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, and OX -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 Spec A: Distinguished open sets 118
3.6. Topological (and Noetherian) properties 119
3.7. The function I(·), taking subsets of Spec A to ideals of A 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 embeddings240
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
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, and OX -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 Spec A: Distinguished open sets 118
3.6. Topological (and Noetherian) properties 119
3.7. The function I(·), taking subsets of Spec A to ideals of A 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 embeddings240
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
