Lectures on the theory of pure motives:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2013]
|
| Schriftenreihe: | University lecture series
Volume 61 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | ix, 149 Seiten Diagramme |
| ISBN: | 9780821894347 |
Internformat
MARC
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| 100 | 1 | |a Murre, Jacob P. |d 1929- |e Verfasser |0 (DE-588)1036574989 |4 aut | |
| 245 | 1 | 0 | |a Lectures on the theory of pure motives |c Jacob P. Murre ; Jan Nagel ; Chris A. M. Peters |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2013] | |
| 264 | 4 | |c © 2013 | |
| 300 | |a ix, 149 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a University lecture series |v Volume 61 | |
| 650 | 4 | |a Motives (Mathematics) | |
| 650 | 7 | |a Algebraic geometry -- Cycles and subschemes -- (Equivariant) Chow groups and rings; motives |2 msc | |
| 650 | 7 | |a Algebraic geometry -- Cycles and subschemes -- Algebraic cycles |2 msc | |
| 650 | 7 | |a Commutative algebra -- Homological methods -- Grothendieck groups, K-theory |2 msc | |
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| 700 | 1 | |a Nagel, Jan |d 1983- |e Verfasser |0 (DE-588)140849505 |4 aut | |
| 700 | 1 | |a Peters, Chris |d 1949- |e Verfasser |0 (DE-588)1036575527 |4 aut | |
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Datensatz im Suchindex
| _version_ | 1804150252118212608 |
|---|---|
| adam_text | Contents
Introduction
Chapter
1.
Algebraic Cycles and Equivalence Relations
1
1.1.
Algebraic Cycles
1
1.2.
Equivalence Relations
3
Appendix A: Survey of Some of the Main Results on Chow Groups
13
А
-l.
Divisors
13
A-2. Classical Results on the
Picard
and
Albanese
Varieties
14
A-3. Higher Codimension
16
Appendix B: Proof of the Theorem of
Voisin-
Voevodsky
19
Chapter
2.
Motives: Construction and First Properties
23
2.1.
Correspondences
23
2.2.
(Pure) Motives
25
2.3.
Examples
27
2.4.
Further Remarks and Properties
28
2.5.
Chow Groups and Cohomology Groups of Motives
29
2.6.
Relations Between the Various Categories of Motives
30
2.7.
Motives of Curves
30
2.8.
Manin s Identity Principle
33
Chapter
3.
On Grothendieck s Standard Conjectures
35
3.1.
The Standard Conjectures
35
3.2.
Jannsen s Theorem
39
Chapter
4.
Finite Dimensionality of Motives
43
4.1.
Introduction
43
4.2.
Preliminaries on Group Representations
43
4.3.
Action of the Symmetric Group on Products
44
4.4.
Dimension of Motives
46
4.5.
The Sign Conjecture and Finite Dimensionality
47
4.6.
Curves
47
Chapter
5.
Properties of Finite Dimensional Motives
51
5.1.
Sums and Tensor Products
51
5.2.
Smash Nilpotence of Morphisms
54
5.3.
Morphisms Between Finite Dimensional Motives
55
5.4.
Surjective Morphisms and Finite Dimensionality
56
5.5.
Finite Dimensionality and Nilpotence
59
iv CONTENTS
5.6. Applications and
Conjectures
61
Chapter
6. Chow-Künneth
Decomposition; the
Picard and Albanese
Motive
67
6.1.
The Künneth
and the
Chow-Künneth
Decomposition
67
6.2.
Picard
and
Albanese
Motives
70
6.3.
The Case of Surfaces
78
Appendix C:
Chow-Künneth
Decomposition in a Special Case
83
Chapter
7.
On the Conjectural Bloch-Beilinson Filtration
85
Introduction
85
7.1.
Bloch-Beilinson Filtration
85
7.2.
Another Set of Conjectures
86
7.3.
Some Evidence for the Conjectures; an Overview
88
7.4.
Threefolds of Type
S x C
89
7.5.
On Some Results of Jannsen and Their Consequences
93
7.6.
Products of Two Surfaces
96
Relations Between the Conjectures
103
Chapter
8.
Relative
Chow-Künneth
Decomposition
105
8.1.
Relative Motives
105
8.2.
Perverse Sheaves and the Decomposition Theorem
109
8.3.
Relative
Chow-Künneth
Decomposition
112
8.4.
Further Examples
113
8.5.
From Relative to Absolute
115
Appendix D: Surfaces Fibered over a Curve
119
Chapter
9.
Beyond Pure Motives
123
Two Auxilliary Notions
124
9.1.
Motivic
Euler
Characteristics
124
9.2.
Motives Via Weights
127
9.3.
Voevodsky s Triangulated Category of Motives
131
Appendix E: The Category of Motivic Complexes
135
Bibliography
139
Index of Notation
145
Index
147
The theory of motives was created by Grothendieck in the
1960s
as he searched for a
universal cohomology theory for algebraic varieties. The theory of pure motives is well
established as far as the construction is concerned. Pure motives are expected to have a
number of additional properties predicted by Grothendieck s standard conjectures, but
these conjectures remain wide open. The theory for mixed motives is still incomplete.
This book deals primarily with the theory of pure motives. The exposition begins with the
fundamentals: Grothendieck s construction of the category of pure motives and examples.
Next, the standard conjectures and the famous theorem of Jannsen on the category of the
numerical motives are discussed. Following this, the important theory of finite dimen¬
sionality is covered. The concept of Chow-Kiinneth decomposition is introduced, with
discussion of the known results and the related conjectures, in particular the conjectures
of Bloch-Beilinson type. We finish with a chapter on relative motives and a chapter giving
a short introduction to Voevodsky s theory of mixed motives.
For additional information
and updates on this book, visit
www.ams.org/bookpages/ulect-ó
I
ISBN
978-0-8218-9434-7
9І780821
894347
ULECT/ei
|
| any_adam_object | 1 |
| author | Murre, Jacob P. 1929- Nagel, Jan 1983- Peters, Chris 1949- |
| author_GND | (DE-588)1036574989 (DE-588)140849505 (DE-588)1036575527 |
| author_facet | Murre, Jacob P. 1929- Nagel, Jan 1983- Peters, Chris 1949- |
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| dewey-full | 516.3/5 |
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| dewey-ones | 516 - Geometry |
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| discipline | Mathematik |
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| institution | BVB |
| isbn | 9780821894347 |
| language | English |
| lccn | 2012048281 |
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| spelling | Murre, Jacob P. 1929- Verfasser (DE-588)1036574989 aut Lectures on the theory of pure motives Jacob P. Murre ; Jan Nagel ; Chris A. M. Peters Providence, Rhode Island American Mathematical Society [2013] © 2013 ix, 149 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier University lecture series Volume 61 Motives (Mathematics) Algebraic geometry -- Cycles and subschemes -- (Equivariant) Chow groups and rings; motives msc Algebraic geometry -- Cycles and subschemes -- Algebraic cycles msc Commutative algebra -- Homological methods -- Grothendieck groups, K-theory msc Motiv Mathematik (DE-588)4197596-0 gnd rswk-swf Motiv Mathematik (DE-588)4197596-0 s DE-604 Nagel, Jan 1983- Verfasser (DE-588)140849505 aut Peters, Chris 1949- Verfasser (DE-588)1036575527 aut Erscheint auch als Online-Ausgabe 978-1-4704-1039-1 University lecture series Volume 61 (DE-604)BV004153846 61 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025925854&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025925854&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Murre, Jacob P. 1929- Nagel, Jan 1983- Peters, Chris 1949- Lectures on the theory of pure motives University lecture series Motives (Mathematics) Algebraic geometry -- Cycles and subschemes -- (Equivariant) Chow groups and rings; motives msc Algebraic geometry -- Cycles and subschemes -- Algebraic cycles msc Commutative algebra -- Homological methods -- Grothendieck groups, K-theory msc Motiv Mathematik (DE-588)4197596-0 gnd |
| subject_GND | (DE-588)4197596-0 |
| title | Lectures on the theory of pure motives |
| title_auth | Lectures on the theory of pure motives |
| title_exact_search | Lectures on the theory of pure motives |
| title_full | Lectures on the theory of pure motives Jacob P. Murre ; Jan Nagel ; Chris A. M. Peters |
| title_fullStr | Lectures on the theory of pure motives Jacob P. Murre ; Jan Nagel ; Chris A. M. Peters |
| title_full_unstemmed | Lectures on the theory of pure motives Jacob P. Murre ; Jan Nagel ; Chris A. M. Peters |
| title_short | Lectures on the theory of pure motives |
| title_sort | lectures on the theory of pure motives |
| topic | Motives (Mathematics) Algebraic geometry -- Cycles and subschemes -- (Equivariant) Chow groups and rings; motives msc Algebraic geometry -- Cycles and subschemes -- Algebraic cycles msc Commutative algebra -- Homological methods -- Grothendieck groups, K-theory msc Motiv Mathematik (DE-588)4197596-0 gnd |
| topic_facet | Motives (Mathematics) Algebraic geometry -- Cycles and subschemes -- (Equivariant) Chow groups and rings; motives Algebraic geometry -- Cycles and subschemes -- Algebraic cycles Commutative algebra -- Homological methods -- Grothendieck groups, K-theory Motiv Mathematik |
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