Lectures on Algebraic Geometry I : Sheaves, Cohomology of Sheaves, and Applications to Riemann Surfaces
Geometry Algebra
Imprint: Springer Spektrum
2011
2nd ed.
EISBN 1283369656
Preface; Contents; Introduction; 1 Categories, Products, Projective and Inductive Limits; 1.1 The Notion of a Category and Examples; 1.2 Functors; 1.3 Products, Projective Limits and Direct Limits in a Category; 1.3.1 The Projective Limit; 1.3.2 The Yoneda Lemma; 1.3.3 Examples; 1.3.4 Representable Functors; 1.3.5 Direct Limits; 1.4 Exercises; 2 Basic Concepts of Homological Algebra; 2.1 The Category ModG of G-modules; 2.2 More Functors; 2.2.1 Invariants, Coinvariants and Exactness; 2.2.2 The First Cohomology Group; 2.2.3 Some Notation; 2.2.4 Exercises; 2.3 The Derived Functors
2.3.1 The Simple Principle2.3.2 Functoriality; 2.3.3 Other Resolutions; 2.3.4 Injective Resolutions of Short Exact Sequences; A Fundamental Remark; The Cohomology and the Long Exact Sequence; The Homology of Groups; 2.4 The Functors Ext and Tor; 2.4.1 The Functor Ext; 2.4.2 The Derived Functor for the Tensor Product; 2.4.3 Exercise; 3 Sheaves; 3.1 Presheaves and Sheaves; 3.1.1 What is a Presheaf ?; 3.1.2 A Remark about Products and Presheaf; 3.1.3 What is a Sheaf ?; 3.1.4 Examples; 3.2 Manifolds as Locally Ringed Spaces; 3.2.1 What Are Manifolds?; 3.2.2 Examples and Exercise
3.3 Stalks and Sheafification3.3.1 Stalks; 3.3.2 The Process of Sheafification of a Presheaf; 3.4 The Functors f* and f *; 3.4.1 The Adjunction Formula; 3.4.2 Extensions and Restrictions; 3.5 Constructions of Sheaves; 4 Cohomology of Sheaves; 4.1 Examples; 4.1.1 Sheaves on Riemann surfaces; 4.1.2 Cohomology of the Circle; 4.2 The Derived Functor; 4.2.1 Injective Sheaves and Derived Functors; 4.2.2 A Direct Definition of H1; 4.3 Fiber Bundles and Non Abelian H1; 4.3.1 Fibrations; Fibre Bundle; Vector Bundles; 4.3.2 Non-Abelian H1; 4.3.3 The Reduction of the Structure Group; Orientation
Local SystemsIsomorphism Classes of Local Systems; Principal G-bundels; 4.4 Fundamental Properties of the Cohomology of Sheaves; 4.4.1 Introduction; 4.4.2 The Derived Functor to f*; 4.4.3 Functorial Properties of the Cohomology; 4.4.4 Paracompact Spaces; 4.4.5 Applications; Cohomology of Spheres; Orientations; Compact Oriented Surfaces; 4.5 Cech Cohomology of Sheaves; 4.5.1 The Cech-Complex; 4.5.2 The Cech Resolution of a Sheaf; 4.6 Spectral Sequences; 4.6.1 Introduction; 4.6.2 The Vertical Filtration; 4.6.3 The Horizontal Filtration; Two Special Cases; Applications of Spectral Sequences
4.6.4 The Derived CategoryThe Composition Rule; Exact Triangles; 4.6.5 The Spectral Sequence of a Fibration; Sphere Bundles an Euler Characteristic; 4.6.6 Cech Complexes and the Spectral Sequence; A Criterion for Degeneration; An Application to Product Spaces; 4.6.7 The Cup Product; 4.6.8 Example: Cup Product for the Comology of Tori; A Connection to the Cohomology of Groups; 4.6.9 An Excursion into Homotopy Theory; 4.7 Cohomology with Compact Supports; 4.7.1 The Definition; 4.7.2 An Example for Cohomology with Compact Supports; The Cohomology with Compact Supports for Open Balls
Formulae for Cup Products
This book and the following second volume is an introduction into modern algebraic geometry. In the first volume the methods of homological algebra, theory of sheaves, and sheaf cohomology are developed. These methods are indispensable for modern algebraic geometry, but they are also fundamental for other branches of mathematics and of great interest in their own. In the last chapter of volume I these concepts are applied to the theory of compact Riemann surfaces. In this chapter the author makes clear how influential the ideas of Abel, Riemann and Jacobi were and that many of the modern methods have been anticipated by them.
2.3.1 The Simple Principle2.3.2 Functoriality; 2.3.3 Other Resolutions; 2.3.4 Injective Resolutions of Short Exact Sequences; A Fundamental Remark; The Cohomology and the Long Exact Sequence; The Homology of Groups; 2.4 The Functors Ext and Tor; 2.4.1 The Functor Ext; 2.4.2 The Derived Functor for the Tensor Product; 2.4.3 Exercise; 3 Sheaves; 3.1 Presheaves and Sheaves; 3.1.1 What is a Presheaf ?; 3.1.2 A Remark about Products and Presheaf; 3.1.3 What is a Sheaf ?; 3.1.4 Examples; 3.2 Manifolds as Locally Ringed Spaces; 3.2.1 What Are Manifolds?; 3.2.2 Examples and Exercise
3.3 Stalks and Sheafification3.3.1 Stalks; 3.3.2 The Process of Sheafification of a Presheaf; 3.4 The Functors f* and f *; 3.4.1 The Adjunction Formula; 3.4.2 Extensions and Restrictions; 3.5 Constructions of Sheaves; 4 Cohomology of Sheaves; 4.1 Examples; 4.1.1 Sheaves on Riemann surfaces; 4.1.2 Cohomology of the Circle; 4.2 The Derived Functor; 4.2.1 Injective Sheaves and Derived Functors; 4.2.2 A Direct Definition of H1; 4.3 Fiber Bundles and Non Abelian H1; 4.3.1 Fibrations; Fibre Bundle; Vector Bundles; 4.3.2 Non-Abelian H1; 4.3.3 The Reduction of the Structure Group; Orientation
Local SystemsIsomorphism Classes of Local Systems; Principal G-bundels; 4.4 Fundamental Properties of the Cohomology of Sheaves; 4.4.1 Introduction; 4.4.2 The Derived Functor to f*; 4.4.3 Functorial Properties of the Cohomology; 4.4.4 Paracompact Spaces; 4.4.5 Applications; Cohomology of Spheres; Orientations; Compact Oriented Surfaces; 4.5 Cech Cohomology of Sheaves; 4.5.1 The Cech-Complex; 4.5.2 The Cech Resolution of a Sheaf; 4.6 Spectral Sequences; 4.6.1 Introduction; 4.6.2 The Vertical Filtration; 4.6.3 The Horizontal Filtration; Two Special Cases; Applications of Spectral Sequences
4.6.4 The Derived CategoryThe Composition Rule; Exact Triangles; 4.6.5 The Spectral Sequence of a Fibration; Sphere Bundles an Euler Characteristic; 4.6.6 Cech Complexes and the Spectral Sequence; A Criterion for Degeneration; An Application to Product Spaces; 4.6.7 The Cup Product; 4.6.8 Example: Cup Product for the Comology of Tori; A Connection to the Cohomology of Groups; 4.6.9 An Excursion into Homotopy Theory; 4.7 Cohomology with Compact Supports; 4.7.1 The Definition; 4.7.2 An Example for Cohomology with Compact Supports; The Cohomology with Compact Supports for Open Balls
Formulae for Cup Products
This book and the following second volume is an introduction into modern algebraic geometry. In the first volume the methods of homological algebra, theory of sheaves, and sheaf cohomology are developed. These methods are indispensable for modern algebraic geometry, but they are also fundamental for other branches of mathematics and of great interest in their own. In the last chapter of volume I these concepts are applied to the theory of compact Riemann surfaces. In this chapter the author makes clear how influential the ideas of Abel, Riemann and Jacobi were and that many of the modern methods have been anticipated by them.
