Harder Lectures on Algebraic Geometry II

Basic Concepts, Coherent Cohomology, Curves and their Jacobians
1. Auflage 2011
ISBN: 978-3-8348-8159-5
Verlag: Vieweg+Teubner (GWV)
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 365 Seiten

This second volume introduces the concept of shemes, reviews some commutative algebra and introduces projective schemes. The finiteness theorem for coherent sheaves is proved, here again the techniques of homological algebra and sheaf cohomology are needed. In the last two chapters, projective curves over an arbitrary ground field are discussed, the theory of Jacobians is developed, and the existence of the Picard scheme is proved.
Finally, the author gives some outlook into further developments- for instance étale cohomology- and states some fundamental theorems.

Autoren/Hrsg.

Harder, Günter

Inhaltsverzeichnis

1;Preface;5
2;Contents;7
3;Introduction;12
4;6 Basic Concepts of the Theory of Schemes;14
4.1;6.1 Affine Schemes;14
4.1.1;6.1.1 Localization;14
4.1.2;6.1.2 The Spectrum of a Ring ;15
4.1.3;6.1.3 The Zariski Topology on Spec(A);19
4.1.4;6.1.4 The Structure Sheaf on Spec(A);21
4.1.5;6.1.5 Quasicoherent Sheaves;24
4.1.6;6.1.6 Schemes as Locally Ringed Spaces;25
4.1.6.1;Closed Subschemes;27
4.1.6.2;Sections;28
4.1.6.3;A remark;28
4.2;6.2 Schemes;29
4.2.1;6.2.1 The Definition of a Scheme;29
4.2.1.1;The gluing;29
4.2.1.2;Closed subschemes again;30
4.2.1.3;Annihilators, supports and intersections;31
4.2.2;6.2.2 Functorial properties;31
4.2.2.1;Affine morphisms;32
4.2.2.2;Sections again;32
4.2.3;6.2.3 Construction of Quasi-coherent Sheaves;32
4.2.3.1;Vector bundles;33
4.2.3.2;Vector Bundles Attached to Locally Free Modules;33
4.2.4;6.2.4 Vector bundles and GLn-torsors.;34
4.2.5;6.2.5 Schemes over a base scheme S.;35
4.2.5.1;Some notions of finiteness;35
4.2.5.2;Fibered products;36
4.2.5.3;Base change;41
4.2.6;6.2.6 Points, T-valued Points and Geometric Points;41
4.2.6.1;Closed Points and Geometric Points on varieties;45
4.2.7;6.2.7 Flat Morphisms;47
4.2.7.1;The Concept of Flatness;48
4.2.7.2;Representability of functors;51
4.2.8;6.2.8 Theory of descend;53
4.2.8.1;Effectiveness for affine descend data;56
4.2.9;6.2.9 Galois descend;57
4.2.9.1;A geometric interpretation;60
4.2.9.2;Descend for general schemes of finite type;61
4.2.10;6.2.10 Forms of schemes;61
4.2.11;6.2.11 An outlook to more general concepts;64
5;7 Some Commutative Algebra;67
5.1;7.1 Finite A-Algebras;67
5.1.1;7.1.1 Rings With Finiteness Conditions;70
5.1.2;7.1.2 Dimension theory for finitely generated k-algebras ;71
5.2;7.2 Minimal prime ideals and decomposition into irreducibles;73
5.2.1;7.2.1 A.ne schemes over k and change of scalars;77
5.2.1.1;What is dim(Z1 n Z2)?;82
5.2.2;7.2.2 Local Irreducibility;83
5.2.2.1;The connected component of the identity of an affine group scheme G/k;84
5.3;7.3 Low Dimensional Rings;85
5.4;7.4 Flat morphisms;92
5.4.1;7.4.1 Finiteness Properties of Tor;92
5.4.2;7.4.2 Construction of flat families;94
5.4.3;7.4.3 Dominant morphisms;96
5.4.3.1;Birational morphisms;100
5.4.3.2;The Artin-Rees Theorem;101
5.4.4;7.4.4 Formal Schemes and Infinitesimal Schemes;102
5.5;7.5 Smooth Points;103
5.5.1;7.5.1 Generic Smoothness;109
5.5.1.1;The singular locus;109
5.5.2;7.5.2 Relative Differentials;111
5.5.3;7.5.3 Examples ;114
5.5.4;7.5.4 Normal schemes and smoothness in codimension one;121
5.5.4.1;Regular local rings;122
5.5.5;7.5.5 Vector fields, derivations and infinitesimal automorphisms;123
5.5.5.1;Automorphisms;126
5.5.6;7.5.6 Group schemes;126
5.5.7;7.5.7 The groups schemes Ga,Gm and µn;128
5.5.8;7.5.8 Actions of group schemes;129
6;8 Projective Schemes;132
6.1;8.1 Geometric Constructions;132
6.1.1;8.1.1 The Projective Space pnA;132
6.1.1.1;Homogenous coordinates;134
6.1.2;8.1.2 Closed subschemes;136
6.1.3;8.1.3 Projective Morphisms and Projective Schemes ;137
6.1.3.1;Locally Free Sheaves on pn;140
6.1.3.2;Opn (d) as Sheaf of Meromorphic Functions;142
6.1.3.3;The Relative Differentials and the Tangent Bundle of pnS;143
6.1.4;8.1.4 Seperated and Proper Morphisms;145
6.1.5;8.1.5 The Valuative Criteria;147
6.1.5.1;The Valuative Criterion for the Projective Space;147
6.1.6;8.1.6 The Construction Proj(R);148
6.1.6.1;A special case of a finiteness result;150
6.1.7;8.1.7 Ample and Very Ample Sheaves;151
6.2;8.2 Cohomology of Quasicoherent Sheaves;157
6.2.1;8.2.1 Cech cohomology;159
6.2.2;8.2.2 The Künneth-formulae;161
6.2.3;8.2.3 The cohomology of the sheaves Opn (r);162
6.3;8.3 Cohomology of Coherent Sheaves;164
6.3.1;8.3.1 The coherence theorem for proper morphisms;169
6.3.2;Digression: Blowing up and contracting;170
6.4;8.4 Base Change;175
6.4.1;8.4.1 Flat families and intersection numbers;182
6.4.1.1;The Theorem of Bertini;190
6.4.2;8.4.2 The hyperplane section and intersection numbers of line bundles;191
7;9 Curves and the Theorem of Riemann-Roch;194
7.1;9.1 Some basic notions;194
7.2;9.2 The local rings at closed points;196
7.2.1;9.2.1 The structure of OC,p;197
7.2.2;9.2.2 Base change;197
7.3;9.3 Curves and their function fields;199
7.3.1;9.3.1 Ramification and the different ideal;201
7.4;9.4 Line bundles and Divisors;204
7.4.1;9.4.1 Divisors on curves;206
7.4.2;9.4.2 Properties of the degree;208
7.4.2.1;Line bundles on non smooth curves have a degree;208
7.4.2.2;Base change for divisors and line bundles;209
7.4.3;9.4.3 Vector bundles over a curve;209
7.4.3.1;Vector bundles on p1;210
7.5;9.5 The Theorem of Riemann-Roch;212
7.5.1;9.5.1 Differentials and Residues;214
7.5.2;9.5.2 The special case C = p1/k;218
7.5.3;9.5.3 Back to the general case;222
7.5.4;9.5.4 Riemann-Roch for vector bundles and for coherent sheaves.;229
7.5.4.1;The structure of K'(C);231
7.6;9.6 Applications of the Riemann-Roch Theorem;232
7.6.1;9.6.1 Curves of low genus;232
7.6.2;9.6.2 The moduli space;234
7.6.3;9.6.3 Curves of higher genus;245
7.6.3.1;The ”moduli space” of curves of genus g;249
7.7;9.7 The Grothendieck-Riemann-Roch Theorem;250
7.7.1;9.7.1 A special case of the Grothendieck -Riemann-Roch theorem;251
7.7.2;9.7.2 Some geometric considerations;252
7.7.3;9.7.3 The Chow ring;255
7.7.3.1;Base extension of the Chow ring;258
7.7.4;9.7.4 The formulation of the Grothendieck-Riemann-Roch Theorem;260
7.7.5;9.7.5 Some special cases of the Grothendieck-Riemann-Roch-Theorem;263
7.7.6;9.7.6 Back to the case p2 : X = C × C -. C;264
7.7.7;9.7.7 Curves over finite fi
elds.;268
7.7.7.1;Elementary properties of the .-function.;269
7.7.7.2;The Riemann hypothesis.;272
8;10 The Picard functor for curves and their Jacobians;276
8.1;10.1 The construction of the Jacobian;276
8.1.1;10.1.1 Generalities and heuristics :;276
8.1.1.1;Rigidification of PIC;278
8.1.2;10.1.2 General properties of the functor PIC;280
8.1.2.1;The locus of triviality;280
8.1.3;10.1.3 Infinitesimal properties;283
8.1.3.1;Differentiating a line bundle along a vector field;285
8.1.3.2;The theorem of the cube.;285
8.1.4;10.1.4 The basic principles of the construction of the Picard scheme of a curve.;289
8.1.5;10.1.5 Symmetric powers;290
8.1.6;10.1.6 The actual construction of the Picard scheme of a curve.;295
8.1.6.1;The gluing;302
8.1.7;10.1.7 The local representability of PICgC/k;305
8.2;10.2 The Picard functor on X and on J;308
8.2.1;10.2.1 Construction of line bundles on X and on J;308
8.2.1.1;The homomorphisms fM;309
8.2.2;10.2.2 The projectivity of X and J;312
8.2.2.1;The morphisms fM are homomorphisms of functors;313
8.2.3;10.2.3 Maps from the curve C to X, local representability of PICX/k , and the self duality of the Jacobian;314
8.2.4;10.2.4 The self duality of the Jacobian;321
8.2.5;10.2.5 General abelian varieties;322
8.3;10.3 The ring of endomorphisms End(J) and the l -adic modules;325
8.4;10.4 Étale Cohomology;345
8.4.1;10.4.1 Étale cohomology groups;346
8.4.1.1;Galois cohomology;347
8.4.1.2;The geometric étale cohomology groups.;349
8.4.2;10.4.2 Schemes over finite fields;355
8.4.2.1;The global case;357
8.4.2.2;The degenerating family of elliptic curves;361
9;Bibliography;368
10;Index;373