ↠ 𝑅/𝐼 tobeaquotienthomomorphism,then𝜑−1 istheusualbijection
fromSpec(𝑅/𝐼)onto𝑉(𝐼).
⋄ In general, 𝜑−1 does not induce MaxSpec(𝑅 ) → MaxSpec(𝑅 ), as illustrated in
2 1
thecase 𝜑 ∶ ℤ ↪ ℚ.
Atthisstage,wecanproveahandyresultconcerningprimeideals.
Proposition 1.1.5 (Prime avoidance). Let 𝐼 and 𝔭 ,…,𝔭 be ideals of 𝑅 such that 𝐼 ⊂
1 𝑛
𝑛
⋃ 𝔭 . Supposethat
𝑖=1 𝑖
⋄ either𝑅containsaninfinitefield,or
⋄ atmosttwooftheideals𝔭 ,…,𝔭 arenon-prime,
1 𝑛
thenthereexists1 ≤ 𝑖 ≤ 𝑛suchthat𝐼 ⊂ 𝔭 .
𝑖
§1.2 Localizationofringsandmodules ⋅ 5 ⋅
Proof. If 𝑅 contains an infinite field 𝐹, the ideals are automatically 𝐹-vector subspaces
𝑟
of 𝑅. Since 𝐼 = ⋃ 𝐼 ∩ 𝔭 whereas an 𝐹-vector space cannot be covered by finitely
𝑖=1 𝑖
manypropersubspaces,theremustexistsome𝑖 with 𝐼 ∩𝔭 = 𝐼.
𝑖
Underthesecondassumption,letusarguebyinductionon𝑛that∀𝑖 𝐼 ⊄ 𝔭 implies
𝑖
𝑛
𝐼 ⊄ ⋃ 𝔭 . The case 𝑛 = 1 is trivial. When 𝑛 ≥ 2, by induction we may choose, for
𝑖=1 𝑖
𝑛
each 𝑖, an element 𝑥 ∈ 𝐼 ∖⋃ 𝔭 . Suppose on the contrary that 𝐼 ⊂ ⋃ 𝔭 , then we
𝑖 𝑗≠𝑖 𝑗 𝑗=1 𝑗
wouldhave𝑥 ∈ 𝔭 ,forall 𝑖 = 1,…,𝑛.
𝑖 𝑖
When𝑛 = 2wehave𝑥 +𝑥 ∉ 𝔭 ∪𝔭 and𝑥 +𝑥 ∈ 𝐼,acontradiction. When𝑛 > 2,
1 2 1 2 1 2
wemayassume𝔭 isprime,therefore
1
𝑛 𝑛
𝑥 +∏𝑥 ∉ ⋃ 𝔭 ,
1 𝑗 𝑖
𝑗=2 𝑖=1
againacontradiction.
Exercise 1.1.6. The following construction from [8, Exercise 3.17] shows that the as-
sumptions of Proposition 1.1.5 cannot be weakened. Take 𝑅 = (ℤ/2ℤ)[𝑋,𝑌]/(𝑋,𝑌)2,
which has a basis {1,𝑋,𝑌} (modulo (𝑋,𝑌)2) as a ℤ/2ℤ-vector space. Show that the
image 𝔪 of (𝑋,𝑌) in 𝑅 is the unique prime ideal, and can be expressed as a union of
threeidealsproperlycontainedin𝔪.
1.2 Localization of rings and modules
Let 𝑆 be a multiplicative subset of 𝑅, which means that (a) 1 ∈ 𝑆, (b) 𝑆 is closed under
multiplication, and (c) 0 ∉ 𝑆. The localization of 𝑅 with respect to 𝑆 is the ring 𝑅[𝑆−1]
formedbyclasses [𝑟,𝑠]with𝑟 ∈ 𝑅,𝑠 ∈ 𝑆,modulotheequivalencerelation
[𝑟,𝑠] = [𝑟′,𝑠′] ⟺ ∃𝑡 ∈ 𝑆, (𝑟𝑠′ −𝑟′𝑠)𝑡 = 0.
You should regard [𝑟,𝑠] as a token for 𝑟/𝑠; the ring structure of 𝑅[𝑆−1] is therefore
evident. Inbrief,localizationamountstoformallyinvertingtheelementsof𝑆,whence
thenotation𝑅[𝑆−1]. Notethatcondition(c)guarantees𝑅[𝑆−1] ≠ {0}.
Exercise 1.2.1. Given 𝑅 and 𝑆, show that 𝑟 ↦ 𝑟/1 yields a natural homomorphism
𝑅 → 𝑅[𝑆−1]andshowthatitskernelequals{𝑟 ∶ ∃𝑠 ∈ 𝑆, 𝑠𝑟 = 0}.
The universal property of 𝑅 → 𝑅[𝑆−1] can be stated using commutative diagrams
asfollows.
⎧ 𝑅 𝑅[𝑆−1]
{ 𝜑 ∶ 𝑅 → 𝑅′ ∶ ringhomomorphism
∀ ,
⎨ ∃!
{ s.t. 𝜑(𝑆) ⊂ (𝑅′)× 𝜑
⎩
𝑅′