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An accessible, motivated introduction to one of the most dynamic areas of mathematics
Decades ago, Mumford wrote that algebraic geometry “seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics.” The revolution has now fully come to pass and has fundamentally changed how we think about many fields of mathematics. This book provides a thorough foundation in the powerful ideas that now shape the landscape, with an informal yet rigorous exposition that builds intuition for understanding the formidable machinery. It begins with a discussion of categorical thinking and sheaves and then develops the notion of schemes and varieties as examples of “geometric spaces” before discussing their specific aspects. The book goes on to cover topics such as dimension and smoothness, vector bundles and their natural generalizations, and important cohomological tools and their applications. Important optional topics are included in starred sections.
- Provides a comprehensive introduction certain to become the standard on the subject
- Features a wealth of exercises that enable students to learn by doing
- Requires few prerequisites, developing the tools students need to succeed, from category theory and sheaf theory to commutative and homological algebra
- Uses an example-driven approach that builds mathematical intuition
- Is a self-contained textbook for graduate students and an essential reference for researchers
An accessible, motivated introduction to one of the most dynamic areas of mathematics
Decades ago, Mumford wrote that algebraic geometry “seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics.” The revolution has now fully come to pass and has fundamentally changed how we think about many fields of mathematics. This book provides a thorough foundation in the powerful ideas that now shape the landscape, with an informal yet rigorous exposition that builds intuition for understanding the formidable machinery. It begins with a discussion of categorical thinking and sheaves and then develops the notion of schemes and varieties as examples of “geometric spaces” before discussing their specific aspects. The book goes on to cover topics such as dimension and smoothness, vector bundles and their natural generalizations, and important cohomological tools and their applications. Important optional topics are included in starred sections.
- Provides a comprehensive introduction certain to become the standard on the subject
- Features a wealth of exercises that enable students to learn by doing
- Requires few prerequisites, developing the tools students need to succeed, from category theory and sheaf theory to commutative and homological algebra
- Uses an example-driven approach that builds mathematical intuition
- Is a self-contained textbook for graduate students and an essential reference for researchers
- Preface
- 0.1 For the Reader
- 0.2 For the Expert
- 0.3 Background and Conventions
- 0.4** The Goals of This Book
- Part I Preliminaries
- 1 Just Enough Category Theory to Be Dangerous
- 1.1 Categories and Functors
- 1.2 Universal Properties Determine an Object up to Unique Isomorphism
- 1.3 Limits and Colimits
- 1.4 Adjoints
- 1.5 An Introduction to Abelian Categories
- 1.6* Spectral Sequences
- 2 Sheaves
- 2.1 Motivating Example: The Sheaf of Smooth Functions
- 2.2 Denition of Sheaf and Presheaf
- 2.3 Morphisms of Presheaves and Sheaves
- 2.4 Properties Determined at the Level of Stalks, and Sheäcation
- 2.5 Recovering Sheaves from a “Sheaf on a Base”
- 2.6 Sheaves of Abelian Groups, and X-Modules, Form Abelian Categories
- 2.7 The Inverse Image Sheaf
- Part II Schemes
- 3 Toward Ane Schemes: The Underlying Set, and Topological Space
- 3.1 Toward Schemes
- 3.2 The Underlying Set of an Ane Scheme
- 3.3 Visualizing Schemes: Generic Points
- 3.4 The Underlying Topological Space of an Ane Scheme
- 3.5 A Base of the Zariski Topology on SpecA: Distinguished Open Sets
- 3.6 Topological (and Noetherian) Properties
- 3.7 The Function I(⋅), Taking Subsets of SpecA to Ideals of A
- 4 The Structure Sheaf, and the Denition of Schemes in General
- 4.1 The Structure Sheaf of an Ane Scheme
- 4.2 Visualizing Schemes: Nilpotents
- 4.3 Denition of Schemes
- 4.4 Three Examples
- 4.5 Projective Schemes, and the Proj Construction
- 5 Some Properties of Schemes
- 5.1 Topological Properties
- 5.2 Reducedness and Integrality
- 5.3 The Ane Communication Lemma, and Properties of Schemes That Can Be Checked “Ane-Locally”
- 5.4 Normality and Factoriality
- 6 Rings Are to Modules as Schemes Are to …
- 6.1 Quasicoherent Sheaves
- 6.2 Characterizing Quasicoherence Using the Distinguished Ane Base
- 6.3 Quasicoherent Sheaves Form an Abelian Category
- 6.4 Finite Type Quasicoherent, Finitely Presented, and Coherent Sheaves
- 6.5 Algebraic Interlude: The Jordan–Hölder Package
- 6.6 Visualizing Schemes: Associated Points and Zerodivisors
- 6.7** Coherent Modules over Non-Noetherian Rings
- Part III Morphisms of Schemes
- 7 Morphisms of Schemes
- 7.1 Motivations for the “Right” Denition of Morphism of Schemes
- 7.2 Morphisms of Ringed Spaces
- 7.3 From Locally Ringed Spaces to Morphisms of Schemes
- 7.4 Maps of Graded Rings and Maps of Projective Schemes
- 7.5 Rational Maps from Reduced Schemes
- 7.6* Representable Functors and Group Schemes
- 7.7** The Grassmannian: First Construction
- 8 Useful Classes of Morphisms of Schemes
- 8.1 “Reasonable” Classes of Morphisms (Such as Open Embeddings)
- 8.2 Another Algebraic Interlude: Lying Over and Nakayama
- 8.3 A Gazillion Finiteness Conditions on Morphisms
- 8.4 Images of Morphisms: Chevalley’s Theorem and Elimination Theory
- 9 Closed Embeddings and Related Notions
- 9.1 Closed Embeddings and Closed Subschemes
- 9.2 Locally Closed Embeddings and Locally Closed Subschemes
- 9.3 Important Examples from Projective Geometry
- 9.4 The (Closed Sub)scheme-Theoretic Image
- 9.5 Slicing by Eective Cartier Divisors, Regular Sequences and Regular Embeddings
- 10 Fibered Products of Schemes, and Base Change
- 10.1 They Exist
- 10.2 Computing Fibered Products in Practice
- 10.3 Interpretations: Pulling Back Families, and Fibers of Morphisms
- 10.4 Properties Preserved by Base Change
- 10.5* Properties Not Preserved by Base Change, and How to Fix Them
- 10.6 Products of Projective Schemes: The Segre Embedding
- 10.7 Normalization
- 11 Separated and Proper Morphisms, and (Finally!) Varieties
- 11.1 Fun with Diagonal Morphisms, and Quasiseparatedness Made Easy
- 11.2 Separatedness, and Varieties
- 11.3 The Locus where Two Morphisms from X to Y Agree, and the “Reduced-to-Separated” Theorem
- 11.4 Proper Morphisms
- Part IV “Geometric” Properties of Schemes
- 12 Dimension
- 12.1 Dimension and Codimension
- 12.2 Dimension, Transcendence Degree, and Noether Normalization
- 12.3 Krull’s Theorems
- 12.4 Dimensions of Fibers of Morphisms of Varieties
- 13 Regularity and Smoothness
- 13.1 The Zariski Tangent Space
- 13.2 Regularity, and Smoothness over a Field
- 13.3 Examples
- 13.4 Bertini’s Theorem
- 13.5 Discrete Valuation Rings, and Algebraic Hartogs’s Lemma
- 13.6 Smooth (and Étale) Morphisms: First Denition
- 13.7* Valuative Criteria for Separatedness and Properness
- 13.8* More Sophisticated Facts about Regular Local Rings
- 13.9* Filtered Rings and Modules, and the Artin-Rees Lemma
- Part V Quasicoherent Sheaves on Schemes, and Their Uses
- 14 More on Quasicoherent and Coherent Sheaves
- 14.1 Vector Bundles “=” Locally Free Sheaves
- 14.2 Locally Free Sheaves on Schemes in Particular
- 14.3 More Pleasant Properties of Finite Type and Coherent Sheaves
- 14.4 Pushforwards of Quasicoherent Sheaves
- 14.5 Pullbacks of Quasicoherent Sheaves: Three Dierent Perspectives
- 14.6 The Quasicoherent Sheaf Corresponding to a Graded Module
- 15 Line Bundles, Maps to Projective Space, and Divisors
- 15.1 Some Line Bundles on Projective Space
- 15.2 Line Bundles and Maps to Projective Space
- 15.3 The Curve-to-Projective Extension Theorem
- 15.4 Hard but Important: Line Bundles and Weil Divisors
- 15.5 The Payo: Many Fun Examples
- 15.6 Eective Cartier Divisors “=” Invertible Ideal Sheaves
- 15.7 The Graded Module Corresponding to a Quasicoherent Sheaf
- 16 Maps to Projective Space, and Properties of Line Bundles
- 16.1 Globally Generated Quasicoherent Sheaves
- 16.2 Ample and Very Ample Line Bundles
- 16.3 Applications to Curves
- 16.4* The Grassmannian as a Moduli Space
- 17 Projective Morphisms, and Relative Versions of Spec and Proj
- 17.1 Relative Spec of a (Quasicoherent) Sheaf of Algebras
- 17.2 Relative Proj of a (Quasicoherent) Sheaf of Graded Algebras
- 17.3 Projective Morphisms
- 18 ech Cohomology of Quasicoherent Sheaves
- 18.1 (Desired) Properties of Cohomology
- 18.2 Denitions and Proofs of Key Properties
- 18.3 Cohomology of Line Bundles on Projective Space
- 18.4 Riemann–Roch, and Arithmetic Genus
- 18.5 A First Glimpse of Serre Duality
- 18.6 Hilbert Functions, Hilbert Polynomials, and Genus
- 18.7 Higher Pushforward (or Direct Image) Sheaves
- 18.8* Serre’s Characterizations of Ampleness and Aneness
- 18.9* From Projective to Proper Hypotheses: Chow’s Lemma and Grothendieck’s Coherence Theorem
- 19 Application: Curves
- 19.1 A Criterion for a Morphism to Be a Closed Embedding
- 19.2 A Series of Crucial Tools
- 19.3 Curves of Genus 0
- 19.4 Classical Geometry Arising from Curves of Positive Genus
- 19.5 Hyperelliptic Curves
- 19.6 Curves of Genus 2
- 19.7 Curves of Genus 3
- 19.8 Curves of Genus 4 and 5
- 19.9 Curves of Genus 1
- 19.10 Elliptic Curves Are Group Varieties
- 19.11 Counterexamples and Pathologies Using Elliptic Curves
- 20* Application: A Glimpse of Intersection Theory
- 20.1 Intersecting n Line Bundles with an n-Dimensional Variety
- 20.2 Intersection Theory on a Surface
- 20.3 The Grothendieck Group of Coherent Sheaves, and an Algebraic Version of Homology
- 20.4** The Nakai–Moishezon and Kleiman Criteria for Ampleness
- 21 Dierentials
- 21.1 Motivation and Game Plan
- 21.2 Denitions and First Properties
- 21.3 Examples
- 21.4 The Riemann–Hurwitz Formula
- 21.5 Understanding Smooth Varieties Using Their Cotangent Bundles
- 21.6 Generic Smoothness, and Consequences
- 21.7 Unramied Morphisms
- 22* Blowing Up
- 22.1 Motivating Example: Blowing Up the Origin in the Plane
- 22.2 Blowing Up, by Universal Property
- 22.3 The Blow-up Exists, and Is Projective
- 22.4 Examples and Computations
- Part VI More Cohomological Tools
- 23 Derived Functors
- 23.1 The Tor Functors
- 23.2 Derived Functors in General
- 23.3 Derived Functors and Spectral Sequences
- 23.4 Derived Functor Cohomology of -Modules
- 23.5 ech Cohomology and Derived Functor Cohomology Agree
- 24 Flatness
- 24.1 Easier Facts
- 24.2 Flatness through Tor
- 24.3 Ideal-Theoretic Criteria for Flatness
- 24.4** Aside: The Koszul Complex and the Hilbert Syzygy Theorem
- 24.5 Topological Implications of Flatness
- 24.6 Local Criteria for Flatness
- 24.7 Flatness Implies Constant Euler Characteristic
- 24.8 Smooth and Étale Morphisms, and Flatness
- 25 Cohomology and Base Change Theorems
- 25.1 Statements and Applications
- 25.2 Proofs of Cohomology and Base Change Theorems
- 25.3 Applying Cohomology and Base Change to Moduli Problems
- 26 Depth and Cohen–Macaulayness
- 26.1 Depth
- 26.2 Cohen–Macaulay Rings and Schemes
- 26.3 Serre’s R1 + S2 Criterion for Normality
- 27 The Twenty-Seven Lines on a Cubic Surface
- 27.1 Preliminary Facts
- 27.2 Every...
| Erscheinungsjahr: | 2025 |
|---|---|
| Fachbereich: | Geometrie |
| Genre: | Importe, Mathematik |
| Rubrik: | Naturwissenschaften & Technik |
| Medium: | Taschenbuch |
| Inhalt: | Einband - flex.(Paperback) |
| ISBN-13: | 9780691268675 |
| ISBN-10: | 0691268673 |
| Sprache: | Englisch |
| Einband: | Kartoniert / Broschiert |
| Autor: | Vakil, Ravi |
| Hersteller: | Princeton University Press |
| Verantwortliche Person für die EU: | Libri GmbH, Europaallee 1, D-36244 Bad Hersfeld, gpsr@libri.de |
| Maße: | 257 x 181 x 43 mm |
| Von/Mit: | Ravi Vakil |
| Erscheinungsdatum: | 16.12.2025 |
| Gewicht: | 1,384 kg |
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