Homotopical algebraic geometry II: geometric stacks and applications
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
Am. Math. Soc.
2008
|
| Series: | Memoirs of the American Mathematical Society
902 = 193,2 |
| Links: | http://www.gbv.de/dms/goettingen/561616604.pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020138033&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | IX, 224 S. |
| ISBN: | 9780821840993 |
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Record in the Search Index
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| adam_text | Contents
Abstract ix
Introduction 1
Reminders on abstract algebraic geometry 1
The setting 2
Linear and commutative algebra in a symmetric monoidal model category 2
Geometric stacks 3
Infinitesimal theory 4
Higher Artin stacks (after C. Simpson) 4
Derived algebraic geometry: Z?~-stacks 4
Complicial algebraic geometry: .D-stacks 6
Brave new algebraic geometry: ^-stacks 6
Relations with other works 7
Acknowledgments 8
Notations and conventions 9
Part 1. General theory of geometric stacks 11
Introduction to Part 1 13
Chapter 1.1. Homotopical algebraic context 15
Chapter 1.2. Preliminaries on linear and commutative algebra in an HA context 25
1.2.1. Derivations and the cotangent complex 25
1.2.2. Hochschild homology 29
1.2.3. Finiteness conditions 30
1.2.4. Some properties of modules 35
1.2.5. Formal coverings 36
1.2.6. Some properties of morphisms 37
1.2.7. Smoothness 41
1.2.8. Infinitesimal lifting properties 41
1.2.9. Standard localizations and Zariski open immersions 43
1.2.10. Zariski open immersions and perfect modules 47
1.2.11. Stable modules 49
1.2.12. Descent for modules and stable modules 54
1.2.13. Comparison with the usual notions 57
Chapter 1.3. Geometric stacks: Basic theory 61
1.3.1. Reminders on model topoi 61
1.3.2. Homotopical algebraic geometry context 65
1.3.3. Main definitions and standard properties 76
1.3.4. Quotient stacks 80
1.3.5. Quotient stacks and torsors 83
vii
1.3.6. Properties of morphisms 86
1.3.7. Quasi-coherent modules, perfect modules and vector bundles 88
Chapter 1.4. Geometric stacks: Infinitesimal theory 97
1.4.1. Tangent stacks and cotangent complexes 97
1.4.2. Obstruction theory 105
1.4.3. Artin conditions 109
Part 2. Applications 121
Introduction to Part 2 123
Chapter 2.1. Geometric n-stacks in algebraic geometry (after C. Simpson) 129
2.1.1. The general theory 129
2.1.2. Comparison with Artin s algebraic stacks 132
Chapter 2.2. Derived algebraic geometry 135
2.2.1. The HA context 135
2.2.2. Flat, smooth, etale and Zariski open morphisms 139
2.2.3. The HAG context: Geometric £ --stacks 151
2.2.4. Truncations 154
2.2.5. Infinitesimal criteria for smooth and etale morphisms 159
2.2.6. Some examples of geometric D~-stacks 164
2.2.6.1. Local systems 164
2.2.6.2. Algebras over an operad 170
2.2.6.3. Mapping D~-stacks 173
Chapter 2.3. Complicial algebraic geometry 177
2.3.1. Two HA contexts 177
2.3.2. Weakly geometric D-stacks 181
2.3.3. Examples of weakly geometric D-stacks 182
2.3.3.1. Perfect modules 182
2.3.3.2. The D-stacks of dg-algebras and dg-categories 183
2.3.4. Geometric .D-stacks 187
2.3.5. Examples of geometric .D-stacks 188
2.3.5.1. D~-stacks and D-stacks 188
2.3.5.2. CW-perfect modules 188
2.3.5.3. CW-dg-algebras 192
2.3.5.4. The D-stack of negative CW-dg-categories 193
Chapter 2.4. Brave new algebraic geometry 199
2.4.1. Two HAG contexts 199
2.4.2. Elliptic cohomology as a Deligne-Mumford 5-stack 203
Appendix A. Classifying spaces of model categories 207
Appendix B. Strictification 211
Appendix C. Represent ability criterion (after J. Lurie) 215
Bibliography 219
Index 223
|
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| illustrated | Not Illustrated |
| indexdate | 2024-12-20T14:29:53Z |
| institution | BVB |
| isbn | 9780821840993 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020138033 |
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| physical | IX, 224 S. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
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| publisher | Am. Math. Soc. |
| record_format | marc |
| series2 | Memoirs of the American Mathematical Society |
| spellingShingle | To∞en, Bertrand Vezzosi, Gabriele Homotopical algebraic geometry II geometric stacks and applications |
| title | Homotopical algebraic geometry II geometric stacks and applications |
| title_auth | Homotopical algebraic geometry II geometric stacks and applications |
| title_exact_search | Homotopical algebraic geometry II geometric stacks and applications |
| title_full | Homotopical algebraic geometry II geometric stacks and applications Bertrand Toën ; Gabriele Vezzosi |
| title_fullStr | Homotopical algebraic geometry II geometric stacks and applications Bertrand Toën ; Gabriele Vezzosi |
| title_full_unstemmed | Homotopical algebraic geometry II geometric stacks and applications Bertrand Toën ; Gabriele Vezzosi |
| title_short | Homotopical algebraic geometry II |
| title_sort | homotopical algebraic geometry ii geometric stacks and applications |
| title_sub | geometric stacks and applications |
| url | http://www.gbv.de/dms/goettingen/561616604.pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020138033&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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