Algebraic geometry 1: Schemes

Cover......Page 1
Algebraic Geometry I: Schemes With Examples and Exercises......Page 4
9783834806765......Page 5
Contents......Page 6
Introduction......Page 10
Leitfaden......Page 12
Acknowledgements......Page 15
1 Prevarieties......Page 16
Affine algebraic sets......Page 17
Affine algebraic sets as spaces with functions......Page 26
Prevarieties......Page 32
Projective varieties......Page 35
Exercises......Page 45
2 Spectrum of a Ring......Page 49
Spectrum of a ring as a topological space......Page 50
Excursion: Sheaves......Page 56
Spectrum of a ring as a locally ringed space......Page 66
Exercises......Page 71
Schemes......Page 75
Examples of schemes......Page 81
Basic properties of schemes and morphisms of schemes......Page 83
Prevarieties as Schemes......Page 87
Subschemes and Immersions......Page 92
Exercises......Page 97
Schemes as functors......Page 102
Fiber products of schemes......Page 106
Base change, Fibers of a morphism......Page 114
Exercises......Page 123
Schemes over a field which is not algebraically closed......Page 127
Dimension of schemes over a field......Page 129
Schemes over fields and extensions of the base field......Page 142
Intersections of plane curves......Page 147
Exercises......Page 150
6 Local Properties of Schemes......Page 154
The tangent space......Page 155
Smooth morphisms......Page 162
Regular schemes......Page 167
Normal schemes......Page 171
Exercises......Page 173
Excursion: OX-modules......Page 178
Quasi-coherent modules on a scheme......Page 190
Properties of quasi-coherent modules......Page 198
Exercises......Page 208
8 Representable Functors......Page 214
Representable Functors......Page 215
The example of the Grassmannian......Page 218
Brauer-Severi schemes......Page 228
Exercises......Page 231
9 Separated morphisms......Page 235
Diagonal of scheme morphisms and separated morphisms......Page 236
Rational maps and function fields......Page 241
Exercises......Page 247
10 Finiteness Conditions......Page 250
Finiteness conditions (noetherian case)......Page 251
Finiteness conditions in the non-noetherian case......Page 258
Schemes over inductive limits of rings......Page 267
Constructible properties......Page 279
Exercises......Page 287
11 Vector bundles......Page 295
Vector bundles and locally free modules......Page 296
Flattening stratification for modules......Page 306
Divisors......Page 307
Vector bundles on P1......Page 322
Exercises......Page 325
Affine morphisms......Page 329
Finite and quasi-finite morphisms......Page 333
Serre’s and Chevalley’s criteria to be affine......Page 343
Normalization......Page 348
Proper morphisms......Page 352
Zariski’s main theorem......Page 358
Exercises......Page 370
13 Projective morphisms......Page 375
Projective spectrum of a graded algebra......Page 376
Embeddings into projective space......Page 393
Blowing-up......Page 415
Exercises......Page 427
Flat morphisms......Page 432
Properties of flat morphisms......Page 438
Faithfully flat descent......Page 448
Dimension and fibers of morphisms......Page 472
Dimension and regularity conditions......Page 482
Hilbert schemes......Page 487
Exercises......Page 489
Morphisms into and from one-dimensional schemes......Page 494
Valuative criteria......Page 496
Curves over fields......Page 500
Divisors on curves......Page 505
Exercises......Page 510
Determinantal varieties......Page 512
Cubic surfaces and a Hilbert modular surface......Page 529
Cyclic quotient singularities......Page 538
Abelian varieties......Page 542
Exercises......Page 549
A The language of categories......Page 550
B Commutative Algebra......Page 556
C Permanence for properties of morphisms of schemes......Page 582
D Relations between properties of morphisms of schemes......Page 585
E Constructible and open properties......Page 587
Bibliography......Page 592
Detailed List of Contents......Page 597
Index of Symbols......Page 607
Index......Page 611