代数几何讲义 第2卷PDF电子书下载

6 Basic Concepts of the Theory of Schemes 1

6.1 Affine Schemes 1

6.1.1 Localization 1

6.1.2 The Spectrum of a Ring 2

6.1.3 The Zariski Topology on Spec（A） 6

6.1.4 The Structure Sheaf on Spec（A） 8

6.1.5 Quasicoherent Sheaves 11

6.1.6 Schemes as Locally Ringed Spaces 12

Closed Subschemes 14

Sections 15

A remark 15

6.2 Schemes 16

6.2.1 The Definition of a Scheme 16

The gluing 16

Closed subschemes again 17

Annihilators,supports and intersections 18

6.2.2 Functorial properties 18

Affine morphisms 19

Sections again 19

6.2.3 Construction of Quasi-coherent Sheaves 19

Vector bundles 20

Vector Bundles Attached to Locally Free Modules 20

6.2.4 Vector bundles and GLn-torsors 21

6.2.5 Schemes over a base scheme S 22

Some notions of finiteness 22

Fibered products 23

Base change 28

6.2.6 Points,T-valued Points and Geometric Points 28

Closed Points and Geometric Points on varieties 32

6.2.7 Flat Morphisms 34

The Concept of Flatness 35

Representability of functors 38

6.2.8 Theory of descend 40

Effectiveness for affine descend data 43

6.2.9 Galois descend 44

A geometric interpretation 47

Descend for general schemes of finite type 48

6.2.10 Forms of schemes 48

6.2.11 An outlook to more general concepts 51

7 Some Commutative Algebra 55

7.1 Finite A-Algebras 55

7.1.1 Rings With Finiteness Conditions 58

7.1.2 Dimension theory for finitely generated k-algebras 59

7.2 Minimal prime ideals and decomposition into irreducibles 61

Associated prime ideals 63

The restriction to the components 63

Decomposition into irreducibles for noetherian schemes 64

Local dimension 65

7.2.1 Affine schemes over k and change of scalars 65

What is dim（Z1∩Z2）？ 70

7.2.2 Local Irreducibility 71

The connected component of the identity of an affine group scheme G/k 72

7.3 Low Dimensional Rings 73

Finite k-Algebras 73

One Dimensional Rings and Basic Results from Algebraic Number Theory 74

7.4 Flat morphisms 80

7.4.1 Finiteness Properties of Tor 80

7.4.2 Construction of fiat families 82

7.4.3 Dominant morphisms 84

Birational morphisms 88

The Artin-Rees Theorem 89

7.4.4 Formal Schemes and Infinitesimal Schemes 90

7.5 Smooth Points 91

The Jacobi Criterion 95

7.5.1 Generic Smoothness 97

The singular locus 97

7.5.2 Relative Differentials 99

7.5.3 Examples 102

7.5.4 Normal schemes and smoothness in codimension one 109

Regular local rings 110

7.5.5 Vector fields,derivations and infinitesimal automorphisms 111

Automorphisms 114

7.5.6 Group schemes 114

7.5.7 The groups schemes Ga,Gm and μn 116

7.5.8 Actions of group schemes 117

8 Projective Schemes 121

8.1 Geometric Constructions 121

8.1.1 The Projective Space IP？ 121

Homogenous coordinates 123

8.1.2 Closed subschemes 125

8.1.3 Projective Morphisms and Projective Schemes 126

Locally Free Sheaves on IPn 129

OIPn（d）as Sheaf of Meromorphic Functions 131

The Relative Differentials and the Tangent Bundle of IP？ 132

8.1.4 Seperated and Proper Morphisms 134

8.1.5 The Valuative Criteria 136

The Valuative Criterion for the Projective Space 136

8.1.6 The Construction Proj（R） 137

A special case of a finiteness result 139

8.1.7 Ample and Very Ample Sheaves 140

8.2 Cohomology of Quasicoherent Sheaves 146

8.2.1 ？ech cohomology 148

8.2.2 The Künneth-formulae 150

8.2.3 The cohomology of the sheaves OIPn （r） 151

8.3 Cohomology of Coherent Sheaves 153

The Hilbert polynomial 157

8.3.1 The coherence theorem for proper morphisms 158

Digression:Blowing up and contracting 159

8.4 Base Change 164

8.4.1 Flat families and intersection numbers 171

The Theorem of Bertini 179

8.4.2 The hyperplane section and intersection numbers of line bundles 180

9 Curves and the Theorem of Riemann-Roch 183

9.1 Some basic notions 183

9.2 The local rings at closed points 185

9.2.1 The structure of ？C,p 186

9.2.2 Base change 186

9.3 Curves and their function fields 188

9.3.1 Ramification and the different ideal 190

9.4 Line bundles and Divisors 193

9.4.1 Divisors on curves 195

9.4.2 Properties of the degree 197

Line bundles on non smooth curves have a degree 197

Base change for divisors and line bundles 198

9.4.3 Vector bundles over a curve 198

Vector bundles on IP1 199

9.5 The Theorem of Riemann-Roch 201

9.5.1 Differentials and Residues 203

9.5.2 The special case C=IP1/k 207

9.5.3 Back to the general case 211

9.5.4 Riemann-Roch for vector bundles and for coherent sheaves 218

The structure of K′（C） 220

9.6 Applications of the Riemann-Roch Theorem 221

9.6.1 Curves of low genus 221

9.6.2 The moduli space 223

9.6.3 Curves of higher genus 234

The“moduli space”of curves of genus g 238

9.7 The Grothendieck-Riemann-Roch Theorem 239

9.7.1 A special case of the Grothendieck-Riemann-Roch theorem 240

9.7.2 Some geometric considerations 241

9.7.3 The Chow ring 244

Base extension of the Chow ring 247

9.7.4 The formulation of the Grothendieck-Riemann-Roch Theorem 249

9.7.5 Some special cases of the Grothendieck-Riemann-Roch-Theorem 252

9.7.6 Back to the case p2:X=C×C→C 253

9.7.7 Curves over finite fields 257

Elementary properties of the ζ-function 258

The Riemann hypothesis 261

10 The Picard functor for curves and their Jacobians 265

Introduction: 265

10.1 The construction of the Jacobian 265

10.1.1 Generalities and heuristics: 265

Rigidification of PIC 267

10.1.2 General properties of the functor PIC 269

The locus of triviality 269

10.1.3 Infinitesimal properties 272

Differentiating a line bundle along a vector field 274

The theorem of the cube 274

10.1.4 The basic principles of the construction of the Picard scheme of a curve 278

10.1.5 Symmetric powers 279

10.1.6 The actual construction of the Picard scheme of a curve 284

The gluing 291

10.1.7 The local representability of PIC？/k 294

10.2 The Picard functor on X and on J 297

Some heuristic remarks 297

10.2.1 Construction of line bundles on X and on J 297

The homomorphisms ？M 298

10.2.2 The projectivity of X and J 301

The morphisms ？M are homomorphisms of functors 302

10.2.3 Maps from the curve C to X,local representability of PIC X/k,PIC J/k and the self duality of the Jacobian 303

10.2.4 The self duality of the Jacobian 310

10.2.5 General abelian varieties 311

10.3 The ring of endomorphisms End（J）and the e-adic modules Te（J） 314

Some heuristics and outlooks 314

The study of End（J） 315

The degree and the trace 318

The Weil Pairing 326

The Neron-Severi groups NS（J）,NS（J×J）and End（J） 328

The ring of correspondences 331

10.4 ？tale Cohomology 334

The cyclotomic character 334

10.4.1 ？tale cohomology groups 335

Galois cohomology 336

The geometric étale cohomology groups 338

10.4.2 Schemes over finite fields 344

The global case 346

The degenerating family of elliptic curves 350

Bibliography 357

Index 362