THE COTANGENT COMPLEX Contents

Contents 1. Introduction 1 2. Advice for the reader 2 3. The cotangent complex of a ring map 2 4. Simplicial resolutions and derived lower shriek 3 5. Constructing a resolution 6 6. Functoriality 11 7. The fundamental triangle 13 8. Localization andétale ring maps 17 9. Smooth ring maps 19 10. Comparison with the naive cotangent complex 19 11. A spectral sequence of Quillen 21 12. Comparison with Lichtenbaum-Schlessinger 24 13. The cotangent complex of a local complete intersection 27 14. Tensor products and the cotangent complex 28 15. Deformations of ring maps and the cotangent complex 30 16. The Atiyah class of a module 31 17. The cotangent complex 31 18. The Atiyah class of a sheaf of modules 34 19. The cotangent complex of a morphism of ringed spaces 35 20. Deformations of ringed spaces and the cotangent complex 35 21. The cotangent complex of a morphism of ringed topoi 36 22. Deformations of ringed topoi and the cotangent complex 37 23. The cotangent complex of a morphism of schemes 38 24. The cotangent complex of a scheme over a ring 39 25. The cotangent complex of a morphism of algebraic spaces 41 26. The cotangent complex of an algebraic space over a ring 42 27. Fibre products of algebraic spaces and the cotangent complex 44 28. Other chapters 45 References 46