Lectures on Algebraic Geometry II (eBook)

Prof. Dr. Günter Harder, Max-Planck-Institute for Mathematics, Bonn

Preface 5
Contents 7
Introduction 12
6 Basic Concepts of the Theory of Schemes 14
6.1 Affine Schemes 14
6.1.1 Localization 14
6.1.2 The Spectrum of a Ring 15
6.1.3 The Zariski Topology on Spec(A) 19
6.1.4 The Structure Sheaf on Spec(A) 21
6.1.5 Quasicoherent Sheaves 24
6.1.6 Schemes as Locally Ringed Spaces 25
Closed Subschemes 27
Sections 28
A remark 28
6.2 Schemes 29
6.2.1 The Definition of a Scheme 29
The gluing 29
Closed subschemes again 30
Annihilators, supports and intersections 31
6.2.2 Functorial properties 31
Affine morphisms 32
Sections again 32
6.2.3 Construction of Quasi-coherent Sheaves 32
Vector bundles 33
Vector Bundles Attached to Locally Free Modules 33
6.2.4 Vector bundles and GLn-torsors. 34
6.2.5 Schemes over a base scheme S. 35
Some notions of finiteness 35
Fibered products 36
Base change 41
6.2.6 Points, T-valued Points and Geometric Points 41
Closed Points and Geometric Points on varieties 45
6.2.7 Flat Morphisms 47
The Concept of Flatness 48
Representability of functors 51
6.2.8 Theory of descend 53
Effectiveness for affine descend data 56
6.2.9 Galois descend 57
A geometric interpretation 60
Descend for general schemes of finite type 61
6.2.10 Forms of schemes 61
6.2.11 An outlook to more general concepts 64
7 Some Commutative Algebra 67
7.1 Finite A-Algebras 67
7.1.1 Rings With Finiteness Conditions 70
7.1.2 Dimension theory for finitely generated k-algebras 71
7.2 Minimal prime ideals and decomposition into irreducibles 73
7.2.1 A.ne schemes over k and change of scalars 77
What is dim(Z1 n Z2)? 82
7.2.2 Local Irreducibility 83
The connected component of the identity of an affine group scheme G/k 84
7.3 Low Dimensional Rings 85
7.4 Flat morphisms 92
7.4.1 Finiteness Properties of Tor 92
7.4.2 Construction of flat families 94
7.4.3 Dominant morphisms 96
Birational morphisms 100
The Artin-Rees Theorem 101
7.4.4 Formal Schemes and Infinitesimal Schemes 102
7.5 Smooth Points 103
7.5.1 Generic Smoothness 109
The singular locus 109
7.5.2 Relative Differentials 111
7.5.3 Examples 114
7.5.4 Normal schemes and smoothness in codimension one 121
Regular local rings 122
7.5.5 Vector fields, derivations and infinitesimal automorphisms 123
Automorphisms 126
7.5.6 Group schemes 126
7.5.7 The groups schemes Ga,Gm and µn 128
7.5.8 Actions of group schemes 129
8 Projective Schemes 132
8.1 Geometric Constructions 132
8.1.1 The Projective Space pnA 132
Homogenous coordinates 134
8.1.2 Closed subschemes 136
8.1.3 Projective Morphisms and Projective Schemes 137
Locally Free Sheaves on pn 140
Opn (d) as Sheaf of Meromorphic Functions 142
The Relative Differentials and the Tangent Bundle of pnS 143
8.1.4 Seperated and Proper Morphisms 145
8.1.5 The Valuative Criteria 147
The Valuative Criterion for the Projective Space 147
8.1.6 The Construction Proj(R) 148
A special case of a finiteness result 150
8.1.7 Ample and Very Ample Sheaves 151
8.2 Cohomology of Quasicoherent Sheaves 157
8.2.1 Cech cohomology 159
8.2.2 The Künneth-formulae 161
8.2.3 The cohomology of the sheaves Opn (r) 162
8.3 Cohomology of Coherent Sheaves 164
8.3.1 The coherence theorem for proper morphisms 169
Digression: Blowing up and contracting 170
8.4 Base Change 175
8.4.1 Flat families and intersection numbers 182
The Theorem of Bertini 190
8.4.2 The hyperplane section and intersection numbers of line bundles 191
9 Curves and the Theorem of Riemann-Roch 194
9.1 Some basic notions 194
9.2 The local rings at closed points 196
9.2.1 The structure of OC,p 197
9.2.2 Base change 197
9.3 Curves and their function fields 199
9.3.1 Ramification and the different ideal 201
9.4 Line bundles and Divisors 204
9.4.1 Divisors on curves 206
9.4.2 Properties of the degree 208
Line bundles on non smooth curves have a degree 208
Base change for divisors and line bundles 209
9.4.3 Vector bundles over a curve 209
Vector bundles on p1 210
9.5 The Theorem of Riemann-Roch 212
9.5.1 Differentials and Residues 214
9.5.2 The special case C = p1/k 218
9.5.3 Back to the general case 222
9.5.4 Riemann-Roch for vector bundles and for coherent sheaves. 229
The structure of K'(C) 231
9.6 Applications of the Riemann-Roch Theorem 232
9.6.1 Curves of low genus 232
9.6.2 The moduli space 234
9.6.3 Curves of higher genus 245
The ”moduli space” of curves of genus g 249
9.7 The Grothendieck-Riemann-Roch Theorem 250
9.7.1 A special case of the Grothendieck -Riemann-Roch theorem 251
9.7.2 Some geometric considerations 252
9.7.3 The Chow ring 255
Base extension of the Chow ring 258
9.7.4 The formulation of the Grothendieck-Riemann-Roch Theorem 260
9.7.5 Some special cases of the Grothendieck-Riemann-Roch-Theorem 263
9.7.6 Back to the case p2 : X = C × C -. C 264
9.7.7 Curves over finite fi 
268 
Elementary properties of the .-function. 269
The Riemann hypothesis. 272
10 The Picard functor for curves and their Jacobians 276
10.1 The construction of the Jacobian 276
10.1.1 Generalities and heuristics : 276
Rigidification of PIC 278
10.1.2 General properties of the functor PIC 280
The locus of triviality 280
10.1.3 Infinitesimal properties 283
Differentiating a line bundle along a vector field 285
The theorem of the cube. 285
10.1.4 The basic principles of the construction of the Picard scheme of a curve. 289
10.1.5 Symmetric powers 290
10.1.6 The actual construction of the Picard scheme of a curve. 295
The gluing 302
10.1.7 The local representability of PICgC/k 305
10.2 The Picard functor on X and on J 308
10.2.1 Construction of line bundles on X and on J 308
The homomorphisms fM 309
10.2.2 The projectivity of X and J 312
The morphisms fM are homomorphisms of functors 313
10.2.3 Maps from the curve C to X, local representability of PICX/k , and the self duality of the Jacobian 314
10.2.4 The self duality of the Jacobian 321
10.2.5 General abelian varieties 322
10.3 The ring of endomorphisms End(J) and the l -adic modules 325
10.4 Étale Cohomology 345
10.4.1 Étale cohomology groups 346
Galois cohomology 347
The geometric étale cohomology groups. 349
10.4.2 Schemes over finite fields 355
The global case 357
The degenerating family of elliptic curves 361
Bibliography 368
Index 373