《代數幾何Ⅱ》是2007年1月科學出版社出版的圖書，作者是沙法列維奇。
基本介紹
- 書名：代數幾何Ⅱ
- 作者：沙法列維奇
- ISBN：9787030183002
- 頁數：262
- 定價：50.00元
- 出版社：科學出版社
- 出版時間：2007-1
- 叢書：國外數學名著系列
- 副標題：代數簇的上同調,代數曲面
內容簡介,圖書目錄,
內容簡介
《國外數學名著系列34:代數幾何2代數簇的上同調,代數曲面》由兩部分構成，前半部分著重介紹代數簇的上同調，後半部分討論代數曲面。本書還包含涉及不同主題的大量例子和見解。作者均為該領域的著名專家，他們盡其所能地嚴謹而系統地闡述了這些論題。
圖書目錄
Introduction
Chapter 1.Homological Machinery
1.Origins of Homological Concepts
1.1 The Idea of Homology
1.2 Homology of Triangulated Spaces
1.3 Singular Homology
1.4 Cohomology
1.5 Sheaves
1.6 Cohomology of Sheaves
1.7 Cohomology of Coherent Sheaves
1.8 Cohomology ofEtale Sheaves
2.Complexes
2.1 Exact Sequences
2.2 Complexes
2.3 A Long Exact Sequence
2.4 Filtered Complexes
2.5 Spectral Sequences
2.6 Bicomplexes
2.7 Mapping Cone
2.8 Products
3.Sheaves
3.1 Presheaves
3.2 Sheaves
3.3 Direct and Inverse Images of Sheaves
3.4 Abelian Sheaves
3.5 Flabby Sheaves
4.Cohomology of Sheaves
4.1 Construction ofCohomology
4.2 Hypercohomology
4.3 Higher Direct Images
4.4 Cohomology of a Covering
4.5 The Acyclicity Criterion for Coverings
Chapter 2.Cohomology of Coherent Sheaves
1.Cohomology of Quasi—Coherent Sheaves
1.1 Quasi—Coherent Sheaves
1.2 Serre's Theorem
1.3 The Koszul Complex
1.4 A Theorem on Affine Coverings
1.5 Cohomological Dimension
1.6 Higher Direct Images
1.7 The Kunneth Formula
1.8 Cohomology of Open Inclusions
2.Cohomology of Projective Space
2.1 Sheaves on Pn and Graded Modules
2.2 Applications to Invertible Sheaves
2.3 Applications to Coherent Sheaves
2.4 Regular Sheaves
2.5 The Euler Characteristic
2.6 Relative Case
3.Cohomology of Proper Morphisms
3.1 The Finiteness Theorem
3.2 The Comparison Theorem
3.3 Sketch'of the Proof
3.4 The Theorem on Formal Functions
3.5 Continuous Families of Sheaves
3.6 The Semicontinuity Theorem
3.7 The Lemma on Equivalent Complex
3.8 The Constancy of Euler Characteristic
4.The Riemann—Roch Theorem
4.1 The Riemann—Roch Theorem for Curves
4.2 The General Riemann Problem
4.3 Chern Classes
4.4 The Riemann—Roch—Hirzebruch Theorem
4.5 The Riemann—Roch—Grothendieck Theorem
4.6 Principle of the Proof
5.Duality
5.1 Heuristic Remarks
5.2 Duality for Curves
5.3 The Serre Duality
5.4 The Hodge Index Theorem
5.5 General Duality
5.6 Duality on Cohen—Macaulay Schemes
6.The de Rham Cohomology
6.1 Definition
6.2 A Degeneration Theorem
6.3 Reduction to Finite Fields
6.4 The Finite Field Case
6.5 The Cartier Operators
6.6 Vanishing Theorems
6.7 Properties of the de Rham Cohomology
6，8 Crystalline Cohomology
Chapter 3.Cohomology of Complex Varieties
1.Complex Varieties as Topological Spaces
1.1 Classical Topology
1.2 Properties ofthe Classical Topology
1.3 Singular （Co）homology
1.4 The Borel—Moore Homology
1.5 The Intersection Theory
1.6 The Lefschetz Formula
2.Cohomology of Coherent Sheaves
2.1 The Analytification Functor
2.2 The Comparison Theorem
2.3 Applications to the de Rham Cohomology
2.4 The Weak Lefschetz Theorem
2.5 The Algebraization Theorem
2.6 The Connectedness Theorem
2.7 The Riemann Existence Theorem
2.8 The Exponential Sequence
3.Weights in Cohomology
3.1 Weight Filtration
3.2 Functoriality of Weights
3.3 Assembling and Sorting out
3.4 Smooth Varieties
3.5 Continuity of Weights
3.6 Existence of Weights
4.Algebraic Approach to Classical Topology
4.1 What the Zariski Topology Gives
4.2 Grothendieck's Idea
4.3 Nice Neighborhoods
4.4 Idealized Reconstruction Procedure
4.5 Algebraic Coverings
4.6 Instructive Example
Chapter 4.Etale Cohomology
1.The Weil Conjectures
1.1 Finite Fields
1.2 Equations over Finite Fields
1.3 Zeta Functions
1.4 Weil's Theorem
1.5 Proof of Weil's Theorem
1.6 The Weil Conjectures
1.7 Weil's Cohomology
2.Algebraic Fundamental Group
2.1 Etale Morphisms
2.2 Etale Coverings
2.3 Algebraic Fundamental Group
2.4 Functorial Properties of the Fundamental Group
2.5 Construction of Coverings
3.Etale Topology
3.1 Etale Presheaves
3.2 Etale Sheaves
3.3 Category of Sheaves
3.4 Stalk of Sheaf at a Point
3.5 Etale Localization
4.Cohomology of Etale Sheaves
4.1 Abelian Sheaves
4.2 Cohomology
4.3 Galois Cohomology
4.4 Cohomology of Coherent Sheaves
4.5 Torsors
4.6 The Kummer Theory
4.7 Acyclicity of Finite Morphisms
5.Cohomology of Algebraic Curves
5.1 Outline of Strategy
5.2 Tsen's Theorem
5.3 Cohomology of
5.4 Cohomology of Complete Curves
5.5 Duality on Complete Curves
5.6 Open Curves
6.Fundamental Theorems
6.1 Constructible Sheaves
6.2 The Base Change Theorem
6.3 Cohomology with Compact Support
6.4 Finiteness Theorem
6.5 Comparison with the Classical Cohomology
6.6 Specialization and Vanishing Cycles
6.7 Acyclicity of Smooth Morphisms
6.8 Etale Monodromy
7.l—Adic Cohomology
7.1 l—Adic Sheaves
7.2 Finiteness
7.3 The Kunneth Formula
7.4 Poincare Duality： Orientation
7.5 Poincare Duality： Pairing
7.6 The Gysin Homomorphism
7.7 The Weak Lefschetz Theorem
7.8 The Lefschetz Trace Formula
7.9 Applications to the Zeta Function
7.10 L—Functions
8.Deligne's Theorem
8.1 Weights
8.2 Main Theorem
8.3 Outline of Proof
8.4 Geometric Applications
8.5 The Hard Lefschetz Theorem
8.6 Theorem on Invariant Subspace
8.7 Tate's Conjecture
Bibliography
References
Chapter 1.Homological Machinery
1.Origins of Homological Concepts
1.1 The Idea of Homology
1.2 Homology of Triangulated Spaces
1.3 Singular Homology
1.4 Cohomology
1.5 Sheaves
1.6 Cohomology of Sheaves
1.7 Cohomology of Coherent Sheaves
1.8 Cohomology ofEtale Sheaves
2.Complexes
2.1 Exact Sequences
2.2 Complexes
2.3 A Long Exact Sequence
2.4 Filtered Complexes
2.5 Spectral Sequences
2.6 Bicomplexes
2.7 Mapping Cone
2.8 Products
3.Sheaves
3.1 Presheaves
3.2 Sheaves
3.3 Direct and Inverse Images of Sheaves
3.4 Abelian Sheaves
3.5 Flabby Sheaves
4.Cohomology of Sheaves
4.1 Construction ofCohomology
4.2 Hypercohomology
4.3 Higher Direct Images
4.4 Cohomology of a Covering
4.5 The Acyclicity Criterion for Coverings
Chapter 2.Cohomology of Coherent Sheaves
1.Cohomology of Quasi—Coherent Sheaves
1.1 Quasi—Coherent Sheaves
1.2 Serre's Theorem
1.3 The Koszul Complex
1.4 A Theorem on Affine Coverings
1.5 Cohomological Dimension
1.6 Higher Direct Images
1.7 The Kunneth Formula
1.8 Cohomology of Open Inclusions
2.Cohomology of Projective Space
2.1 Sheaves on Pn and Graded Modules
2.2 Applications to Invertible Sheaves
2.3 Applications to Coherent Sheaves
2.4 Regular Sheaves
2.5 The Euler Characteristic
2.6 Relative Case
3.Cohomology of Proper Morphisms
3.1 The Finiteness Theorem
3.2 The Comparison Theorem
3.3 Sketch'of the Proof
3.4 The Theorem on Formal Functions
3.5 Continuous Families of Sheaves
3.6 The Semicontinuity Theorem
3.7 The Lemma on Equivalent Complex
3.8 The Constancy of Euler Characteristic
4.The Riemann—Roch Theorem
4.1 The Riemann—Roch Theorem for Curves
4.2 The General Riemann Problem
4.3 Chern Classes
4.4 The Riemann—Roch—Hirzebruch Theorem
4.5 The Riemann—Roch—Grothendieck Theorem
4.6 Principle of the Proof
5.Duality
5.1 Heuristic Remarks
5.2 Duality for Curves
5.3 The Serre Duality
5.4 The Hodge Index Theorem
5.5 General Duality
5.6 Duality on Cohen—Macaulay Schemes
6.The de Rham Cohomology
6.1 Definition
6.2 A Degeneration Theorem
6.3 Reduction to Finite Fields
6.4 The Finite Field Case
6.5 The Cartier Operators
6.6 Vanishing Theorems
6.7 Properties of the de Rham Cohomology
6，8 Crystalline Cohomology
Chapter 3.Cohomology of Complex Varieties
1.Complex Varieties as Topological Spaces
1.1 Classical Topology
1.2 Properties ofthe Classical Topology
1.3 Singular （Co）homology
1.4 The Borel—Moore Homology
1.5 The Intersection Theory
1.6 The Lefschetz Formula
2.Cohomology of Coherent Sheaves
2.1 The Analytification Functor
2.2 The Comparison Theorem
2.3 Applications to the de Rham Cohomology
2.4 The Weak Lefschetz Theorem
2.5 The Algebraization Theorem
2.6 The Connectedness Theorem
2.7 The Riemann Existence Theorem
2.8 The Exponential Sequence
3.Weights in Cohomology
3.1 Weight Filtration
3.2 Functoriality of Weights
3.3 Assembling and Sorting out
3.4 Smooth Varieties
3.5 Continuity of Weights
3.6 Existence of Weights
4.Algebraic Approach to Classical Topology
4.1 What the Zariski Topology Gives
4.2 Grothendieck's Idea
4.3 Nice Neighborhoods
4.4 Idealized Reconstruction Procedure
4.5 Algebraic Coverings
4.6 Instructive Example
Chapter 4.Etale Cohomology
1.The Weil Conjectures
1.1 Finite Fields
1.2 Equations over Finite Fields
1.3 Zeta Functions
1.4 Weil's Theorem
1.5 Proof of Weil's Theorem
1.6 The Weil Conjectures
1.7 Weil's Cohomology
2.Algebraic Fundamental Group
2.1 Etale Morphisms
2.2 Etale Coverings
2.3 Algebraic Fundamental Group
2.4 Functorial Properties of the Fundamental Group
2.5 Construction of Coverings
3.Etale Topology
3.1 Etale Presheaves
3.2 Etale Sheaves
3.3 Category of Sheaves
3.4 Stalk of Sheaf at a Point
3.5 Etale Localization
4.Cohomology of Etale Sheaves
4.1 Abelian Sheaves
4.2 Cohomology
4.3 Galois Cohomology
4.4 Cohomology of Coherent Sheaves
4.5 Torsors
4.6 The Kummer Theory
4.7 Acyclicity of Finite Morphisms
5.Cohomology of Algebraic Curves
5.1 Outline of Strategy
5.2 Tsen's Theorem
5.3 Cohomology of
5.4 Cohomology of Complete Curves
5.5 Duality on Complete Curves
5.6 Open Curves
6.Fundamental Theorems
6.1 Constructible Sheaves
6.2 The Base Change Theorem
6.3 Cohomology with Compact Support
6.4 Finiteness Theorem
6.5 Comparison with the Classical Cohomology
6.6 Specialization and Vanishing Cycles
6.7 Acyclicity of Smooth Morphisms
6.8 Etale Monodromy
7.l—Adic Cohomology
7.1 l—Adic Sheaves
7.2 Finiteness
7.3 The Kunneth Formula
7.4 Poincare Duality： Orientation
7.5 Poincare Duality： Pairing
7.6 The Gysin Homomorphism
7.7 The Weak Lefschetz Theorem
7.8 The Lefschetz Trace Formula
7.9 Applications to the Zeta Function
7.10 L—Functions
8.Deligne's Theorem
8.1 Weights
8.2 Main Theorem
8.3 Outline of Proof
8.4 Geometric Applications
8.5 The Hard Lefschetz Theorem
8.6 Theorem on Invariant Subspace
8.7 Tate's Conjecture
Bibliography
References
