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