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· 제목 : The Rising Sea: Foundations of Algebraic Geometry (Hardcover)
· 분류 : 외국도서 > 과학/수학/생태 > 수학 > 기하학 > 대수기하학
· ISBN : 9780691268668
· 쪽수 : 688쪽
· 출판일 : 2025-10-21
· 분류 : 외국도서 > 과학/수학/생태 > 수학 > 기하학 > 대수기하학
· ISBN : 9780691268668
· 쪽수 : 688쪽
· 출판일 : 2025-10-21
목차
- 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 De?nition of Sheaf and Presheaf
- 2.3 Morphisms of Presheaves and Sheaves
- 2.4 Properties Determined at the Level of Stalks, and Shea??cation
- 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 A?ne Schemes: The Underlying Set, and Topological Space
- 3.1 Toward Schemes
- 3.2 The Underlying Set of an A?ne Scheme
- 3.3 Visualizing Schemes: Generic Points
- 3.4 The Underlying Topological Space of an A?ne 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 De?nition of Schemes in General
- 4.1 The Structure Sheaf of an A?ne Scheme
- 4.2 Visualizing Schemes: Nilpotents
- 4.3 De?nition 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 A?ne Communication Lemma, and Properties of Schemes That Can Be Checked “A?ne-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 A?ne 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–Holder 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” De?nition 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 E?ective 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 Etale) Morphisms: First De?nition
- 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 Di?erent 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 E?ective 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 ?ech Cohomology of Quasicoherent Sheaves
- 18.1 (Desired) Properties of Cohomology
- 18.2 De?nitions 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 A?neness
- 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 Di?erentials
- 21.1 Motivation and Game Plan
- 21.2 De?nitions 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 Unrami?ed 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 ?ech 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** 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 Etale Morphisms, and Flatness
- 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
- 26 Depth and Cohen–Macaulayness
- 26.1 Depth
- 26.2 Cohen–Macaulay Rings and Schemes
- 26.3 Serre’s R1 + S2 Criterion for Normality
- 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
- 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
- 29∗ Proof of Serre Duality
- 29.1 Desiderata
- 29.2 Ext Groups and Ext Sheaves for ??-Modules
- 29.3 Serre Duality for Projective k-Schemes
- 29.4 The Adjunction Formula for the ωX, and ωX = ??X
- Bibliography
- Index
