Autorentext

Ravi Vakil

Klappentext

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

Inhalt

• 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 Preliminaries1 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 Sequences2 Sheaves

• 2.1 Motivating Example: The Sheaf of Smooth Functions

• 2.2 De nition of Sheaf and Presheaf

• 2.3 Morphisms of Presheaves and Sheaves

• 2.4 Properties Determined at the Level of Stalks, and Sheä cation

• 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 Schemes3 Toward A ne Schemes: The Underlying Set, and Topological Space

• 3.1 Toward Schemes

• 3.2 The Underlying Set of an A ne Scheme

• 3.3 Visualizing Schemes: Generic Points

• 3.4 The Underlying Topological Space of an A ne 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 A4 The Structure Sheaf, and the De nition of Schemes in General

• 4.1 The Structure Sheaf of an A ne Scheme

• 4.2 Visualizing Schemes: Nilpotents

• 4.3 De nition of Schemes

• 4.4 Three Examples

• 4.5 Projective Schemes, and the Proj Construction5 Some Properties of Schemes

• 5.1 Topological Properties

• 5.2 Reducedness and Integrality

• 5.3 The A ne Communication Lemma, and Properties of Schemes That Can Be Checked “A ne-Locally”

• 5.4 Normality and Factoriality6 Rings Are to Modules as Schemes Are to …

• 6.1 Quasicoherent Sheaves

• 6.2 Characterizing Quasicoherence Using the Distinguished A ne 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 Schemes7 Morphisms of Schemes

• 7.1 Motivations for the “Right” De nition 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 Construction8 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 Theory9 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 E ective Cartier Divisors, Regular Sequences and Regular Embeddings10 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 Normalization11 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 Schemes12 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 Varieties13 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 De nition

• 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 Uses14 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 Di erent Perspectives

• 14.6 The Quasicoherent Sheaf Corresponding to a Graded Module15 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 E ective Cartier Divisors “=” Invertible Ideal Sheaves

• 15.7 The Graded Module Corresponding to a Quasicoherent Sheaf16 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 Space17 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 Morphisms18 ech Cohomology of Quasicoherent Sheaves

• 18.1 (Desired) Properties of Cohomology

• 18.2 De nitions and Proofs of Key Properties

• 18.3 Cohomology of Line Bundles on Projective Space

• 18.4 Riemann–Roch, a…