Motivic homotopy theory: lectures at a Summer School in Nordfjordeid, Norway, August 2002
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 foun...
Saved in:
| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2007
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| 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. 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. |
| Physical Description: | X, 220 S. graph. Darst. |
| ISBN: | 9783540458951 3540458956 |
Staff View
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| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2007 | |
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| 520 | 3 | |a 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. | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Prerequisites in Algebraic Topology the
Nordfjordeid
Summer
School on Motivic Homotopy Theory
Bjom Ian
Dundas
................................................ 1
Preface
........................................................ 3
I Basic Properties and Examples
............................ 5
1
Topologica!
Spacos ...........................................
6
1.1
Singular Homology
....................................... 6
1.2
Weak Equivalences
....................................... 8
1.3
Mapping Spaces
......................................... 9
2
Simplicia!
Sots
............................................... 9
2.1
The Category
Δ
......................................... 10
2.2
Simplicial Sets vs.
Topologica!
Spaces
...................... 12
2.3
Weak Equivalences
....................................... 14
3
Some Const ructions in
S
...................................... 15
4
Simplicia! 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
Simplicia!
Sets
................................ 22
6.3
Stallie
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)fibratious
................................................ 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
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
Simplicia]
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^Strueture
........................................ 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 Ts
->
Spts
...............................
62
References
......................................................
Index
..........................................................
65
Background from Algebraic Geometry
Marc
Levine
..................................................... ■*
I Elementary Algebraic Geometry
........................... *
1
The Spectrum of a Commutative Ring
.......................... *
1.1
Ideals and Spec
..........................................
1.2
The Zariski Topology
.....................................
73
1.3
Fimctorial Properties
.....................................
1.4
Naive Algebraic Geometry and Hubert 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 Spares
............................................... 81
2.1
Presheaves and Sheaves
ou 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/,.
........................................... 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
étale
Morphisius
...................
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
............................................
Ill
6.5
Blowing Up a Subscheme
.................................112
II 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 Abeliau 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
Ame 0stvœr
...............147
1
Introduction
.................................................148
2
Motivic Stable Homotopy Theory
..............................148
2.1
Spaces
..................................................148
2.2
The Motivic s-Stable Homotopy Category SH*
(к)
...........150
2.3
The Motivic Stable Homotopy Category SH(fc)
..............153
3
Cohomology Theories
.........................................162
3.1
The Motivic Eilenberg-MacLaiie 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
|
| adam_txt |
Contents
Prerequisites in Algebraic Topology the
Nordfjordeid
Summer
School on Motivic Homotopy Theory
Bjom Ian
Dundas
. 1
Preface
. 3
I Basic Properties and Examples
. 5
1
Topologica!
Spacos .
6
1.1
Singular Homology
. 6
1.2
Weak Equivalences
. 8
1.3
Mapping Spaces
. 9
2
Simplicia!
Sots
. 9
2.1
The Category
Δ
. 10
2.2
Simplicial Sets vs.
Topologica!
Spaces
. 12
2.3
Weak Equivalences
. 14
3
Some Const ructions in
S
. 15
4
Simplicia! 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
Simplicia!
Sets
. 22
6.3
Stallie
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)fibratious
. 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
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
Simplicia]
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^Strueture
. 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 Ts
->
Spts
.
62
References
.
Index
.
65
Background from Algebraic Geometry
Marc
Levine
. "■*
I Elementary Algebraic Geometry
. ' *
1
The Spectrum of a Commutative Ring
. ' *
1.1
Ideals and Spec
.
1.2
The Zariski Topology
.
73
1.3
Fimctorial Properties
. '
1.4
Naive Algebraic Geometry and Hubert'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 Spares
. 81
2.1
Presheaves and Sheaves
ou 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/,.
. 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
étale
Morphisius
.
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
.
Ill
6.5
Blowing Up a Subscheme
.112
II 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 Abeliau 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
Ame 0stvœr
.147
1
Introduction
.148
2
Motivic Stable Homotopy Theory
.148
2.1
Spaces
.148
2.2
The Motivic s-Stable Homotopy Category SH*'
(к)
.150
2.3
The Motivic Stable Homotopy Category SH(fc)
.153
3
Cohomology Theories
.162
3.1
The Motivic Eilenberg-MacLaiie 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 |
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| genre | (DE-588)1071861417 Konferenzschrift 2002 Nordfjordeid gnd-content |
| genre_facet | Konferenzschrift 2002 Nordfjordeid |
| geographic | Nordfjordeid <2002> swd |
| geographic_facet | Nordfjordeid <2002> |
| id | DE-604.BV023167402 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:56:43Z |
| indexdate | 2024-07-09T21:12:07Z |
| institution | BVB |
| isbn | 9783540458951 3540458956 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016351885 |
| oclc_num | 180948078 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-11 DE-384 DE-19 DE-BY-UBM |
| owner_facet | DE-355 DE-BY-UBR DE-11 DE-384 DE-19 DE-BY-UBM |
| physical | X, 220 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Springer |
| record_format | marc |
| series2 | Universitext |
| spelling | Motivic homotopy theory lectures at a Summer School in Nordfjordeid, Norway, August 2002 Bjørn Ian Dundas ... Berlin [u.a.] Springer 2007 X, 220 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext 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. Homotopie Homotopietheorie swd Homotopy theory Homotopietheorie (DE-588)4128142-1 gnd rswk-swf Nordfjordeid <2002> swd (DE-588)1071861417 Konferenzschrift 2002 Nordfjordeid gnd-content Homotopietheorie (DE-588)4128142-1 s DE-604 Dundas, Bjørn Ian Sonstige oth Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016351885&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Motivic homotopy theory lectures at a Summer School in Nordfjordeid, Norway, August 2002 Homotopie Homotopietheorie swd Homotopy theory Homotopietheorie (DE-588)4128142-1 gnd |
| subject_GND | (DE-588)4128142-1 (DE-588)1071861417 |
| title | Motivic homotopy theory lectures at a Summer School in Nordfjordeid, Norway, August 2002 |
| title_auth | Motivic homotopy theory lectures at a Summer School in Nordfjordeid, Norway, August 2002 |
| title_exact_search | Motivic homotopy theory lectures at a Summer School in Nordfjordeid, Norway, August 2002 |
| title_exact_search_txtP | Motivic homotopy theory lectures at a Summer School in Nordfjordeid, Norway, August 2002 |
| title_full | Motivic homotopy theory lectures at a Summer School in Nordfjordeid, Norway, August 2002 Bjørn Ian Dundas ... |
| title_fullStr | Motivic homotopy theory lectures at a Summer School in Nordfjordeid, Norway, August 2002 Bjørn Ian Dundas ... |
| title_full_unstemmed | Motivic homotopy theory lectures at a Summer School in Nordfjordeid, Norway, August 2002 Bjørn Ian Dundas ... |
| title_short | Motivic homotopy theory |
| title_sort | motivic homotopy theory lectures at a summer school in nordfjordeid norway august 2002 |
| title_sub | lectures at a Summer School in Nordfjordeid, Norway, August 2002 |
| topic | Homotopie Homotopietheorie swd Homotopy theory Homotopietheorie (DE-588)4128142-1 gnd |
| topic_facet | Homotopie Homotopietheorie Homotopy theory Nordfjordeid <2002> Konferenzschrift 2002 Nordfjordeid |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016351885&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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