Algebra I: Commutative Algebra - Userpage - Freie Universität Berlin

167 Pages·2013·2.62 MB·English

Algebra I:
Commutative Algebra
Alexander Schmitt
Berlin, Winter 2012/2013
1
i
Table of Contents
Table ofContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
I BasicTheory ofRings andtheir Ideals . . . . . . . . . . . . . . . . . . . . . 1
I.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I.2 Ideals and QuotientRings . . . . . . . . . . . . . . . . . . . . . . . . . . 8
I.3 Zero Divisors,NilpotentElements,and Units . . . . . . . . . . . . . . . 11
I.4 PrimeIdeals andMaximalIdeals . . . . . . . . . . . . . . . . . . . . . . 12
I.5 IrreducibleElementsandPrimeElements . . . . . . . . . . . . . . . . . 19
I.6 Factorial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
I.7 TheNilradical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
I.8 Operationson Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
I.9 AlgebraicSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
II NoetherianRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
II.1 Chain Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
II.2 ArtinianRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
II.3 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
II.4 Primary Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
III The Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
III.1 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
III.2 FiniteRing Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
III.3 TheNullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
III.4 NoetherNormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
III.5 NormalRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
IV DimensionTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
IV.1 KrullDimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
IV.2 TheGoing-UpTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
IV.3 TheTranscendenceDegreeofaField . . . . . . . . . . . . . . . . . . . . 122
IV.4 TheDimensionofan AlgebraicVariety . . . . . . . . . . . . . . . . . . 123
IV.5 Krull’sPrincipalIdeal Theorem . . . . . . . . . . . . . . . . . . . . . . . 130
i
Preface
IV.6 EmbeddingDimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
IV.7 SingularPointsofAlgebraicVarieties . . . . . . . . . . . . . . . . . . . 135
IV.8 Regularityand Normality . . . . . . . . . . . . . . . . . . . . . . . . . . 148
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
ii
Preface
The present text contains notes on my course “Algebra I” at Freie Universita¨tBerlin dur-
ing the winterterm 2012/2013. The course “AlgebraI” is part of a cycleof three courses
providingan introductionto algebraic geometry. It is also meant as a continuationof my
course“Algebraund Zahlentheorie”(see[30]).
The basic objects we will be studying are commutative rings and their ideals. There
are many different motivations for looking at these objects. A ring can be, for example,
viewed as a domain of numbers with which we would like to compute. In certain situa-
tions, ideals are valuable generalizations of numbers.1 A ring can also consist of regular
functions on an affine algebraic variety. If we consider algebraic varieties over an alge-
braically closed field such as the field of complex numbers, then this algebra completely
determinesthevariety. Ideals inthealgebracorrespond tosubvarieties,e.g., pointsofthe
variety. Itisoneofthemeritsofcommutativealgebrathatitprovidesaunifiedframework
for, amongotherthings,arithmeticand algebro-geometricinvestigations.
The first chapter presents the language of rings and their ideals. Many operations on
rings and ideals which will be used throughout the text are presented. I would like to
highlighttwo topics: The first oneis the section on factorial rings. It showshow we may
generalizethemaintheoremofelementarynumbertheory,i.e.,theuniquefactorizationof
naturalnumbersintopowersofprimenumbers,toothersettingsandproblemswhichwill
usuallyoccur. Thesecondoneisthespectrumofaring. Itattachestoacommutativering
ageometricobject. Thisisa fundamentalconstructionofmodernalgebraicgeometry.
Noetherianrings are ringswhich satisfya crucial finiteness condition. Thiscondition
is fulfilled by important rings occuring in number theory and algebraic geometry, such
as orders in number fields and coordinate algebras of algebraic varieties. A central result
is the decomposition of ideals in noetherian rings into primary ideals and its uniqueness
properties. Thisisavastgeneralizationofthemaintheoremofelementarynumbertheory
and has alsoan importantgeometricinterpretation.
As the main topic of the courses “Algebra I-III” is algebraic geometry, the remaining
two chapters deal with subjects of a more geometric nature. The third chapter focusses
on Hilbert’s Nullstellensatz. The Nullstellensatzprovides the dictionary between finitely
generated algebras over an algebraically closed field k and their (radical) ideals on the
onehandandalgebraicvarietiesdefinedoverkandtheirsubvarietiesontheotherhand. It
explains thefundamental roleof commutativealgebra in algebraic geometry. We present
an elementary proof due to Munshi. Another central topic is Noether’s normalization
1ThisisanideaofKummer.
iii
Preface
theorem which supplies important information on the structure of algebraic varieties and
willbeusedoverandoveragaininthefourthchapter. Thethirdchapteralsodevelopsthe
notionofmodulesoveraring.
The fourth chapter begins the study of the geometry of algebraic varieties. The Krull
dimension of a ring is introduced and investigated. If the ring in question is the coordi-
nate algebra of an affine algebraic variety, this provides a basic geometric invariant. We
will check various properties which we intuitively expect from the notion of dimension.
Finally, we will study singularities of algebraic varieties. In this context, we will also
discusstherelationbetweentheintricatenotionofnormalityofringsandaffinealgebraic
varietiesand singularities.
Dr. Juan Pons Llopisand AnnaWißdorfproofread themanuscriptand suggestedvar-
ious corrections and improvements. The biographicaldata ofmathematicianswere taken
fromWikipedia. Thesenotesareheavilybasedonthebooks[1]and[14]. Otherimportant
sources are[4], [8], and [11].
AlexanderSchmitt
Berlin, March 2013
iv
I
Basic Theory of Rings and their Ideals
Inanintroductorycourseonlinearalgebra,oneusuallyworksoverfields. Incommutative
algebra,fieldsarereplacedbymoregeneralobjects,namelycommutativeringswithiden-
tity element. In linear algebra, you rarely talk aboutthe fields themselves. Certainly, you
can do someexplicitcomputationsoverthefield of rational numbers or finitefields, and,
for the theory of the Jordan normal form, you need the ground field to be algebraically
closed. Apart from that, you don’tworry muchabout the“internal”structureofthe field.
In various respects, rings have a much richer structure than fields. To begin with, you
should think of the ring of integers or the polynomial ring k[x] over a field k. In these
Z
rings, you may investigate when a number or a polynomial divides another one. This
leads you to certain “indivisible” objects which you call prime numbers or irreducible
polynomials. In fields, there are no counterparts to these concepts, because any non-zero
element is a unit. During the development of algebra, it turned out that it is more useful
to work with ideals than with the ring elements themselves.1 In this section, we will first
define rings and look at somebasicexamples. Then, wewill developthe notionofideals
andexplainhowtocomputewiththem. AsamotivationinKummer’s2spirit,wewillalso
look at prime factorization. In that context, we will see its failure in some rings and the
class offactorial ringsinwhich primefactorizationsdo exist.
I.1 Rings
A ring is a tuple(R,0,+, )which consistsof an abelian group (R,0,+)(see [30], Defini-
·
tionII.1.1 and II.1.4)andamap
: R R R
· × −→
(a,b) a b
7−→ ·
1ThetermidealgoesbacktoKummer’snotionof“idealeZahl”,anextensionoftheconceptofnumber
orringelementthatpermitstogeneralizetheprimefactorizationintheringofintegerstosomeringssuch
as [√ 5](seeSectionI.5).
Z2E−rnstEduardKummer(1810-1893)wasaGermanmathematician.
1
I. BasicTheoryofRingsandtheirIdeals
whichis associative,i.e.,
a,b,c R : a (b c) = (a b) c,
∀ ∈ · · · ·
such thatthedistributivelawshold,i.e.,
a,b,c R : a (b+c) = a b+a c,
∀ ∈ · · ·
(a+b) c = a c+b c.
· · ·
We willrefer to “+” as theadditionand to “”as themultiplication. In thesequel,we
·
willwriteRrather than(R,0,+, )forthedatumofa ring.
·
A ring Ris called commutative,if
a,b R : a b = b a.
∀ ∈ · ·
An element1 inaring Ris an identityelement,if
a R : 1 a = a = a 1.
∀ ∈ · ·
I.1.1 Note. A ring R can have at most one identity element. For, if 1,1 R are identity
′
∈
elements,wehave
1 = 1 1 = 1 .
′ ′
·
Let R be a ring. For a R, we let a be theadditiveinverseof a, i.e., theelement for
∈ −
whicha+( a) = 0 holds.
−
I.1.2Properties. Let Rbea ring.
i)Fora R,we havea 0 = 0 a = 0.
∈ · ·
ii)Fora,b R,wehave( a) b = (a b) = a ( b). Inparticular,( a) ( b) = a b.
∈ − · − · · − − · − ·
Proof. i)We use0 = 0+0, sothat
0 a = (0+0) a = 0 a+0 a.
· · · ·
Add (0 a)tobothsides toget
− ·
0 = 0+0 a = 0 a.
· ·
Similarly,oneshowsa 0 = 0.
·
ii)Usingi),we see
a ( b)+a b = a ( b+b) = a 0 = 0.
· − · · − ·
Thus,a ( b) = (a b). In thesamevein,oneshows( a) b = (a b). (cid:3)
· − − · − · − ·
I.1.3Examples. i)R = 0 withtheonlypossibleadditionandmultiplicationisaringwith
{ }
identity element 1 = 0. (It is not a field (compare [30], Definition III.1.1.)!) Note that R
is the only ring with identity element in which 1 = 0 holds. Indeed, let R , 0 be a ring
{ }
withidentityelement 1and pick a R 0 . Then,
∈ \{ }
1 a = a , 0 = 0 a.
· ·
2
I.1. Rings
Hence, 1 , 0.
ii)Let (R,0,+)bean abelian groupand define
: R R R
· × −→
(a,b) 0
7−→
Then, (R,0,+, )isaring. If R , 0 ,then Rhas no identityelement.
· { }
iii)Theintegersform thering .
Z
iv)Fieldsare rings,e.g., , , , , q = pn, n 1and p a primenumber.
q
Q R C F ≥
v)SupposeR isaring and X is aset. Weintroduce
Map(X,R) := f : X R f isaset theoreticmap .
−→ |
(cid:8) (cid:9)
For f,g Map(X,R), weform
∈
f +g: X R
−→
x f(x)+g(x)
7−→
and
f g: X R
· −→
x f(x) g(x).
7−→ ·
Moreover,weset
0: X R
−→
x 0.
7−→
The tuple (Map(X,R),0,+, ) is a ring. Observe that Map(X,R) is commutativeif R is so
·
andthatMap(X,R)possessesanidentityelementifRdoes. Infact,if1 Ristheidentity
∈
element ofR,then
1: X R
−→
x 1
7−→
is the identity element in Map(X,R). Note, however, that Map(X,R) will, in general, not
beafield, evenifR isone(seeExampleI.3.1, iv).
vi)Suppose X is atopologicalspace. Then,
C(X, ) := f : X f iscontinuous
R −→ R|
(cid:8) (cid:9)
togetherwith0,1, +, as inv)is acommutativeringwithidentityelement.
·
vii) Suppose X is an open subset (compare [31], Definition III.2.1, ii), and
⊂ C
III.3.10, i). Then,
O(X) := f : X f isholomorphic
−→ C|
togetherwith0,1, +, as inv)is(cid:8)acommuativering withidentity(cid:9) element.
·
viii)Let(G,0,+)bean abeliangroup. We lookat
End(G) := f : G G f isagroup homomorphism
−→ |
(cid:8) (cid:9)
3
I. BasicTheoryofRingsandtheirIdeals
and define
0: G G
−→
g 0
7−→
and, for f,h End(G),
∈
f +h: G G
−→
g f(g)+h(g).
7−→
Compositionofmapsprovidesus withthemultiplication: For f,h End(G), we set
∈
f h: G G
◦ −→
g f h(g) .
7−→
(cid:0) (cid:1)
Weleaveittothereadertoverifythat(End(G),0,+, )isaringwithidentityelementid .
G
·
It is,ingeneral, notacommutativering(see ExerciseI.1.4).
ix)Letkbeafield. Thevectorspace M (k)of(n n)-matriceswithentriesinkformsa
n
×
ring with respect to componentwiseadditionand matrix multiplication(see [33], Kapitel
III). The unit matrix is the identity element. Note that M (k) is non-commutative if
n n
E
and onlyifn 2.
≥
x) Suppose R , ..., R are rings. The cartesian product R R equipped with
1 n 1 n
× ··· ×
componentwiseaddition and multiplicationis again a ring. It is called the direct product
oftheringsR , ...,R .
1 n
I.1.4 Exercise. Describe End( ) in terms of (2 2)-matrices and give two elements
Z× Z ×
ofthatring whichdo notcommutewitheach other.
Let R be aring with identityelement. A subset S R is asubring,if S is a subgroup
⊂
of(R,+,0)(see [30], DefinitionII.4.1), 1 S and a b S, ifa,b S.
∈ · ∈ ∈
From nowon,all ringsaresupposedtobecommutative andto possessan
identity element.
Let R,S betworings. A homomorphismfromR to S isa map f : R S,such that
−→
⋆ a,b R : f(a + b) = f(a) + f(b), i.e., f is a homomorphism of the underlying
∀ ∈
abelian groups,
⋆ a,b R : f(a b) = f(a) f(b),
∀ ∈ · ·
⋆ f(1) = 1.
I.1.5 Remark. One may be tempted to believe that the third condition is a consequence
of the second. However, the proof for the analogous statement in group theory ([30],
Lemma II.3.4, i) requires the existence of inverse elements. This cannot be assumed for
multiplication. Indeed,choosingR = 0 andS , 0 ,weseethatitmaybefalse. Amore
{ } { }
sophisticatedexample,usingtheconstructioninExampleI.1.3, x), isthefollowing:
ϕ:
Z −→ Z×Z
k (k,0).
7−→
4