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 DeFinition of Sheaf and Presheaf 2.3 Morphisms of Presheaves and Sheaves 2.4 Properties Determined at the Level of Stalks, and SheaFiFication 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 DeFinition of Schemes in General 4.1 The Structure Sheaf of an Ane Scheme 4.2 Visualizing Schemes: Nilpotents 4.3 DeFinition 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" DeFinition 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 DeFinition 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 cech Cohomology of Quasicoherent Sheaves 18.1 (Desired) Properties of Cohomology 18.2 DeFinitions 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 DeFinitions 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 UnramiFied 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 cech 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** As.