Motivic homotopy theory : lectures at a summer school in Nordfjordeid, Norway, August 2002 / B. I. Dundas ... [and others].

Format:

Book

Contributor:

Dundas, B. I. (Bjørn Ian)

Series:

Universitext

Language:

English

Subjects (All):

Homotopy theory.

Physical Description:

x, 220 pages : illustrations ; 24 cm.

Place of Publication:

Berlin : Springer, [2007]

Summary:

This book is based on lectures given at a summer school on motivic homotopy theory at the Sophus Lie Centre in Nordfjordeid, Norway, in August 2002. Aimed at graduate students in algebraic topology and algebraic geometry, it contains background material from both of these fields, as well as the foundations of motivic homotopy theory. It will serve as a good introduction as well as a convenient reference for a broad group of mathematicians to this important and fascinating new subject.

Contents:

Prerequisites in Algebraic Topology the Nordfjordeid Summer School on Motivic Homotopy Theory / Bjorn Ian Dundas 1

I Basic Properties and Examples 5

1 Topological Spaces 6

1.1 Singular Homology 6

1.2 Weak Equivalences 8

1.3 Mapping Spaces 9

2 Simplicial Sets 9

2.1 The Category [Delta] 10

2.2 Simplicial Sets vs. Topological Spaces 12

2.3 Weak Equivalences 14

3 Some Constructions in S 15

4 Simplicial Abelian Groups 16

4.1 Simplicial Abelian Groups vs. Chain Complexes 17

4.2 The Normalized Chain Complex 17

5 The Pointed Case 18

6 Spectra 20

6.2 Relation to Simplicial Sets 22

6.3 Stable Equivalences 22

6.4 Homology Theories 23

6.5 Relation to Chain Complexes 24

II Deeper Structure: Simplicial Sets 27

0.1 Realization as an Extension Through Presheaves 28

1 (Co)fibrations 30

1.1 Simplicial Sets are Built Out of Simplices 30

1.2 Lifting Properties and Factorizations 31

1.3 Small Objects 33

1.4 Fibrations 34

2 Combinatorial Homotopy Groups 37

2.1 Homotopies and Fibrant Objects 37

III Model Categories 41

0.1 Liftings 41

1 The Axioms 42

1.1 Simple Consequences 43

1.2 Proper Model Categories 45

1.3 Quillen Functors 46

2 Functor Categories: The Projective Structure 47

3 Cofibrantly Generated Model Categories 48

4 Simplicial Model Categories 50

5 Spectra 51

5.1 Pointwise Structure 51

5.2 Stable Structure 52

IV Motivic Spaces and Spectra 55

1 Motivic Spaces 55

1.1 The A[superscript 1]-Structure 57

2 Motivic Functors 57

2.1 Two Questions 57

2.2 Algebraic Structure 58

2.3 The Motivic Eilenberg-Mac Lane Spectrum 59

2.4 Wanted 60

3 Model Structures of Motivic Functors and Relation to Spectra 60

3.1 The Homotopy Functor Model Structure 60

3.2 Motivic Spectra 62

3.3 The Connection F[subscript S] [RightArrow] Spt[subscript S] 62

Background from Algebraic Geometry / Marc Levine 69

I Elementary Algebraic Geometry 71

1 The Spectrum of a Commutative Ring 71

1.1 Ideals and Spec 71

1.2 The Zariski Topology 73

1.3 Functorial Properties 74

1.4 Naive Algebraic Geometry and Hilbert's Nullstellensatz 75

1.5 Krull Dimension, Height One Primes and the UFD Property 77

1.6 Open Subsets and Localization 79

2 Ringed Spaces 81

2.1 Presheaves and Sheaves on a Space 81

2.2 The Sheaf of Regular Functions on Spec A 82

2.3 Local Rings and Stalks 84

3 The Category of Schemes 85

3.1 Objects and Morphisms 86

3.2 Gluing Constructions 88

3.3 Open and Closed Subschemes 89

3.4 Fiber Products 90

4 Schemes and Morphisms 91

4.1 Noetherian Schemes 91

4.2 Irreducible Schemes, Reduced Schemes and Generic Points 92

4.3 Separated Schemes and Morphisms 94

4.4 Finite Type Morphisms 95

4.5 Proper, Finite and Quasi-Finite Morphisms 96

4.6 Flat Morphisms 97

4.7 Valuative Criteria 97

5 The Category Sch[subscript k] 98

5.1 R-Valued Points 98

5.2 Group-Schemes and Bundles 99

5.3 Dimension 100

5.4 Flatness and Dimension 102

5.5 Smooth Morphisms and etale Morphisms 102

5.6 The Jacobian Criterion 105

6 Projective Schemes and Morphisms 105

6.1 The Functor Proj 106

6.2 Properness 109

6.3 Projective and Quasi-Projective Morphisms 110

6.4 Globalization 111

6.5 Blowing Up a Subscheme 112

II Sheaves for a Grothendieck Topology 115

7 Limits 115

7.2 Functoriality of Limits 117

7.3 Representability and Exactness 117

7.4 Cofinality 118

8 Presheaves 118

8.1 Limits and Exactness 119

8.2 Functoriality and Generators for Presheaves 119

8.3 Generators for Presheaves 120

8.4 PreShv[superscript Ab] (C) as an Abelian Category 121

9 Sheaves 123

9.1 Grothendieck Pre-Topologies and Topologies 123

9.2 Sheaves on a Site 126

Voevodsky's Nordfjordeid Lectures: Motivic Homotopy Theory / Vladimir Voevodsky, Oliver Rondigs, Paid Arne Ostvoer 147

2 Motivic Stable Homotopy Theory 148

2.1 Spaces 148

2.2 The Motivic s-Stable Homotopy Category SH[Characters not reproducible] (k) 150

2.3 The Motivic Stable Homotopy Category SH(K) 153

3 Cohomology Theories 162

3.1 The Motivic Eilenberg-MacLane Spectrum HZ 162

3.2 The Algebraic K-Theory Spectrum KGL 164

3.3 The Algebraic Cobordism Spectrum MGL 165

4 The Slice Filtration 166

5.1 The Nisnevich Topology 172

5.2 Model Structures for Spaces 180

5.3 Model Structures for Spectra of Spaces 203.

Notes:

Includes bibliographical references and index.

ISBN:

3540458956

9783540458951

OCLC:

76809209