Vor Ort
Merkzettel Warenkorb
Produktbild: The Rising Sea

The Rising Sea Foundations of Algebraic Geometry

52,99 €

inkl. gesetzl. MwSt., Versandkostenfrei

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

21.10.2025

Abbildungen

100 b/w illus.

Verlag

University Presses

Seitenzahl

688

Maße (L/B/H)

25,7/18,1/4,3 cm

Gewicht

1384 g

Sprache

Englisch

ISBN

978-0-691-26867-5

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

21.10.2025

Abbildungen

100 b/w illus.

Verlag

University Presses

Seitenzahl

688

Maße (L/B/H)

25,7/18,1/4,3 cm

Gewicht

1384 g

Sprache

Englisch

ISBN

978-0-691-26867-5

Herstelleradresse

Libri GmbH
Europaallee 1
36244 Bad Hersfeld
DE

Email: GPSR Kontakt

Ein neues Kapitel für Ihre Bücher

Ein neues Kapitel für Ihre Bücher

Schenken Sie Ihren alten Schätzen ein zweites Leben und erhalten dafür eine Thalia Geschenkkarte.

Jetzt verkaufen
Jetzt verkaufen

Noch keine Bewertungen vorhanden

Verfassen Sie die erste Bewertung zu diesem Artikel

Helfen Sie anderen Kundinnen und Kunden durch Ihre Meinung.

Kundinnen und Kunden meinen

Bewertungen (0)
  • 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