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: D¯-stacks ........................... 4
Complicial algebraic geometry: D-stacks ......................... 6
Brave new algebraic geometry: S-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
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 D¯-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 S-stack ....... 203
Appendix A. Classifying spaces of model categories ............ 207
Appendix B. Strictification ................................... 211
Appendix C. Representability criterion (after J. Lurie) ....... 215
Bibliography .................................................. 219
Index ......................................................... 223
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