Lectures on Algebraic Geometry II: Basic Concepts, Coherent Cohomology, Curves and their Jacobians

Cover......Page 1
Lectures on\rAlgebraic Geometry II......Page 4
ISBN 9783834804327......Page 5
Preface......Page 6
Contents......Page 8
Introduction......Page 13
6.1.1 Localization......Page 16
6.1.2 The Spectrum of a Ring......Page 17
6.1.3 The Zariski Topology on Spec(A)......Page 21
6.1.4 The Structure Sheaf on Spec(A)......Page 23
6.1.5 Quasicoherent Sheaves......Page 26
6.1.6 Schemes as Locally Ringed Spaces......Page 27
Closed Subschemes......Page 29
A remark......Page 30
The gluing......Page 31
Closed subschemes again......Page 32
6.2.2 Functorial properties......Page 33
6.2.3 Construction of Quasi-coherent Sheaves......Page 34
Vector Bundles Attached to Locally Free Modules......Page 35
6.2.4 Vector bundles and GLn-torsors.......Page 36
Some notions of finiteness......Page 37
Fibered products......Page 38
6.2.6 Points, T-valued Points and Geometric Points......Page 43
Closed Points and Geometric Points on varieties......Page 47
6.2.7 Flat Morphisms......Page 49
The Concept of Flatness......Page 50
Representability of functors......Page 53
6.2.8 Theory of descend......Page 55
Effectiveness for affine descend data......Page 58
6.2.9 Galois descend......Page 59
A geometric interpretation......Page 62
6.2.10 Forms of schemes......Page 63
6.2.11 An outlook to more general concepts......Page 66
7.1 Finite A-Algebras......Page 70
7.1.1 Rings With Finiteness Conditions......Page 73
7.1.2 Dimension theory for finitely generated k-algebras......Page 74
7.2 Minimal prime ideals and decomposition into irreducibles......Page 76
7.2.1 A.ne schemes over k and change of scalars......Page 80
What is dim(Z1 ∩ Z2)?......Page 85
7.2.2 Local Irreducibility......Page 86
The connected component of the identity of an affine group scheme G/k......Page 87
7.3 Low Dimensional Rings......Page 88
7.4.1 Finiteness Properties of Tor......Page 95
7.4.2 Construction of flat families......Page 97
7.4.3 Dominant morphisms......Page 99
Birational morphisms......Page 103
The Artin-Rees Theorem......Page 104
7.4.4 Formal Schemes and Infinitesimal Schemes......Page 105
7.5 Smooth Points......Page 106
The singular locus......Page 112
7.5.2 Relative Differentials......Page 114
7.5.3 Examples......Page 117
7.5.4 Normal schemes and smoothness in codimension one......Page 124
Regular local rings......Page 125
7.5.5 Vector fields, derivations and infinitesimal automorphisms......Page 126
7.5.6 Group schemes......Page 129
7.5.7 The groups schemes Ga,Gm and μn......Page 131
7.5.8 Actions of group schemes......Page 132
8.1.1 The Projective Space pnA......Page 136
Homogenous coordinates......Page 138
8.1.2 Closed subschemes......Page 140
8.1.3 Projective Morphisms and Projective Schemes......Page 141
Locally Free Sheaves on pn......Page 144
Opn (d) as Sheaf of Meromorphic Functions......Page 146
The Relative Differentials and the Tangent Bundle of pnS......Page 147
8.1.4 Seperated and Proper Morphisms......Page 149
The Valuative Criterion for the Projective Space......Page 151
8.1.6 The Construction Proj(R)......Page 152
A special case of a finiteness result......Page 154
8.1.7 Ample and Very Ample Sheaves......Page 155
8.2 Cohomology of Quasicoherent Sheaves......Page 161
8.2.1 Čech cohomology......Page 163
8.2.2 The Künneth-formulae......Page 165
8.2.3 The cohomology of the sheaves Opn (r)......Page 166
8.3 Cohomology of Coherent Sheaves......Page 168
8.3.1 The coherence theorem for proper morphisms......Page 173
Digression: Blowing up and contracting......Page 174
8.4 Base Change......Page 179
8.4.1 Flat families and intersection numbers......Page 186
The Theorem of Bertini......Page 194
8.4.2 The hyperplane section and intersection numbers of line bundles......Page 195
9.1 Some basic notions......Page 198
9.2 The local rings at closed points......Page 200
9.2.2 Base change......Page 201
9.3 Curves and their function fields......Page 203
9.3.1 Ramification and the different ideal......Page 205
9.4 Line bundles and Divisors......Page 208
9.4.1 Divisors on curves......Page 210
Line bundles on non smooth curves have a degree......Page 212
9.4.3 Vector bundles over a curve......Page 213
Vector bundles on p1......Page 214
9.5 The Theorem of Riemann-Roch......Page 216
9.5.1 Differentials and Residues......Page 218
9.5.2 The special case C = p1/k......Page 222
9.5.3 Back to the general case......Page 226
9.5.4 Riemann-Roch for vector bundles and for coherent sheaves.......Page 233
The structure of K\'(C)......Page 235
9.6.1 Curves of low genus......Page 236
9.6.2 The moduli space......Page 238
9.6.3 Curves of higher genus......Page 249
The ”moduli space” of curves of genus g......Page 253
9.7 The Grothendieck-Riemann-Roch Theorem......Page 254
9.7.1 A special case of the Grothendieck -Riemann-Roch theorem......Page 255
9.7.2 Some geometric considerations......Page 256
9.7.3 The Chow ring......Page 259
Base extension of the Chow ring......Page 262
9.7.4 The formulation of the Grothendieck-Riemann-Roch Theorem......Page 264
9.7.5 Some special cases of the Grothendieck-Riemann-Roch-Theorem......Page 267
9.7.6 Back to the case p2 : X = C × C -. C......Page 268
9.7.7 Curves over finite fi\relds.......Page 272
Elementary properties of the ζ-function.......Page 273
The Riemann hypothesis.......Page 276
10.1.1 Generalities and heuristics :......Page 280
Rigidification of PIC......Page 282
The locus of triviality......Page 284
10.1.3 Infinitesimal properties......Page 287
The theorem of the cube.......Page 289
10.1.4 The basic principles of the construction of the Picard scheme of a curve.......Page 293
10.1.5 Symmetric powers......Page 294
10.1.6 The actual construction of the Picard scheme of a curve.......Page 299
The gluing......Page 306
10.1.7 The local representability of PICgC/k......Page 309
10.2.1 Construction of line bundles on X and on J......Page 312
The homomorphisms φM......Page 313
10.2.2 The projectivity of X and J......Page 316
The morphisms φM are homomorphisms of functors......Page 317
10.2.3 Maps from the curve C to X, local representability of PICX/k , and the self duality of the Jacobian......Page 318
10.2.4 The self duality of the Jacobian......Page 325
10.2.5 General abelian varieties......Page 326
10.3 The ring of endomorphisms End(J) and the l -adic modules......Page 329
10.4 Étale Cohomology......Page 349
10.4.1 Étale cohomology groups......Page 350
Galois cohomology......Page 351
The geometric étale cohomology groups.......Page 353
10.4.2 Schemes over finite fields......Page 359
The global case......Page 361
The degenerating family of elliptic curves......Page 365
Bibliography......Page 372
Index......Page 377