Catalogue


Motivic homotopy theory [electronic resource] : lectures at a summer school in Nordfjordeid, Norway, August 2002 /
B.I. Dundas.
imprint
Berlin ; New York : Springer, c2007.
description
x, 220 p. : ill. ; 24 cm.
ISBN
3540458956 (hardcover : alk. paper)
format(s)
Book
More Details
series title
imprint
Berlin ; New York : Springer, c2007.
isbn
3540458956 (hardcover : alk. paper)
restrictions
Licensed for access by U. of T. users.
catalogue key
8000032
 
Includes bibliographical references and indexes.
A Look Inside
About the Author
Author Affiliation
Bjorn Ian Dundas is Professor of Mathematics at the University of Bergen Marc N. Levine is Professor of Mathematics at Northeastern University Paul Arne Ostvaer is Associate Professor of Mathematics at the University of Oslo Oliver Rondigs has an assistant position at the University of Bielefeld Vladimir Voevodsky is Professor of Mathematics at the Institute for Advanced Study in Princeton
Reviews
Review Quotes
From the reviews:"This research monograph on motivic homotopy theory contains material based on lectures at a summer school at the Sophus Lie Centre in Nordfjordeid, Norway, in August 2002. With a similar scope as the summer school it is aimed at graduate students and researchers in algebraic topology and algebraic geometry. … They provide an excellent introduction as well as a convenient reference for anybody who wants to learn more about this important and fascinating new subject." (Frank Neumann, Mathematical Reviews, Issue 2008 k)
To find out how to look for other reviews, please see our guides to finding book reviews in the Sciences or Social Sciences and Humanities.
Summaries
Main Description
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. Vladimir Voevodsky is one of the founders of the theory and received the Fields medal for his work, and the other authors have all done important work in the subject.
Main Description
This book is based on lectures given at a summer school on motivic homotopy theoryat 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.
Long Description
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.
Table of Contents
Prerequisites in Algebraic Topology the Nordfjordeid Summer School on Motivic Homotopy Theoryp. 1
Prefacep. 3
Basic Properties and Examplesp. 5
Topological Spacesp. 6
Singular Homologyp. 6
Weak Equivalencesp. 8
Mapping Spacesp. 9
Simplicial Setsp. 9
The Category [Delta]p. 10
Simplicial Sets vs. Topological Spacesp. 12
Weak Equivalencesp. 14
Some Constructions in Sp. 15
Simplicial Abelian Groupsp. 16
Simplicial Abelian Groups vs. Chain Complexesp. 17
The Normalized Chain Complexp. 17
The Pointed Casep. 18
Spectrap. 20
Introductionp. 20
Relation to Simplicial Setsp. 22
Stable Equivalencesp. 22
Homology Theoriesp. 23
Relation to Chain Complexesp. 24
Deeper Structure: Simplicial Setsp. 27
Realization as an Extension Through Presheavesp. 28
(Co)fibrationsp. 30
Simplicial Sets are Built Out of Simplicesp. 30
Lifting Properties and Factorizationsp. 31
Small Objectsp. 33
Fibrationsp. 34
Combinatorial Homotopy Groupsp. 37
Homotopies and Fibrant Objectsp. 37
Model Categoriesp. 41
Liftingsp. 41
The Axiomsp. 42
Simple Consequencesp. 43
Proper Model Categoriesp. 45
Quillen Functorsp. 46
Functor Categories: The Projective Structurep. 47
Cofibrantly Generated Model Categoriesp. 48
Simplicial Model Categoriesp. 50
Spectrap. 51
Pointwise Structurep. 51
Stable Structurep. 52
Motivic Spaces and Spectrap. 55
Motivic Spacesp. 55
The A[superscript 1]-Structurep. 57
Motivic Functorsp. 57
Two Questionsp. 57
Algebraic Structurep. 58
The Motivic Eilenberg-Mac Lane Spectrump. 59
Wantedp. 60
Model Structures of Motivic Functors and Relation to Spectrap. 60
The Homotopy Functor Model Structurep. 60
Motivic Spectrap. 62
The Connection F[subscript S] [RightArrow] Spt[subscript S]p. 62
Referencesp. 63
Indexp. 65
Background from Algebraic Geometryp. 69
Elementary Algebraic Geometryp. 71
The Spectrum of a Commutative Ringp. 71
Ideals and Specp. 71
The Zariski Topologyp. 73
Functorial Propertiesp. 74
Naive Algebraic Geometry and Hilbert's Nullstellensatzp. 75
Krull Dimension, Height One Primes and the UFD Propertyp. 77
Open Subsets and Localizationp. 79
Ringed Spacesp. 81
Presheaves and Sheaves on a Spacep. 81
The Sheaf of Regular Functions on Spec Ap. 82
Local Rings and Stalksp. 84
The Category of Schemesp. 85
Objects and Morphismsp. 86
Gluing Constructionsp. 88
Open and Closed Subschemesp. 89
Fiber Productsp. 90
Schemes and Morphismsp. 91
Noetherian Schemesp. 91
Irreducible Schemes, Reduced Schemes and Generic Pointsp. 92
Separated Schemes and Morphismsp. 94
Finite Type Morphismsp. 95
Proper, Finite and Quasi-Finite Morphismsp. 96
Flat Morphismsp. 97
Valuative Criteriap. 97
The Category Sch[subscript k]p. 98
R-Valued Pointsp. 98
Group-Schemes and Bundlesp. 99
Dimensionp. 100
Flatness and Dimensionp. 102
Smooth Morphisms and etale Morphismsp. 102
The Jacobian Criterionp. 105
Projective Schemes and Morphismsp. 105
The Functor Projp. 106
Propernessp. 109
Projective and Quasi-Projective Morphismsp. 110
Globalizationp. 111
Blowing Up a Subschemep. 112
Sheaves for a Grothendieck Topologyp. 115
Limitsp. 115
Definitionsp. 115
Functoriality of Limitsp. 117
Representability and Exactnessp. 117
Cofinalityp. 118
Presheavesp. 118
Limits and Exactnessp. 119
Functoriality and Generators for Presheavesp. 119
Generators for Presheavesp. 120
PreShv[superscript Ab] (C) as an Abelian Categoryp. 121
Sheavesp. 123
Grothendieck Pre-Topologies and Topologiesp. 123
Sheaves on a Sitep. 126
Referencesp. 140
Indexp. 143
Voevodsky's Nordfjordeid Lectures: Motivic Homotopy Theoryp. 147
Introductionp. 148
Motivic Stable Homotopy Theoryp. 148
Spacesp. 148
The Motivic s-Stable Homotopy Category SH[Characters not reproducible] (k)p. 150
The Motivic Stable Homotopy Category SH(K)p. 153
Cohomology Theoriesp. 162
The Motivic Eilenberg-MacLane Spectrum HZp. 162
The Algebraic K-Theory Spectrum KGLp. 164
The Algebraic Cobordism Spectrum MGLp. 165
The Slice Filtrationp. 166
Appendixp. 172
The Nisnevich Topologyp. 172
Model Structures for Spacesp. 180
Model Structures for Spectra of Spacesp. 203
Referencesp. 218
Indexp. 221
Table of Contents provided by Ingram. All Rights Reserved.

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