Ravi Vakil is the Robert Grimmett Professor of Mathematics at Stanford University and president of the American Mathematical Society. He is the author of A Mathematical Mosaic: Patterns and Problem Solving.

"Winner of the PROSE Award in Mathematics and Statistics, Association of American Publishers"

• Preface0.1 For the Reader0.2 For the Expert0.3 Background and Conventions0.4** The Goals of This BookPart I Preliminaries1 Just Enough Category Theory to Be Dangerous1.1 Categories and Functors1.2 Universal Properties Determine an Object up to Unique Isomorphism1.3 Limits and Colimits1.4 Adjoints1.5 An Introduction to Abelian Categories1.6* Spectral Sequences2 Sheaves2.1 Motivating Example: The Sheaf of Smooth Functions2.2 Definition of Sheaf and Presheaf2.3 Morphisms of Presheaves and Sheaves2.4 Properties Determined at the Level of Stalks, and Sheafification2.5 Recovering Sheaves from a “Sheaf on a Base”2.6 Sheaves of Abelian Groups, and 𝒪X-Modules, Form Abelian Categories2.7 The Inverse Image SheafPart II Schemes3 Toward Affine Schemes: The Underlying Set, and Topological Space3.1 Toward Schemes3.2 The Underlying Set of an Affine Scheme3.3 Visualizing Schemes: Generic Points3.4 The Underlying Topological Space of an Affine Scheme3.5 A Base of the Zariski Topology on SpecA: Distinguished Open Sets3.6 Topological (and Noetherian) Properties3.7 The Function I(⋅), Taking Subsets of SpecA to Ideals of A4 The Structure Sheaf, and the Definition of Schemes in General4.1 The Structure Sheaf of an Affine Scheme4.2 Visualizing Schemes: Nilpotents4.3 Definition of Schemes4.4 Three Examples4.5 Projective Schemes, and the Proj Construction5 Some Properties of Schemes5.1 Topological Properties5.2 Reducedness and Integrality5.3 The Affine Communication Lemma, and Properties of Schemes That Can Be Checked “Affine-Locally”5.4 Normality and Factoriality6 Rings Are to Modules as Schemes Are to …6.1 Quasicoherent Sheaves6.2 Characterizing Quasicoherence Using the Distinguished Affine Base6.3 Quasicoherent Sheaves Form an Abelian Category6.4 Finite Type Quasicoherent, Finitely Presented, and Coherent Sheaves6.5 Algebraic Interlude: The Jordan–Hölder Package6.6 Visualizing Schemes: Associated Points and Zerodivisors6.7** Coherent Modules over Non-Noetherian RingsPart III Morphisms of Schemes7 Morphisms of Schemes7.1 Motivations for the “Right” Definition of Morphism of Schemes7.2 Morphisms of Ringed Spaces7.3 From Locally Ringed Spaces to Morphisms of Schemes7.4 Maps of Graded Rings and Maps of Projective Schemes7.5 Rational Maps from Reduced Schemes7.6* Representable Functors and Group Schemes7.7** The Grassmannian: First Construction8 Useful Classes of Morphisms of Schemes8.1 “Reasonable” Classes of Morphisms (Such as Open Embeddings)8.2 Another Algebraic Interlude: Lying Over and Nakayama8.3 A Gazillion Finiteness Conditions on Morphisms8.4 Images of Morphisms: Chevalley’s Theorem and Elimination Theory9 Closed Embeddings and Related Notions9.1 Closed Embeddings and Closed Subschemes9.2 Locally Closed Embeddings and Locally Closed Subschemes9.3 Important Examples from Projective Geometry9.4 The (Closed Sub)scheme-Theoretic Image9.5 Slicing by Effective Cartier Divisors, Regular Sequences and Regular Embeddings10 Fibered Products of Schemes, and Base Change10.1 They Exist10.2 Computing Fibered Products in Practice10.3 Interpretations: Pulling Back Families, and Fibers of Morphisms10.4 Properties Preserved by Base Change10.5* Properties Not Preserved by Base Change, and How to Fix Them10.6 Products of Projective Schemes: The Segre Embedding10.7 Normalization11 Separated and Proper Morphisms, and (Finally!) Varieties11.1 Fun with Diagonal Morphisms, and Quasiseparatedness Made Easy11.2 Separatedness, and Varieties11.3 The Locus where Two Morphisms from X to Y Agree, and the “Reduced-to-Separated” Theorem11.4 Proper MorphismsPart IV “Geometric” Properties of Schemes12 Dimension12.1 Dimension and Codimension12.2 Dimension, Transcendence Degree, and Noether Normalization12.3 Krull’s Theorems12.4 Dimensions of Fibers of Morphisms of Varieties13 Regularity and Smoothness13.1 The Zariski Tangent Space13.2 Regularity, and Smoothness over a Field13.3 Examples13.4 Bertini’s Theorem13.5 Discrete Valuation Rings, and Algebraic Hartogs’s Lemma13.6 Smooth (and Étale) Morphisms: First Definition13.7* Valuative Criteria for Separatedness and Properness13.8* More Sophisticated Facts about Regular Local Rings13.9* Filtered Rings and Modules, and the Artin-Rees LemmaPart V Quasicoherent Sheaves on Schemes, and Their Uses14 More on Quasicoherent and Coherent Sheaves14.1 Vector Bundles “=” Locally Free Sheaves14.2 Locally Free Sheaves on Schemes in Particular14.3 More Pleasant Properties of Finite Type and Coherent Sheaves14.4 Pushforwards of Quasicoherent Sheaves14.5 Pullbacks of Quasicoherent Sheaves: Three Different Perspectives14.6 The Quasicoherent Sheaf Corresponding to a Graded Module15 Line Bundles, Maps to Projective Space, and Divisors15.1 Some Line Bundles on Projective Space15.2 Line Bundles and Maps to Projective Space15.3 The Curve-to-Projective Extension Theorem15.4 Hard but Important: Line Bundles and Weil Divisors15.5 The Payoff: Many Fun Examples15.6 Effective Cartier Divisors “=” Invertible Ideal Sheaves15.7 The Graded Module Corresponding to a Quasicoherent Sheaf16 Maps to Projective Space, and Properties of Line Bundles16.1 Globally Generated Quasicoherent Sheaves16.2 Ample and Very Ample Line Bundles16.3 Applications to Curves16.4* The Grassmannian as a Moduli Space17 Projective Morphisms, and Relative Versions of Spec and Proj17.1 Relative Spec of a (Quasicoherent) Sheaf of Algebras17.2 Relative Proj of a (Quasicoherent) Sheaf of Graded Algebras17.3 Projective Morphisms18 Čech Cohomology of Quasicoherent Sheaves18.1 (Desired) Properties of Cohomology18.2 Definitions and Proofs of Key Properties18.3 Cohomology of Line Bundles on Projective Space18.4 Riemann–Roch, and Arithmetic Genus18.5 A First Glimpse of Serre Duality18.6 Hilbert Functions, Hilbert Polynomials, and Genus18.7 Higher Pushforward (or Direct Image) Sheaves18.8* Serre’s Characterizations of Ampleness and Affineness18.9* From Projective to Proper Hypotheses: Chow’s Lemma and Grothendieck’s Coherence Theorem19 Application: Curves19.1 A Criterion for a Morphism to Be a Closed Embedding19.2 A Series of Crucial Tools19.3 Curves of Genus 019.4 Classical Geometry Arising from Curves of Positive Genus19.5 Hyperelliptic Curves19.6 Curves of Genus 219.7 Curves of Genus 319.8 Curves of Genus 4 and 519.9 Curves of Genus 119.10 Elliptic Curves Are Group Varieties19.11 Counterexamples and Pathologies Using Elliptic Curves20* Application: A Glimpse of Intersection Theory20.1 Intersecting n Line Bundles with an n-Dimensional Variety20.2 Intersection Theory on a Surface20.3 The Grothendieck Group of Coherent Sheaves, and an Algebraic Version of Homology20.4** The Nakai–Moishezon and Kleiman Criteria for Ampleness21 Differentials21.1 Motivation and Game Plan21.2 Definitions and First Properties21.3 Examples21.4 The Riemann–Hurwitz Formula21.5 Understanding Smooth Varieties Using Their Cotangent Bundles21.6 Generic Smoothness, and Consequences21.7 Unramified Morphisms22* Blowing Up22.1 Motivating Example: Blowing Up the Origin in the Plane22.2 Blowing Up, by Universal Property22.3 The Blow-up Exists, and Is Projective22.4 Examples and ComputationsPart VI More Cohomological Tools23 Derived Functors23.1 The Tor Functors23.2 Derived Functors in General23.3 Derived Functors and Spectral Sequences23.4 Derived Functor Cohomology of 𝒪-Modules23.5 Čech Cohomology and Derived Functor Cohomology Agree24 Flatness24.1 Easier Facts24.2 Flatness through Tor24.3 Ideal-Theoretic Criteria for Flatness24.4** Aside: The Koszul Complex and the Hilbert Syzygy Theorem24.5 Topological Implications of Flatness24.6 Local Criteria for Flatness24.7 Flatness Implies Constant Euler Characteristic24.8 Smooth and Étale Morphisms, and Flatness25 Cohomology and Base Change Theorems25.1 Statements and Applications25.2 Proofs of Cohomology and Base Change Theorems25.3 Applying Cohomology and Base Change to Moduli Problems26 Depth and Cohen–Macaulayness26.1 Depth26.2 Cohen–Macaulay Rings and Schemes26.3 Serre’s R1 + S2 Criterion for Normality27 The Twenty-Seven Lines on a Cubic Surface27.1 Preliminary Facts27.2 Every Smooth Cubic Surface (over k) Contains 27 Lines27.3 Every Smooth Cubic Surface (over k) is a Blown-Up Plane28 Power Series and the Theorem on Formal Functions28.1 Algebraic Preliminaries28.2 Types of Singularities28.3 The Theorem on Formal Functions28.4 Zariski’s Connectedness Lemma and Stein Factorization28.5 Zariski’s Main Theorem28.6 Castelnuovo’s Criterion for Contracting (−1)-Curves28.7** Proof of the Theorem on Formal Functions 28.3.229∗ Proof of Serre Duality29.1 Desiderata29.2 Ext Groups and Ext Sheaves for 𝒪-Modules29.3 Serre Duality for Projective k-Schemes29.4 The Adjunction Formula for the ωX, and ωX = 𝒦XBibliographyIndex