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THE RISING SEA

Foundations of Algebraic Geometry

math216.wordpress.com

November 18, 2017 draft

c ⃝2010-2017 by Ravi Vakil. Note to reader: the index and formatting have yet to be properly dealt with. There remain many issues still to be dealt with in the main part of the notes (including many of your corrections and suggestions).

Contents

Preface

11

0.1. For the reader

12

0.2. For the expert

16

0.3. Background and conventions

17

0.4.⋆⋆The goals of this book

18

Part I. Preliminaries

21

Chapter 1. Some category theory

23

1.1. Motivation

23

1.2. Categories and functors

25

1.3. Universal properties determine an object up to unique isomorphism

31

1.4. Limits and colimits

39

1.5. Adjoints

43

1.6. An introduction to abelian categories

47

1.7.⋆Spectral sequences

57

Chapter 2. Sheaves

71

2.1. Motivating example: The sheaf of differentiable functions

71

2.2. Definition of sheaf and presheaf

73

2.3. Morphisms of presheaves and sheaves

78

2.4. Properties determined at the level of stalks, and sheafification

82

2.5. Recovering sheaves from a “sheaf on a base"

86

2.6. Sheaves of abelian groups, andOX-modules, form abelian categories

89

2.7. The inverse image sheaf

92

Part II. Schemes

97
Chapter 3. Toward affine schemes: the underlying set, and topological space 99

3.1. Toward schemes

99

3.2. The underlying set of affine schemes

101

3.3. Visualizing schemes I: generic points

113

3.4. The underlying topological space of an affine scheme

115

3.5. A base of the Zariski topology on SpecA: Distinguished open sets

118

3.6. Topological (and Noetherian) properties

119

3.7. The functionI(), taking subsets of SpecAto ideals ofA

127
Chapter 4. The structure sheaf, and the definition of schemes in general 129

4.1. The structure sheaf of an affine scheme

129

4.2. Visualizing schemes II: nilpotents

133
3

4.3. Definition of schemes

136

4.4. Three examples

139

4.5. Projective schemes, and the Proj construction

145

Chapter 5. Some properties of schemes

153

5.1. Topological properties

153

5.2. Reducedness and integrality

155

5.3. Properties of schemes that can be checked “affine-locally"

157

5.4. Normality and factoriality

161

5.5. The crucial points of a scheme that control everything: Associated

points and primes 166

Part III. Morphisms

175

Chapter 6. Morphisms of schemes

177

6.1. Introduction

177

6.2. Morphisms of ringed spaces

178

6.3. From locally ringed spaces to morphisms of schemes

180

6.4. Maps of graded rings and maps of projective schemes

186

6.5. Rational maps from reduced schemes

188

6.6.⋆Representable functors and group schemes

194

6.7.⋆⋆The Grassmannian (initial construction)

199

Chapter 7. Useful classes of morphisms of schemes

201

7.1. An example of a reasonable class of morphisms: Open embeddings

201

7.2. Algebraic interlude: Lying Over and Nakayama

203

7.3. A gazillion finiteness conditions on morphisms

207

7.4. Images of morphisms: Chevalley"s Theorem and elimination theory

216

Chapter 8. Closed embeddings and related notions

225

8.1. Closed embeddings and closed subschemes

225

8.2. More projective geometry

230

8.3. The (closed sub)scheme-theoretic image

236

8.4. Effective Cartier divisors, regular sequences and regular embeddings

240
Chapter 9. Fibered products of schemes, and base change 247

9.1. They exist

247

9.2. Computing fibered products in practice

253

9.3. Interpretations: Pulling back families, and fibers of morphisms

256

9.4. Properties preserved by base change

262

9.5.⋆Properties not preserved by base change, and how to fix them

263

9.6. Products of projective schemes: The Segre embedding

271

9.7. Normalization

273
Chapter 10. Separated and proper morphisms, and (finally!) varieties 279

10.1. Separated morphisms (and quasiseparatedness done properly)

279

10.2. Rational maps to separated schemes

289

10.3. Proper morphisms

293
Part IV. “Geometric" properties: Dimension and smoothness 301

Chapter 11. Dimension

303

11.1. Dimension and codimension

303

11.2. Dimension, transcendence degree, and Noether normalization

307

11.3. Codimension one miracles: Krull"s and Hartogs"s Theorems

315

11.4. Dimensions of fibers of morphisms of varieties

322

11.5.⋆⋆Proof of Krull"s Principal Ideal and Height Theorems

327

Chapter 12. Regularity and smoothness

331

12.1. The Zariski tangent space

331

12.2. Regularity, and smoothness over a field

337

12.3. Examples

342

12.4. Bertini"s Theorem

346

12.5. Another (co)dimension one miracle: Discrete valuation rings

349

12.6. Smooth (and

´etale) morphisms (first definition)

354

12.7.⋆Valuative criteria for separatedness and properness

358

12.8.⋆More sophisticated facts about regular local rings

362

12.9.⋆Filtered rings and modules, and the Artin-Rees Lemma

364

Part V. Quasicoherent sheaves

367

Chapter 13. Quasicoherent and coherent sheaves

369

13.1. Vector bundles and locally free sheaves

369

13.2. Quasicoherent sheaves

375

13.3. Characterizing quasicoherence using the distinguished affine base

377

13.4. Quasicoherent sheaves form an abelian category

381

13.5. Module-like constructions

383

13.6. Finite type and coherent sheaves

386

13.7. Pleasant properties of finite type and coherent sheaves

389

13.8.⋆⋆Coherent modules over non-Noetherian rings

393
Chapter 14. Line bundles: Invertible sheaves and divisors 397

14.1. Some line bundles on projective space

397
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