Motivic Homotopy Theory: Lectures at a Summer School in Nordfjordeid, Norway, August 2002

227 Pages·2007·3.484 MB·English

by  Dundas, Bjørn Ian.;Levine, Marc.;Østvær, Paul Arne.;Röndigs, Oliver.;Voevodsky, Vladimir

B.I. Dundas
M. Levine
P.A. Østvær
O. Röndigs
V. Voevodsky
Motivic
Homotopy Theory
Lectures at a Summer School
in Nordfjordeid, Norway, August 2002
ABC
BjørnIanDundas OliverRöndigs
DepartmentofMathematics FakultätfürMathematik
UniversityofOslo UniversitätBielefeld
POBox1053,Blindern Postfach100131
0316Oslo 33501Bielefeld
Norway Germany
E-mail:[email protected] E-mail:[email protected]
VladimirVoevodsky
MarcLevine
SchoolofMathematics
NortheasternUniversity
PrincetonUniversity
DepartmentofMathematics
Princeton,NJ08540
360HuntingtonAvenue
USA
Boston,MA02115
E-mail:[email protected]
USA
E-mail:[email protected]
Editor:
PaulArneØstvær BjørnJahren
DepartmentofMathematics DepartmentofMathematics
UniversityofOslo UniversityofOslo
POBox1053,Blindern Box1053Blindern
0316Oslo 0316Oslo
Norway Norway
E-mail:[email protected] E-mail:[email protected]
MathematicsSubjectClassification(2000):14-xx,18-xx,19-xx,55-xx
LibraryofCongressControlNumber:2006933719
ISBN-10 3-540-45895-6 SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-45895-1SpringerBerlinHeidelbergNewYork
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Preface
This book is based on lectures given at a summer school held in Nordfjordeid
ontheNorwegianwestcoastinAugust2002.Inthelittletownwiththespec-
tacular surroundings where Sophus Lie was born in 1842, the municipality,
in collaboration with the mathematics departments at the universities, has
established the “Sophus Lie conference center”. The purpose is to help orga-
nizing conferences and summer schools at a local boarding school during its
summervacation,andthealgebraistsandalgebraicgeometersinNorwayhad
already organized such summer schools for a number of years. In 2002 a joint
project with the algebraic topologists was proposed, and a natural choice of
topic was Motivic homotopy theory, which depends heavily on both algebraic
topology and algebraic geometry and has had deep impact in both fields.
The organizing committee consisted of Bjørn Jahren and Kristian Ranes-
tad,Oslo,AlexeiRudakov,TrondheimandSteinArildStrømme,Bergen,and
the summer school was partly funded by NorFA — Nordisk Forskerutdan-
ningsakademi. It was primarily intended for Norwegian graduate students,
but it attracted students from a number of other countries as well.
These summer schools traditionally go on for one week, with three series
of lectures given by internationally known experts. Motivic homotopy theory
was an obvious choice for one of the series, and, especially considering the
diverse background of the participants, the two remaining series were chosen
to cover necessary background material from algebraic topology and model
categories, and from algebraic geometry. The background lectures were given
by Bjørn I. Dundas and Marc Levine, both of whom have done important
workintheirrespectiveareasinconnectionwiththemaintopicoftheschool.
Motivichomotopytheorywastaughtbyoneofthefoundersofthesubjectand
certainlyitsmostprominentfigure:VladimirVoevodsky.Wewereveryhappy
to have such great and inspiring experts come and share their knowledge and
insight with a new generation of students.
After the summer school, Dundas and Levine agreed to write up their
lecture series for publication, and Voevodsky agreed to let Oliver Röndigs
and Paul Arne Østvær write up his. Röndigs and Østvær have also added an
VI Preface
extensive appendix with a more detailed discussion of the homotopy theory
andmodelstructuresinvolved.InthisvolumethecontributionsofDundasand
Levinearepresentedfirst,sincetheycontaintheprerequisitesforVoevodsky’s
lectures.Theyarebasicallyindependentandcanbereadinanyorder,orjust
referred to while reading the third part, depending on the background of the
reader.
Finally, we would like to thank Springer Verlag for offering to publishthis
book.Weapologizethatthishastakenlongerthanexpected,butnowthatthe
lectures are available, our hope is that many students will find it useful and
convenient to find both an introduction to the fascinating subject of motivic
homotopy theory and the background material in one place.
Oslo, August 2006 Bjørn Jahren
Contents
Prerequisites in Algebraic Topology the Nordfjordeid Summer
School on Motivic Homotopy Theory
Bjørn Ian Dundas................................................ 1
Preface ........................................................ 3
II 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 ∆ ......................................... 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
6.1 Introduction ............................................ 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
IIII 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
VIII Contents
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
IIIIII 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
IIVV Motivic Spaces and Spectra................................ 55
1 Motivic Spaces............................................... 55
1.1 The A1-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 →Spt ............................... 62
S S
References ...................................................... 63
Index.......................................................... 65
Background from Algebraic Geometry
Marc Levine..................................................... 69
II 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
Contents IX
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 SpecA .................. 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 ........................................... 98
k
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 étale 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
IIII Sheaves for a Grothendieck Topology ......................115
7 Limits ......................................................115
7.1 Definitions..............................................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 PreShvAb(C) as an Abelian Category......................121
X Contents
9 Sheaves .....................................................123
9.1 Grothendieck Pre-Topologies and Topologies ................123
9.2 Sheaves on a Site ........................................126
References ......................................................140
Index..........................................................143
Voevodsky’s Nordfjordeid Lectures: Motivic Homotopy
Theory
Vladimir Voevodsky, Oliver Röndigs, Paul Arne Østvær...............147
1 Introduction .................................................148
2 Motivic Stable Homotopy Theory ..............................148
2.1 Spaces..................................................148
A1
2.2 The Motivic s-Stable Homotopy Category SH (k)...........150
s
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 Appendix ...................................................172
5.1 The Nisnevich Topology ..................................172
5.2 Model Structures for Spaces...............................180
5.3 Model Structures for Spectra of Spaces .....................203
References ......................................................218
Index..........................................................221