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Produktbild: The Rising Sea

The Rising Sea Foundations of Algebraic Geometry

167,99 €

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Beschreibung

Produktdetails

Einband

Gebundene Ausgabe

Erscheinungsdatum

21.10.2025

Abbildungen

100 b/w illus.

Verlag

University Presses

Seitenzahl

688

Maße (L/B/H)

26/18,3/4,1 cm

Gewicht

1468 g

Sprache

Englisch

ISBN

978-0-691-26866-8

Beschreibung

Produktdetails

Einband

Gebundene Ausgabe

Erscheinungsdatum

21.10.2025

Abbildungen

100 b/w illus.

Verlag

University Presses

Seitenzahl

688

Maße (L/B/H)

26/18,3/4,1 cm

Gewicht

1468 g

Sprache

Englisch

ISBN

978-0-691-26866-8

Herstelleradresse

Libri GmbH
Europaallee 1
36244 Bad Hersfeld
DE

Email: gpsr@libri.de

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  • Produktbild: The Rising Sea

    • 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 Sheä¿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–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” 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 Étale) 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 Étale 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 ¿X, and ¿X = ¿X
                                                              • Bibliography
                                                              • Index