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6 Basic Concepts of the Theory of Schemes1

6.1 Affine Schemes1

6.1.1 Localization1

6.1.2 The Spectrum of a Ring2

6.1.3 The Zariski Topology on Spec（A）6

6.1.4 The Structure Sheaf on Spec（A）8

6.1.5 Quasicoherent Sheaves11

6.1.6 Schemes as Locally Ringed Spaces12

Closed Subschemes14

Sections15

A remark15

6.2 Schemes16

6.2.1 The Definition of a Scheme16

The gluing16

Closed subschemes again17

Annihilators,supports and intersections18

6.2.2 Functorial properties18

Affine morphisms19

Sections again19

6.2.3 Construction of Quasi-coherent Sheaves19

Vector bundles20

Vector Bundles Attached to Locally Free Modules20

6.2.4 Vector bundles and GLn-torsors21

6.2.5 Schemes over a base scheme S22

Some notions of finiteness22

Fibered products23

Base change28

6.2.6 Points,T-valued Points and Geometric Points28

Closed Points and Geometric Points on varieties32

6.2.7 Flat Morphisms34

The Concept of Flatness35

Representability of functors38

6.2.8 Theory of descend40

Effectiveness for affine descend data43

6.2.9 Galois descend44

A geometric interpretation47

Descend for general schemes of finite type48

6.2.10 Forms of schemes48

6.2.11 An outlook to more general concepts51

7 Some Commutative Algebra55

7.1 Finite A-Algebras55

7.1.1 Rings With Finiteness Conditions58

7.1.2 Dimension theory for finitely generated k-algebras59

7.2 Minimal prime ideals and decomposition into irreducibles61

Associated prime ideals63

The restriction to the components63

Decomposition into irreducibles for noetherian schemes64

Local dimension65

7.2.1 Affine schemes over k and change of scalars65

What is dim（Z1∩Z2）？70

7.2.2 Local Irreducibility71

The connected component of the identity of an affine group scheme G/k72

7.3 Low Dimensional Rings73

Finite k-Algebras73

One Dimensional Rings and Basic Results from Algebraic Number Theory74

7.4 Flat morphisms80

7.4.1 Finiteness Properties of Tor80

7.4.2 Construction of fiat families82

7.4.3 Dominant morphisms84

Birational morphisms88

The Artin-Rees Theorem89

7.4.4 Formal Schemes and Infinitesimal Schemes90

7.5 Smooth Points91

The Jacobi Criterion95

7.5.1 Generic Smoothness97

The singular locus97

7.5.2 Relative Differentials99

7.5.3 Examples102

7.5.4 Normal schemes and smoothness in codimension one109

Regular local rings110

7.5.5 Vector fields,derivations and infinitesimal automorphisms111

Automorphisms114

7.5.6 Group schemes114

7.5.7 The groups schemes Ga,Gm and μn116

7.5.8 Actions of group schemes117

8 Projective Schemes121

8.1 Geometric Constructions121

8.1.1 The Projective Space IP？121

Homogenous coordinates123

8.1.2 Closed subschemes125

8.1.3 Projective Morphisms and Projective Schemes126

Locally Free Sheaves on IPn129

OIPn（d）as Sheaf of Meromorphic Functions131

The Relative Differentials and the Tangent Bundle of IP？132

8.1.4 Seperated and Proper Morphisms134

8.1.5 The Valuative Criteria136

The Valuative Criterion for the Projective Space136

8.1.6 The Construction Proj（R）137

A special case of a finiteness result139

8.1.7 Ample and Very Ample Sheaves140

8.2 Cohomology of Quasicoherent Sheaves146

8.2.1 ？ech cohomology148

8.2.2 The Künneth-formulae150

8.2.3 The cohomology of the sheaves OIPn （r）151

8.3 Cohomology of Coherent Sheaves153

The Hilbert polynomial157

8.3.1 The coherence theorem for proper morphisms158

Digression:Blowing up and contracting159

8.4 Base Change164

8.4.1 Flat families and intersection numbers171

The Theorem of Bertini179

8.4.2 The hyperplane section and intersection numbers of line bundles180

9 Curves and the Theorem of Riemann-Roch183

9.1 Some basic notions183

9.2 The local rings at closed points185

9.2.1 The structure of ？C,p186

9.2.2 Base change186

9.3 Curves and their function fields188

9.3.1 Ramification and the different ideal190

9.4 Line bundles and Divisors193

9.4.1 Divisors on curves195

9.4.2 Properties of the degree197

Line bundles on non smooth curves have a degree197

Base change for divisors and line bundles198

9.4.3 Vector bundles over a curve198

Vector bundles on IP1199

9.5 The Theorem of Riemann-Roch201

9.5.1 Differentials and Residues203

9.5.2 The special case C=IP1/k207

9.5.3 Back to the general case211

9.5.4 Riemann-Roch for vector bundles and for coherent sheaves218

The structure of K′（C）220

9.6 Applications of the Riemann-Roch Theorem221

9.6.1 Curves of low genus221

9.6.2 The moduli space223

9.6.3 Curves of higher genus234

The“moduli space”of curves of genus g238

9.7 The Grothendieck-Riemann-Roch Theorem239

9.7.1 A special case of the Grothendieck-Riemann-Roch theorem240

9.7.2 Some geometric considerations241

9.7.3 The Chow ring244

Base extension of the Chow ring247

9.7.4 The formulation of the Grothendieck-Riemann-Roch Theorem249

9.7.5 Some special cases of the Grothendieck-Riemann-Roch-Theorem252

9.7.6 Back to the case p2:X=C×C→C253

9.7.7 Curves over finite fields257

Elementary properties of the ζ-function258

The Riemann hypothesis261

10 The Picard functor for curves and their Jacobians265

Introduction:265

10.1 The construction of the Jacobian265

10.1.1 Generalities and heuristics:265

Rigidification of PIC267

10.1.2 General properties of the functor PIC269

The locus of triviality269

10.1.3 Infinitesimal properties272

Differentiating a line bundle along a vector field274

The theorem of the cube274

10.1.4 The basic principles of the construction of the Picard scheme of a curve278

10.1.5 Symmetric powers279

10.1.6 The actual construction of the Picard scheme of a curve284

The gluing291

10.1.7 The local representability of PIC？/k294

10.2 The Picard functor on X and on J297

Some heuristic remarks297

10.2.1 Construction of line bundles on X and on J297

The homomorphisms ？M298

10.2.2 The projectivity of X and J301

The morphisms ？M are homomorphisms of functors302

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 Jacobian303

10.2.4 The self duality of the Jacobian310

10.2.5 General abelian varieties311

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

Some heuristics and outlooks314

The study of End（J）315

The degree and the trace318

The Weil Pairing326

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

The ring of correspondences331

10.4 ？tale Cohomology334

The cyclotomic character334

10.4.1 ？tale cohomology groups335

Galois cohomology336

The geometric étale cohomology groups338

10.4.2 Schemes over finite fields344

The global case346

The degenerating family of elliptic curves350

Bibliography357

Index362