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Siegfried Bosch

AlgebraicGeometry andCommutativeAlgebra

Prof. Dr. Siegfried BoschMathematisches InstitutWestfälische Wilhelms-UniversitätMünster, Germany

ISSN 0172-5939 ISSN 2191-6675 (electronic)UniversitextISBN 978-1-4471-4828-9 ISBN 978-1-4471-4829-6 (eBook)DOI 10.1007/978-1-4471-4829-6Springer London Heidelberg New York Dordrecht

Library of Congress Control Number: 2012953696

Mathematics Subject Classification: 13-02, 13Axx, 13Bxx, 13Cxx, 13Dxx, 13Exx, 13Hxx, 13Nxx, 14-02, 14Axx, 14B25, 14C20, 14F05, 14F10, 14K05, 14L15

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Preface

The domain of Algebraic Geometry is a fascinating branch of Mathematicsthat combines methods from Algebra and Geometry. In fact, it transcendsthe limited scope of pureAlgebra, in particular Commutative Algebra, bymeans of geometrical construction principles. Looking at its history, the theoryhas behaved more like an evolving process than a completed workpiece, as quiteoften the challenge of new problems has caused extensions and revisions. Forexample, the concept of schemes invented by Grothendieck in the late 1950smade it possible to introduce geometric methods even into fields that formerlyseemed to be far fromGeometry, like algebraicNumber Theory. This pavedthe way to spectacular new achievements, such as the proof by Wiles and Taylorof Fermat’s Last Theorem, a famous problem that was open for more than 350years.

The purpose of the present book is to explain the basics of modern Al-gebraic Geometry to non-experts, thereby creating a platform from whichone can take off towards more advanced regions. Several times I have givencourses and seminars on the subject, requiring just two semesters of LinearAlgebra for beginners as a prerequisite. Usually I did one semester of Com-mutative Algebra and then continued with two semesters of AlgebraicGeometry. Each semester consisted of a combination of traditional lecturestogether with an attached seminar where the students presented additional ma-terial by themselves, extending the theory, supplying proofs that were skippedin the lectures, or solving exercise problems. The material covered in this waycorresponds roughly to the contents of the present book. Just as for my stu-dents, the necessary prerequisites are limited to basic knowledge in LinearAlgebra, 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 Elements de Geometrie Algebrique byGrothendieck and Dieudonne, four volumes of increasing size that were latercontinued by seven volumes of Seminaire de Geometrie Algebrique. The seriesis 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 acertain amount of expertise in the field. To approach Algebraic Geometryfrom a more economic point of view, I think it is necessary to learn about itsbasic principles. If these are well understood, many results become easier to di-gest, including proofs, and getting lost in a multitude of details can be avoided.

Preface

Therefore it is not my intention to cover as many topics as possible in my book.Instead I have chosen to concentrate on a certain selection of main themes thatare explained with all their underlying structures and without making use ofany artificial shortcuts. In spite of thematic restrictions, I am aiming at a self-contained exposition up to a level where more specialized literature comes intoreach.

Anyone willing to enter Algebraic Geometry should begin with certainbasic facts in Commutative Algebra. So the first part of the book is con-cerned with this subject. It begins with a general chapter on rings and moduleswhere, among other things, I explain the fundamental process of localization, aswell as certain finiteness conditions for modules, like being Noetherian or coher-ent. Then follows a classical chapter on Noetherian (and Artinian) rings, includ-ing the discussion of primary decompositions and of Krull dimensions, as wellas a classical chapter on integral ring extensions. In another chapter I explainthe process of coefficient extension for modules by means of tensor products,as well as its reverse, descent. In particular, a complete proof of Grothendieck’sfundamental theorem on faithfully flat descent for modules is given. Moreover,as it is quite useful at this place, I cast a cautious glimpse on categories and theirfunctors, including functorial morphisms. The first part of the book ends by achapter on Ext and Tor modules where the general machinery of homologicalmethods is explained.

The second part deals with Algebraic Geometry in the stricter sense ofthe word. Here I have limited myself to four general themes, each of them dealtwith in a chapter by itself, namely the construction of affine schemes, techniquesof global schemes, etale and smooth morphisms, and projective and properschemes, including the correspondence between ample and very ample invert-ible sheaves and its application to abelian varieties. There is nothing really newin these chapters, although the style in which I present the material is differentfrom other treatments. In particular, this concerns the handling of smooth mor-phisms via the Jacobian Condition, as well as the definition of ample invertiblesheaves via the use of quasi-affine schemes. This is the way in which M. Raynaudliked 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 itscontents and give an overview. As I cannot deliver a comprehensive accountalready at this point, I try to spotlight the main aspects, usually illustratingthese by a typical example. It is recommended to resort to the introductorysections at various times during the study of the corresponding chapter, in or-der to gradually increase the level of understanding for the strategy and pointof view employed at different stages. The latter is an important part of thelearning process, since Mathematics, like Algebraic Geometry, consists ofa 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 areliable feeling for both of these components. The selection of exercise problemsat the end of each section is meant to provide additional assistance for this.

VI

Preface

Preliminary versions of my manuscripts on Commutative Algebra andAlgebraic Geometry were made available to several generations of students.It was a great pleasure for me to see them getting excited about the subject, andI am very grateful for all their comments and other sort of feedback, includinglists of typos, such as the ones by David Krumm and Claudius Zibrowius. Specialthanks go to Christian Kappen, who worked carefully on earlier versions of thetext, as well as to Martin Brandenburg, who was of invaluable help duringthe final process. Not only did he study the whole manuscript meticulously,presenting an abundance of suggestions for improvements, he also contributedto the exercises and acted as a professional coach for the students attendingmy seminars on the subject. It is unfortunate that the scope of the book didnot permit me to put all his ingenious ideas into effect. Last, but not least,let me thank my young colleagues Matthias Strauch and Clara Loh, who runseminars on the material of the book together with me and who helped mesetting up appropriate themes for the students. In addition, Clara Loh suggestednumerous improvements for the manuscript, including matters of typesettingand language. Also the figure for gluing schemes in the beginning of Section 7.1is due to her.

Munster, December 2011 Siegfried Bosch

VII

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

Contents

Part B. Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . 193Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

6 Affine Schemes and Basic Constructions . . . . . . . . . . . . . . . . 2016.1 The Spectrum of a Ring . . . . . . . . . . . . . . . . . . . . . . 2036.2 Functorial Properties of Spectra . . . . . . . . . . . . . . . . . . 2126.3 Presheaves and Sheaves . . . . . . . . . . . . . . . . . . . . . . . 2166.4 Inductive and Projective Limits . . . . . . . . . . . . . . . . . . 2226.5 Morphisms of Sheaves and Sheafification . . . . . . . . . . . . . 2326.6 Construction of Affine Schemes . . . . . . . . . . . . . . . . . . 2416.7 The Affine n-Space . . . . . . . . . . . . . . . . . . . . . . . . . 2556.8 Quasi-Coherent Modules . . . . . . . . . . . . . . . . . . . . . . 2576.9 Direct and Inverse Images of Module Sheaves . . . . . . . . . . 266

7 Techniques of Global Schemes . . . . . . . . . . . . . . . . . . . . . 2777.1 Construction of Schemes by Gluing . . . . . . . . . . . . . . . . 2827.2 Fiber Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 2947.3 Subschemes and Immersions . . . . . . . . . . . . . . . . . . . . 3047.4 Separated Schemes . . . . . . . . . . . . . . . . . . . . . . . . . 3127.5 Noetherian Schemes and their Dimension . . . . . . . . . . . . . 3187.6 Cech Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 3227.7 Grothendieck Cohomology . . . . . . . . . . . . . . . . . . . . . 330

8 Etale and Smooth Morphisms . . . . . . . . . . . . . . . . . . . . . . 3418.1 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 3438.2 Sheaves of Differential Forms . . . . . . . . . . . . . . . . . . . . 3568.3 Morphisms of Finite Type and of Finite Presentation . . . . . . 3608.4 Unramified Morphisms . . . . . . . . . . . . . . . . . . . . . . . 3658.5 Smooth Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . 374

9 Projective Schemes and Proper Morphisms . . . . . . . . . . . . . . 3999.1 Homogeneous Prime Spectra as Schemes . . . . . . . . . . . . . 4039.2 Invertible Sheaves and Serre Twists . . . . . . . . . . . . . . . . 4189.3 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4319.4 Global Sections of Invertible Sheaves . . . . . . . . . . . . . . . 4469.5 Proper Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . 4629.6 Abelian Varieties are Projective . . . . . . . . . . . . . . . . . . 477

Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

Glossary of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

X