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The Rising Sea - Foundations of Algebraic Geometry
Contents
Index
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
Chapter 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
Chapter 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 OX-modules, form abelian categories
2.7. The inverse image sheaf
Part II. Schemes
Chapter 3. Toward affine schemes: the underlying set, and topological space
3.1. Toward schemes
3.2. The underlying set of an affine scheme
3.3. Visualizing schemes: Generic points
3.4. The underlying topological space of an affine scheme
3.5. A base of the Zariski topology on Spec A: Distinguished open sets
3.6. Topological (and Noetherian) properties
3.7. The function I(), taking subsets of Spec A to ideals of A
Chapter 4. The structure sheaf, and the definition of schemes in general
4.1. The structure sheaf of an affine scheme
4.2. Visualizing schemes: Nilpotents
4.3. Definition of schemes
4.4. Three examples
4.5. Projective schemes, and the Proj construction
Chapter 5. Some properties of schemes
5.1. Topological properties
5.2. Reducedness and integrality
5.3. The Affine Communication Lemma, and properties of schemes that can be checked ``affine-locally''
5.4. Normality and factoriality
Chapter 6. Rings are to modules as schemes are to ...
6.1. Quasicoherent sheaves
6.2. Characterizing quasicoherence using the distinguished affine 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
Chapter 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
Chapter 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
Chapter 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 effective Cartier divisors, regular sequences and regular embeddings
Chapter 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
Chapter 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
Chapter 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
Chapter 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
Chapter 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 different perspectives
14.6. The quasicoherent sheaf corresponding to a graded module
Chapter 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 pay-off: Many fun examples
15.6. Effective Cartier divisors ``='' invertible ideal sheaves
15.7. The graded module corresponding to a quasicoherent sheaf
Chapter 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
Chapter 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
Chapter 18. ÄŒech 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 affineness
18.9. From projective to proper hypotheses: Chow's Lemma and Grothendieck's Coherence Theorem
Chapter 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
Chapter 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
Chapter 21. Differentials
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
Chapter 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
Chapter 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 O-modules
23.5. ÄŒech cohomology and derived functor cohomology agree
Chapter 24. Flatness
24.1. Easier facts
24.2. Flatness through Tor
24.3. Ideal-theoretic criteria for flatness
24.4. Aside: The Koszul complex and the Hilbert Syzygy Theorem
24.5. Topological implications of flatness
24.6. Local criteria for flatness
24.7. Flatness implies constant Euler characteristic
24.8. Smooth and étale morphisms, and flatness
Chapter 25. Cohomology and base change theorems
25.1. Statements and applications
25.2. Proofs of cohomology and base change theorems
25.3. Applying cohomology and base change to moduli problems
Chapter 26. Depth and Cohen-Macaulayness
26.1. Depth
26.2. Cohen-Macaulay rings and schemes
26.3. Serre's R1+S2 criterion for normality
Chapter 27. The twenty-seven lines on a cubic surface
27.1. Preliminary facts
27.2. Every smooth cubic surface (over k) contains 27 lines
27.3. Every smooth cubic surface (over k) is a blown up plane
Chapter 28. Power series and the Theorem on Formal Functions
28.1. Algebraic preliminaries
28.2. Types of singularities
28.3. The Theorem on Formal Functions
28.4. Zariski's Connectedness Lemma and Stein factorization
28.5. Zariski's Main Theorem
28.6. Castelnuovo's Criterion for contracting (-1)-curves
28.7. Proof of the Theorem on Formal Functions 28.3.2
Chapter 29. Proof of Serre duality
29.1. Desiderata
29.2. Ext groups and Ext sheaves for O-modules
29.3. Serre duality for projective k-schemes
29.4. The adjunction formula for X, and X = KX
Bibliography
Index
Contents
Ravi Vakil's Book Homepage
