Noncommutative motives:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2015]
|
| Series: | University lecture series / American Mathematical Society
volume 63 |
| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Physical Description: | x, 114 Seiten Illustrationen |
| ISBN: | 9781470423971 |
Staff View
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Record in the Search Index
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| adam_text | The theory of motives began in the eariy 1960s when Gxothendieck envisioned the existence
of a ‘^universal cohomology theory of algebraic varieties *. The theory of noncommutative
motives is more recent It began in the 1980s when the Moscow school (Beilinson, Bondal,
Kapranov, Mania, and others) began the study of algebraic varieties via their derived
categories of coherent sheaves, and continued in the 2000s when Kontsevich conjectured
the existence of a “universal invariant of noncommutative algebraic varieties”.
This book, prefaced by Yuri I Mania, gives a rigorous overview of some of the main
advances in the theory of noncommutative motives. It is divided into three main parts. The
first part, which is of independent interest, is devoted to the study of DG categories from
a homotopical viewpoint. The second part, written with an emphasis on examples and
applications, covers the theory of noncommutative pure motives, noncommutative stan-
dard conjectures, noncommutative motivic Galois groups, and also the relations between
these notions and their commutative counterparts. The last part is devoted to the theory of
nopcommutative mixed motives. The rigorous formalization of this latter theory requires
the language of Grothendieck derivators, which, for die reader s convenience, is revised
in a brief appendix.
Contents
Preface lx
Introduction 1
Chapter 1. Differential graded categories 3
1.1* Definitions 3
1.2* Quasi-equivalences S
1.3. Dxinfeld’s D G quotient 11
1.4. Fretriangulated equivalences 13
1.5. Bcndal-KapranovJ8 pretriangulated envelope 14
1.6* Marita equivalences IS
1.7. Kontsevidbt*s smooth proper dg categories 16
Chapter 2. Additive invariants 21
2.1. Definitions 21
2.2. Examples 22
2.3. Universal additive invariant 26
2.4. Computations 29
2.5. Lefechetz’s fixed point formula 32
Chapter 3. Background on pure motives 35
Chapter 4* Noncomxnutative pure motives 41
4.1. NoxKoaunutative Chow motives 41
4.2. Relation with Chow motives 41
4.3. Relation with Merionjev-Ptoin^ motives 46
4*4. Noscopmp«^ ·· 47
4.5. NomnmmuUtive homological motives 47
4.6. Nonoommutativc numerical motives 48
4.7. Kontoevich’s noncommutative numerical motives 48
4.8. Sexmoimplxaty 50
4.9. Hcmconunutative Artin motives 52
4.10. FVmctoraiity 53
4.11. Well restriction 54
Chapter 5. Noncommutative (standard) conjectures 57
5.1. Standard conjecture of type 57
5.2. Standard conjecture of type 58
5.3. Noncommutative nilpotence conjecture 59
5.4. Kimuia^finitenees 59
5.5. All together 60
vM
CONTENTS
Viii
Chapter 6. Noncommutative motivic Galois groups 63
6.1. Definitions 63
6.2. Relation with motivic Galois groups 65
6.3. Unconditional version 65
6.4. Base-change short exact sequence 66
Chapter 7. Jacobians of noncommutative Chow motives 69
Chapter 8. Localizing invariants 71
8.1. Definitions 71
8.2. Examples 72
8.3. Universal localizing invariant 73
8.4. Additivity 77
8.5. A1֊homotopy 79
8.6. Algebraic iL-theory 82
8.7. Witt vectors 84
8.8. Natural transformations 85
Chapter 9. Noncommutative mixed motives 87
9.1. Definitions 87
9.2. Relation with noncommutative Chow motives 89
9.3. Weight structure 89
9.4. Relation with Morel-Voevodsky’s motivic homotopy theory 90
9.5. Relation with Voevodsky’s geometric mixed motives 91
9.6. Relation with Levine’s mixed motives 93
9.7. Noncommutative mixed Artin motives 93
9.8. Kimura-finiteness 94
9.9. Coefficients 95
Chapter 10. Noncommutative motivic Hopf dg algebras 97
10.1. Definitions 97
10.2. Base-change short exact sequence 98
Appendix A. Grothendieck derivators 99
A.l. Definitions 99
A.2. Left Bousfield localization 101
A.3. Stabilization and spectral enrichment 102
A.4. Filtered homotopy colimits 102
A.5. Symmetric monoidal structures 103
Bibliography 105
Index 113
|
| any_adam_object | 1 |
| author | Tabuada, Gonçalo 1979- |
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| author_variant | g t gt |
| building | Verbundindex |
| bvnumber | BV043045520 |
| classification_rvk | SI 165 SK 240 SK 320 |
| classification_tum | MAT 143f |
| ctrlnum | (OCoLC)932080826 (DE-599)BVBBV043045520 |
| dewey-full | 516.3/5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/5 |
| dewey-search | 516.3/5 |
| dewey-sort | 3516.3 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV043045520 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:15:53Z |
| institution | BVB |
| isbn | 9781470423971 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028470048 |
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| physical | x, 114 Seiten Illustrationen |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | American Mathematical Society |
| record_format | marc |
| series2 | University lecture series / American Mathematical Society |
| spelling | Tabuada, Gonçalo 1979- Verfasser (DE-588)1079187995 aut Noncommutative motives Gonçalo Tabuada Providence, Rhode Island American Mathematical Society [2015] © 2015 x, 114 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier University lecture series / American Mathematical Society volume 63 Nichtkommutative Geometrie (DE-588)4171742-9 gnd rswk-swf Motiv Mathematik (DE-588)4197596-0 gnd rswk-swf Motiv Mathematik (DE-588)4197596-0 s Nichtkommutative Geometrie (DE-588)4171742-9 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-2627-9 American Mathematical Society University lecture series volume 63 (DE-604)BV004153846 63 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028470048&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028470048&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tabuada, Gonçalo 1979- Noncommutative motives Nichtkommutative Geometrie (DE-588)4171742-9 gnd Motiv Mathematik (DE-588)4197596-0 gnd |
| subject_GND | (DE-588)4171742-9 (DE-588)4197596-0 |
| title | Noncommutative motives |
| title_auth | Noncommutative motives |
| title_exact_search | Noncommutative motives |
| title_full | Noncommutative motives Gonçalo Tabuada |
| title_fullStr | Noncommutative motives Gonçalo Tabuada |
| title_full_unstemmed | Noncommutative motives Gonçalo Tabuada |
| title_short | Noncommutative motives |
| title_sort | noncommutative motives |
| topic | Nichtkommutative Geometrie (DE-588)4171742-9 gnd Motiv Mathematik (DE-588)4197596-0 gnd |
| topic_facet | Nichtkommutative Geometrie Motiv Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028470048&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028470048&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV004153846 |
| work_keys_str_mv | AT tabuadagoncalo noncommutativemotives |