Algebraic Geometry and Commutative Algebra

507 Pages·2013·2.77 MB·English

by  Siegfried Bosch (auth.)

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Siegfried Bosch
Algebraic
Geometry and
Commutative
Algebra
Prof.Dr.SiegfriedBosch
MathematischesInstitut
WestfälischeWilhelms-Universität
Münster,Germany
ISSN0172-5939 ISSN2191-6675(electronic)
Universitext
ISBN978-1-4471-4828-9 ISBN978-1-4471-4829-6(eBook)
DOI10.1007/978-1-4471-4829-6
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Preface
ThedomainofAlgebraic GeometryisafascinatingbranchofMathematics
that combines methods from Algebra and Geometry. In fact, it transcends
thelimitedscopeofpureAlgebra,inparticularCommutativeAlgebra,by
means of geometrical construction principles. Looking at its history, the theory
hasbehavedmorelikeanevolvingprocessthanacompletedworkpiece,asquite
often the challenge of new problems has caused extensions and revisions. For
example, the concept of schemes invented by Grothendieck in the late 1950s
made it possible to introduce geometric methods even into fields that formerly
seemedtobefarfromGeometry,likealgebraicNumberTheory.Thispaved
thewaytospectacularnewachievements,suchastheproofbyWilesandTaylor
of Fermat’s Last Theorem, a famous problem that was open for more than 350
years.
The purpose of the present book is to explain the basics of modern Al-
gebraic Geometry to non-experts, thereby creating a platform from which
one can take off towards more advanced regions. Several times I have given
courses and seminars on the subject, requiring just two semesters of Linear
Algebra for beginners as a prerequisite. Usually I did one semester of Com-
mutative Algebra and then continued with two semesters of Algebraic
Geometry. Each semester consisted of a combination of traditional lectures
togetherwithanattachedseminarwherethestudentspresentedadditionalma-
terial by themselves, extending the theory, supplying proofs that were skipped
in the lectures, or solving exercise problems. The material covered in this way
corresponds roughly to the contents of the present book. Just as for my stu-
dents, the necessary prerequisites are limited to basic knowledge in Linear
Algebra, supplemented by a few facts from classical Galois theory of fields.
Explaining Algebraic Geometry from scratch is not an easy task.
Of course, there are the celebrated E´l´ements de G´eom´etrie Alg´ebrique by
Grothendieck and Dieudonn´e, four volumes of increasing size that were later
continued by seven volumes of S´eminaire de G´eom´etrie Alg´ebrique. The series
is like an extensive encyclopaedia where the basic facts are dealt with in striv-
ing generality, but which is hard work for someone who has not yet acquired a
certain amount of expertise in the field. To approach Algebraic Geometry
from a more economic point of view, I think it is necessary to learn about its
basicprinciples.Ifthesearewellunderstood,manyresultsbecomeeasiertodi-
gest,includingproofs,andgettinglostinamultitudeofdetailscanbeavoided.
VI Preface
Thereforeitisnotmyintentiontocoverasmanytopicsaspossibleinmybook.
InsteadIhavechosentoconcentrateonacertainselectionofmainthemesthat
are explained with all their underlying structures and without making use of
any artificial shortcuts. In spite of thematic restrictions, I am aiming at a self-
contained exposition up to a level where more specialized literature comes into
reach.
AnyonewillingtoenterAlgebraic Geometryshouldbeginwithcertain
basic facts in Commutative Algebra. So the first part of the book is con-
cernedwiththissubject.Itbeginswithageneralchapteronringsandmodules
where,amongotherthings,Iexplainthefundamentalprocessoflocalization,as
wellascertainfinitenessconditionsformodules,likebeingNoetherianorcoher-
ent.ThenfollowsaclassicalchapteronNoetherian(andArtinian)rings,includ-
ing the discussion of primary decompositions and of Krull dimensions, as well
as a classical chapter on integral ring extensions. In another chapter I explain
the process of coefficient extension for modules by means of tensor products,
aswellasitsreverse,descent.Inparticular,acompleteproofofGrothendieck’s
fundamental theorem on faithfully flat descent for modules is given. Moreover,
asitisquiteusefulatthisplace,Icastacautiousglimpseoncategoriesandtheir
functors, including functorial morphisms. The first part of the book ends by a
chapter on Ext and Tor modules where the general machinery of homological
methods is explained.
ThesecondpartdealswithAlgebraic Geometryinthestrictersenseof
theword.HereIhavelimitedmyselftofourgeneralthemes,eachofthemdealt
withinachapterbyitself,namelytheconstructionofaffineschemes,techniques
of global schemes, ´etale and smooth morphisms, and projective and proper
schemes, including the correspondence between ample and very ample invert-
iblesheavesanditsapplicationtoabelianvarieties.Thereisnothingreallynew
inthesechapters,although thestyleinwhichIpresentthematerialisdifferent
fromothertreatments.Inparticular,thisconcernsthehandlingofsmoothmor-
phisms via the Jacobian Condition, as well as the definition of ample invertible
sheavesviatheuseofquasi-affineschemes.ThisisthewayinwhichM.Raynaud
liked to see these things and I am largely indebted to him for these ideas.
Each chapter is preceded by an introductory section where I motivate its
contents and give an overview. As I cannot deliver a comprehensive account
already at this point, I try to spotlight the main aspects, usually illustrating
these by a typical example. It is recommended to resort to the introductory
sections at various times during the study of the corresponding chapter, in or-
der to gradually increase the level of understanding for the strategy and point
of view employed at different stages. The latter is an important part of the
learning process, since Mathematics, like Algebraic Geometry, consists of
a well-balanced combination of philosophy on the one hand and detailed argu-
mentation or even hard computation on the other. It is necessary to develop a
reliablefeelingforbothofthesecomponents.Theselectionofexerciseproblems
at the end of each section is meant to provide additional assistance for this.
Preface VII
Preliminary versions of my manuscripts on Commutative Algebra and
AlgebraicGeometryweremadeavailabletoseveralgenerationsofstudents.
Itwasagreatpleasureformetoseethemgettingexcitedaboutthesubject,and
I am very grateful for all their comments and other sort of feedback, including
listsoftypos,suchastheonesbyDavidKrummandClaudiusZibrowius.Special
thanks go to Christian Kappen, who worked carefully on earlier versions of the
text, as well as to Martin Brandenburg, who was of invaluable help during
the final process. Not only did he study the whole manuscript meticulously,
presenting an abundance of suggestions for improvements, he also contributed
to the exercises and acted as a professional coach for the students attending
my seminars on the subject. It is unfortunate that the scope of the book did
not permit me to put all his ingenious ideas into effect. Last, but not least,
let me thank my young colleagues Matthias Strauch and Clara L¨oh, who run
seminars on the material of the book together with me and who helped me
settingupappropriatethemesforthestudents.Inaddition,ClaraL¨ohsuggested
numerous improvements for the manuscript, including matters of typesetting
andlanguage.AlsothefigureforgluingschemesinthebeginningofSection7.1
is due to her.
Mu¨nster, December 2011 Siegfried Bosch
Contents
Part A. Commutative Algebra . . . . . . . . . . . . . . . . . . . . . 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Rings and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1 Rings and Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Local Rings and Localization of Rings . . . . . . . . . . . . . . 18
1.3 Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.5 Finiteness Conditions and the Snake Lemma . . . . . . . . . . . 38
2 The Theory of Noetherian Rings . . . . . . . . . . . . . . . . . . . . 55
2.1 Primary Decomposition of Ideals . . . . . . . . . . . . . . . . . 57
2.2 Artinian Rings and Modules . . . . . . . . . . . . . . . . . . . . 66
2.3 The Artin–Rees Lemma . . . . . . . . . . . . . . . . . . . . . . 71
2.4 Krull Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3 Integral Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1 Integral Dependence . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2 Noether Normalization and Hilbert’s Nullstellensatz . . . . . . . 91
3.3 The Cohen–Seidenberg Theorems . . . . . . . . . . . . . . . . . 96
4 Extension of Coefficients and Descent . . . . . . . . . . . . . . . . . 103
4.1 Tensor Products. . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2 Flat Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3 Extension of Coefficients . . . . . . . . . . . . . . . . . . . . . . 123
4.4 Faithfully Flat Descent of Module Properties . . . . . . . . . . . 131
4.5 Categories and Functors . . . . . . . . . . . . . . . . . . . . . . 138
4.6 Faithfully Flat Descent of Modules and their Morphisms . . . . 143
5 Homological Methods: Ext and Tor . . . . . . . . . . . . . . . . . . . 157
5.1 Complexes, Homology, and Cohomology . . . . . . . . . . . . . 159
5.2 The Tor Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.3 Injective Resolutions . . . . . . . . . . . . . . . . . . . . . . . . 181
5.4 The Ext Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 187
X Contents
Part B. Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . 193
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6 Affine Schemes and Basic Constructions . . . . . . . . . . . . . . . . 201
6.1 The Spectrum of a Ring . . . . . . . . . . . . . . . . . . . . . . 203
6.2 Functorial Properties of Spectra . . . . . . . . . . . . . . . . . . 212
6.3 Presheaves and Sheaves. . . . . . . . . . . . . . . . . . . . . . . 216
6.4 Inductive and Projective Limits . . . . . . . . . . . . . . . . . . 222
6.5 Morphisms of Sheaves and Sheafification . . . . . . . . . . . . . 232
6.6 Construction of Affine Schemes . . . . . . . . . . . . . . . . . . 241
6.7 The Affine n-Space . . . . . . . . . . . . . . . . . . . . . . . . . 255
6.8 Quasi-Coherent Modules . . . . . . . . . . . . . . . . . . . . . . 257
6.9 Direct and Inverse Images of Module Sheaves . . . . . . . . . . 266
7 Techniques of Global Schemes . . . . . . . . . . . . . . . . . . . . . 277
7.1 Construction of Schemes by Gluing . . . . . . . . . . . . . . . . 282
7.2 Fiber Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
7.3 Subschemes and Immersions . . . . . . . . . . . . . . . . . . . . 304
7.4 Separated Schemes . . . . . . . . . . . . . . . . . . . . . . . . . 312
7.5 Noetherian Schemes and their Dimension . . . . . . . . . . . . . 318
7.6 Cˇech Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 322
7.7 Grothendieck Cohomology . . . . . . . . . . . . . . . . . . . . . 330
8 E´tale and Smooth Morphisms . . . . . . . . . . . . . . . . . . . . . . 341
8.1 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 343
8.2 Sheaves of Differential Forms. . . . . . . . . . . . . . . . . . . . 356
8.3 Morphisms of Finite Type and of Finite Presentation . . . . . . 360
8.4 Unramified Morphisms . . . . . . . . . . . . . . . . . . . . . . . 365
8.5 Smooth Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . 374
9 Projective Schemes and Proper Morphisms . . . . . . . . . . . . . . 399
9.1 Homogeneous Prime Spectra as Schemes . . . . . . . . . . . . . 403
9.2 Invertible Sheaves and Serre Twists . . . . . . . . . . . . . . . . 418
9.3 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
9.4 Global Sections of Invertible Sheaves . . . . . . . . . . . . . . . 446
9.5 Proper Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . 462
9.6 Abelian Varieties are Projective . . . . . . . . . . . . . . . . . . 477
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
Glossary of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
Commutative Algebra
Introduction
ThemainsubjectinLinear Algebraisthestudyofvectorspacesoverfields.
However,quiteoftenitisconvenienttoreplacethebasefieldofavectorspaceby
aringofmoregeneraltypeandtoconsidervectorspaces,orbetter,modules over
such a ring. Several methods and basic constructions from Linear Algebra
extend to the theory of modules. On the other hand, it is easy to observe that
the theory of modules is influenced by a number of new phenomena, which
basically are due to the fact that modules do not admit bases, at least in the
general case.
The theory of modules over a given ring R depends fundamentally on the
structure of this ring. Roughly speaking, the multitude of module phenomena
increases with the level of generality of the ring R. If R is a field, we end up
with the well-known theory of vector spaces. If R is merely a principal ideal
domain, then the Theorem of Elementary Divisors and the Main Theorem on
finitely generated modules over principal ideal domains are two central results
fromthetheoryofmodulesoversuchrings.Anotherimportantcaseistheoneof
modulesoverNoetherianrings.Inthefollowingwewanttolookatfairlygeneral
situations:themainobjectiveofCommutative Algebraisthestudyofrings
and modules in this case.
The motivation of Commutative Algebra stems from two basic classes
of rings. The first covers rings that are of interest from the viewpoint of Num-
ber Theory, so-called rings of integral algebraic numbers. Typical members
of this class are the ring of integers Z, the ring Z(cid:2)(cid:3)i(cid:4)(cid:5) = Z+iZ ⊂ C of integral
Ga√uß numbers,√as well as√various other rings of integral algebraic numbers like
Z(cid:2)(cid:3) 2(cid:4)(cid:5) = Z+ 2Z ⊂ Q( 2) ⊂ R. The second class of rings we are thinking
of involvesrings that are of interestfrom the viewpoint ofAlgebraic Geom-
etry. The main example is the polynomial ring k(cid:2)(cid:3)t ,...,t (cid:4)(cid:5) in m variables
1 m
t ,...,t over a field k. However, it is natural to pass to rings of more gen-
1 m
eraltype,so-calledk-algebrasoffinitetype,ork-algebrasoffinitepresentation,
where k is an arbitrary ring (commutative and with unit element 1).
Letusdiscusssomeexamplesdemonstratinghowringsofthejustmentioned
typesoccurinanaturalway.WestartwithaproblemfromNumber Theory.