Lectures on Algebraic Geometry II Basic Concepts Coherent Cohomology Curves and their Jacobians 1st Edition by Gunter Harder ISBN 3834826863 9783834826862 by Günter Harder 9783834804327, 3834804320 instant download after payment.

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ISBN 10: 3834826863 
ISBN 13: 9783834826862
Author: Gunter Harder

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.

Lectures on Algebraic Geometry II Basic Concepts Coherent Cohomology Curves and their Jacobians 1st Table of contents:

6 Basic Concepts of the Theory of Schemes
6.1 Affine Schemes
6.1.1 Localization
6.1.2 The Spectrum of a Ring
6.1.3 The Zariski Topology on Spec(A)
6.1.4 The Structure Sheaf on Spec(A)
6.1.5 Quasicoherent Sheaves
6.1.6 Schemes as Locally Ringed Spaces
Closed Subschemes
Sections
A remark
6.2 Schemes
6.2.1 The Definition of a Scheme
The gluing
Closed subschemes again
Annihilators, supports and intersections
6.2.2 Functorial properties
Affine morphisms
Sections again
6.2.3 Construction of Quasi-coherent Sheaves
Vector bundles
Vector Bundles Attached to Locally Free Modules
6.2.4 Vector bundles and GLn-torsors.
6.2.5 Schemes over a base scheme S.
Some notions of finiteness
Fibered products
Base change
6.2.6 Points, T-valued Points and Geometric Points
Closed Points and Geometric Points on varieties
6.2.7 Flat Morphisms
The Concept of Flatness
Representability of functors
6.2.8 Theory of descend
Effectiveness for affine descend data
6.2.9 Galois descend
A geometric interpretation
Descend for general schemes of finite type
6.2.10 Forms of schemes
6.2.11 An outlook to more general concepts
7 Some Commutative Algebra
7.1 Finite A-Algebras
7.1.1 Rings With Finiteness Conditions
7.1.2 Dimension theory for finitely generated k-algebras
7.2 Minimal prime ideals and decomposition into irreducibles
7.2.1 A.ne schemes over k and change of scalars
What is dim(Z1 ∩ Z2)?
7.2.2 Local Irreducibility
The connected component of the identity of an affine group scheme G/k
7.3 Low Dimensional Rings
7.4 Flat morphisms
7.4.1 Finiteness Properties of Tor
7.4.2 Construction of flat families
7.4.3 Dominant morphisms
Birational morphisms
The Artin-Rees Theorem
7.4.4 Formal Schemes and Infinitesimal Schemes
7.5 Smooth Points
7.5.1 Generic Smoothness
The singular locus
7.5.2 Relative Differentials
7.5.3 Examples
7.5.4 Normal schemes and smoothness in codimension one
Regular local rings
7.5.5 Vector fields, derivations and infinitesimal automorphisms
Automorphisms
7.5.6 Group schemes
7.5.7 The groups schemes Ga,Gm and μn
7.5.8 Actions of group schemes
8 Projective Schemes
8.1 Geometric Constructions
8.1.1 The Projective Space pnA
Homogenous coordinates
8.1.2 Closed subschemes
8.1.3 Projective Morphisms and Projective Schemes
Locally Free Sheaves on pn
Opn (d) as Sheaf of Meromorphic Functions
The Relative Differentials and the Tangent Bundle of pnS
8.1.4 Seperated and Proper Morphisms
8.1.5 The Valuative Criteria
The Valuative Criterion for the Projective Space
8.1.6 The Construction Proj(R)
A special case of a finiteness result
8.1.7 Ample and Very Ample Sheaves
8.2 Cohomology of Quasicoherent Sheaves
8.2.1 Čech cohomology
8.2.2 The Künneth-formulae
8.2.3 The cohomology of the sheaves Opn (r)
8.3 Cohomology of Coherent Sheaves
8.3.1 The coherence theorem for proper morphisms
Digression: Blowing up and contracting
8.4 Base Change
8.4.1 Flat families and intersection numbers
The Theorem of Bertini
8.4.2 The hyperplane section and intersection numbers of line bundles
9 Curves and the Theorem of Riemann-Roch
9.1 Some basic notions
9.2 The local rings at closed points
9.2.1 The structure of OC,p
9.2.2 Base change
9.3 Curves and their function fields
9.3.1 Ramification and the different ideal
9.4 Line bundles and Divisors
9.4.1 Divisors on curves
9.4.2 Properties of the degree
Line bundles on non smooth curves have a degree
Base change for divisors and line bundles
9.4.3 Vector bundles over a curve
Vector bundles on p1
9.5 The Theorem of Riemann-Roch
9.5.1 Differentials and Residues
9.5.2 The special case C = p1/k
9.5.3 Back to the general case
9.5.4 Riemann-Roch for vector bundles and for coherent sheaves.
The structure of K'(C)
9.6 Applications of the Riemann-Roch Theorem
9.6.1 Curves of low genus
9.6.2 The moduli space
9.6.3 Curves of higher genus
The ”moduli space” of curves of genus g
9.7 The Grothendieck-Riemann-Roch Theorem
9.7.1 A special case of the Grothendieck -Riemann-Roch theorem
9.7.2 Some geometric considerations
9.7.3 The Chow ring
Base extension of the Chow ring
9.7.4 The formulation of the Grothendieck-Riemann-Roch Theorem
9.7.5 Some special cases of the Grothendieck-Riemann-Roch-Theorem
9.7.6 Back to the case p2 : X = C × C -. C
9.7.7 Curves over finite fi elds.
Elementary properties of the ζ-function.
The Riemann hypothesis.
10 The Picard functor for curves and their Jacobians
10.1 The construction of the Jacobian
10.1.1 Generalities and heuristics :
Rigidification of PIC
10.1.2 General properties of the functor PIC
The locus of triviality
10.1.3 Infinitesimal properties
Differentiating a line bundle along a vector field
The theorem of the cube.
10.1.4 The basic principles of the construction of the Picard scheme of a curve.
10.1.5 Symmetric powers
10.1.6 The actual construction of the Picard scheme of a curve.
The gluing
10.1.7 The local representability of PICgC/k
10.2 The Picard functor on X and on J
10.2.1 Construction of line bundles on X and on J
The homomorphisms φM
10.2.2 The projectivity of X and J
The morphisms φM are homomorphisms of functors
10.2.3 Maps from the curve C to X, local representability of PICX/k , and the self duality of the Ja
10.2.4 The self duality of the Jacobian
10.2.5 General abelian varieties
10.3 The ring of endomorphisms End(J) and the l -adic modules
10.4 Étale Cohomology
10.4.1 Étale cohomology groups
Galois cohomology
The geometric étale cohomology groups.

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