| Description |
1 online resource |
| Contents |
Intro -- Preface -- Introduction -- Contents -- 1 Beginning Concepts -- 1 The Definition of Algebraic Varieties -- 1.1 Affine Algebraic Sets -- 1.1.1 Complex Numbers -- 1.1.2 Nullstellensatz -- 1.2 Zariski Topology -- 1.2.1 Topology -- 1.2.2 Zariski and Analytic Topology -- 1.3 Affine and Projective Varieties -- 1.3.1 Affine and Quasi-Affine Varieties -- 1.3.2 Projective and Quasi-Projective Varieties -- 1.4 Regular Functions on Different Types of Varieties -- 1.4.1 Regular Functions on a Quasiaffine and Quasiprojective Variety -- 1.4.2 Regular Functions on AnC |
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1.4.3 Regular Functions on an Affine Variety -- 1.4.4 Regular Functions on the Complement of a Set of Zeros Z(f) -- 1.4.5 Regular Functions on a Quasiaffine Variety -- 1.4.6 Regular Functions on a Projective Variety -- 1.5 Sheaves -- 2 Categories, and the Category of Algebraic Varieties -- 2.1 Categories, Functors and Algebraic Structures -- 2.1.1 The Definition of a Category, and an Example: The Category of Sets -- 2.1.2 Categories of Algebraic Structures -- 2.1.3 Isomorphisms -- 2.1.4 Functors and Natural Transformations -- 2.2 Categories: Topological Spaces and Algebraic Varieties |
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2.2.1 The Category of Topological Spaces -- 2.2.2 The Category of Algebraic Varieties -- 2.3 The Morphisms into an Affine Variety -- 2.3.1 Quasiaffine Varieties which are not Isomorphic to Affine Varieties -- 2.3.2 Quasiaffine Varieties which are Isomorphic to Affine Varieties -- 3 Rational Maps, Smooth Maps and Dimension -- 3.1 Definition of a Rational Map -- 3.2 The Field of Rational Functions -- 3.2.1 The Category of Varieties and Dominant Rational Maps -- 3.3 Standard Smooth Homomorphisms of Commutative Rings -- 3.4 Smooth Morphisms of Varieties -- 3.4.1 Smooth Varieties |
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3.5 Regular Rings and Dimension -- 3.5.1 Smooth Varieties and Regular Rings -- 4 Computing with Polynomials -- 4.1 Divisibility of Polynomials -- 4.2 Gröbner Basis -- 4.3 Nullstellensatz -- 5 Introduction to Commutative Algebra -- 5.1 Primary Decomposition -- 5.2 Artinian Rings -- 5.3 Dimension -- 5.4 Regular Local Rings -- 5.5 Dimension of Affine Varieties -- 6 Exercises -- 2 Schemes -- 1 Sheaves and Schemes -- 1.1 Sheaves Revisited -- 1.2 Ringed Spaces and Locally Ringed Spaces -- 1.3 Schemes -- 1.4 Category Theory Revisited: Adjoints, Limits and Colimits, Universality |
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1.5 Abelian Categories: Abelian Sheaves -- 2 Beginning Properties and Examples of Schemes -- 2.1 Connected, Irreducible, Reduced and Integral Schemes -- 2.2 Properties of Affine Schemes -- 2.3 Open and Closed Subschemes -- 2.4 Limits of Schemes -- 2.5 Gluing of Schemes: Colimits -- 2.6 A Diagram of Schemes Which Does Not Have a Limit -- 2.7 Proj Schemes -- 2.8 The Affine and Projective Space Over a Scheme: Projective Schemes -- 3 Finiteness Properties of Schemes and Morphisms of Schemes -- 3.1 Quasicompactness -- 3.2 Noetherian Schemes |
| Summary |
The goal of this book is to provide an introduction to algebraic geometry accessible to students. Starting from solutions of polynomial equations, modern tools of the subject soon appear, motivated by how they improve our understanding of geometrical concepts. In many places, analogies and differences with related mathematical areas are explained. The text approaches foundations of algebraic geometry in a complete and self-contained way, also covering the underlying algebra. The last two chapters include a comprehensive treatment of cohomology and discuss some of its applications in algebraic geometry |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Online resource; title from PDF title page (SpringerLink, viewed April 2, 2021) |
| Subject |
Geometry, Algebraic.
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Geometría algebraica
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Geometry, Algebraic
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| Form |
Electronic book
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| Author |
Kriz, Sophie, author
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| ISBN |
9783030626440 |
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303062644X |
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