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Cotangent complexes and deformations in ringed topoi

Homotopical algebra in a ringed topos and the cohomology of diagrams, as foundations for the cotangent complex of morphisms of ringed topoi: homotopy sheaves and the free abelian Whitehead theorem; derived functors of non-additive functors of complexes, the Deligne-Lazard theorem and the shift formulas; décalage, nerves of internal categories and the homotopy type of a category; strict simplicial objects and Deligne's contractibility theorem; derived colimits; fibred topoi, their total topoi, direct images and modules; the cohomology of total topoi and of simplicial topoi, and cohomological descent; the second total topos and local systems; left derived functors of left adjoints; a duality formula for diagrams; equivariant sheaves on simplicial topoi, the nerves of induced and coinduced modules, and the cohomology and Ext groups of equivariant modules; spectral diagrams, the stabilization of non-additive functors and stable derived functors; simplicial rings and modules, their calculus of fractions, triangles, Ext groups, derived tensor products and their spectral sequences; differential graded modules in a topos and their comparison with simplicial modules; graded monoids, the nerves of their actions, graded rings and the spectra of modules; diagonal rings and the adjunction formulas; the stable ring of a ring and the Mac Lane resolution; Ext groups of modules as cohomology of spectral nerves; quasi-coherent and perfect complexes on big fppf and fpqc sites; deformations of diagrams of schemes; equivariant cotangent complexes; equivariant extensions and deformations; flat group schemes as local complete intersections; torsors, co-Lie complexes and the Atiyah class; deformations of torsors; the co-Lie complex of a group scheme; the adjoint co-Lie complex of a group scheme; deformations of flat group schemes and of homomorphisms; Lie complexes of commutative group schemes; Lie complexes of module schemes; deformations of module schemes; graded modules and the graded cotangent complex; the Atiyah class of a module; deformations of modules and of their morphisms; filtered derived categories; filtered tensor products on a ringed topos; filtered internal Hom and the formulas of Cartan; perfect filtered complexes and duality; traces and cup products of perfect filtered complexes; filtered and graded powers of complexes; filtered Koszul complexes and the shift isomorphisms; diagonal maps of symmetric and exterior powers; L-series of endomorphisms of perfect complexes; the Atiyah class of filtered complexes; Chern classes of perfect complexes; Chern classes with supports from partial connections; weakly regular ideals; connectivity of powers of simplicial ideals; the fundamental spectral sequence; weak complete intersections; weakly regular ideals of noetherian rings; regular local rings and complete intersections; the arrow topos and relative Ext groups; deforming a morphism together with its target; de Rham complexes and Lie algebroids; formal categories with divided powers and the Čech–Alexander complex; the de Rham complex of a formal category with divided powers; décalage and the formal Poincaré lemma; cohomology of flat formal categories with divided powers; cotangent complexes of simplicial rings; graded simplicial rings and the graded transitivity triangle; localization morphisms and the cotangent complex; the derived de Rham complex; derived de Rham complexes in characteristic p; the completed derived de Rham complex of a complete intersection; the simplicial derived category of a topos; local fibrations and Kan's subdivision functor; simplicial groups and right fractions; hypercoverings and Ext groups of free simplicial modules; Giraud's characterization of topoi; deforming the target of a morphism of diagrams, equivariantly and for group and module schemes.

87 lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

  1. Homotopy sheaves and the free abelian Whitehead theorem
  2. Non-additive functors of complexes in a ringed topos
  3. Koszul complexes and the shift formulas
  4. Décalage, nerves and the homotopy type of a category
  5. Strict simplicial objects and Deligne's contractibility theorem
  6. Derived colimits over a small category
  7. Fibred topoi and their total topoi
  8. Morphisms of fibred topoi and modules on total topoi
  9. Cohomology of total topoi
  10. Cohomology of simplicial topoi and cohomological descent
  11. Cohomology of the second total topos and local systems
  12. Left derived functors of left adjoints on total topoi
  13. Derived left adjoints on the second total topos and local inductive limits
  14. A duality formula for diagrams in a topos
  15. Equivariant sheaves on simplicial topoi
  16. Giraud's characterization of topoi
  17. Nerves of induced and coinduced modules
  18. Cohomology of equivariant modules
  19. Mixed Hom complexes and Ext groups of equivariant modules
  20. Multisimplicial objects and spectral diagrams
  21. Stabilization of non-additive functors
  22. Stable derived functors
  23. Simplicial rings and modules in a topos
  24. Triangles and Ext groups of simplicial modules
  25. Derived tensor products of simplicial modules
  26. Spectral sequences for derived tensor products
  27. Differential graded modules in a topos
  28. Graded monoids and the nerves of their actions
  29. Graded rings and the spectra of modules
  30. Diagonal rings and adjunction formulas
  31. The stable ring of a ring and the Mac Lane resolution
  32. Ext groups of modules as cohomology of spectral nerves
  33. Quasi-coherent and perfect complexes on big fppf sites
  34. Deformations of diagrams of schemes
  35. Equivariant cotangent complexes
  36. Equivariant extensions and deformations
  37. Flat group schemes are local complete intersections
  38. Torsors, co-Lie complexes and the Atiyah class
  39. Deformations of torsors
  40. The co-Lie complex of a group scheme
  41. The adjoint co-Lie complex of a group scheme
  42. Deformations of flat group schemes and of homomorphisms
  43. Lie complexes of commutative group schemes
  44. Lie complexes of module schemes
  45. Deformations of module schemes
  46. Graded modules and the graded cotangent complex
  47. The Atiyah class of a module
  48. Deformations of modules and of their morphisms
  49. Filtered derived categories
  50. Filtered tensor products on a ringed topos
  51. Filtered internal Hom and the formulas of Cartan
  52. Perfect filtered complexes and duality
  53. Traces and cup products of perfect filtered complexes
  54. Filtered and graded powers of complexes
  55. Filtered Koszul complexes and the shift isomorphisms
  56. Diagonal maps of symmetric and exterior powers
  57. L-series of endomorphisms of perfect complexes
  58. The Atiyah class of filtered complexes
  59. Chern classes of perfect complexes
  60. Chern classes with supports from partial connections
  61. Cotangent complexes of simplicial rings
  62. Graded simplicial rings and the graded transitivity triangle
  63. Localization morphisms and the cotangent complex
  64. Weakly regular ideals
  65. Connectivity of powers of simplicial ideals
  66. The fundamental spectral sequence
  67. Weak complete intersections
  68. Weakly regular ideals of noetherian rings
  69. Regular local rings and complete intersections
  70. The arrow topos and relative Ext groups
  71. Deforming a morphism together with its target
  72. De Rham complexes and Lie algebroids
  73. Formal categories with divided powers and the Čech–Alexander complex
  74. The de Rham complex of a formal category with divided powers
  75. Décalage and the formal Poincaré lemma
  76. Cohomology of flat formal categories with divided powers
  77. The derived de Rham complex
  78. Derived de Rham complexes in characteristic p
  79. The completed derived de Rham complex of a complete intersection
  80. The simplicial derived category of a topos
  81. Local fibrations and Kan's subdivision functor
  82. Simplicial groups and right fractions
  83. Hypercoverings and Ext groups of free simplicial modules
  84. Deforming the target of a morphism of diagrams
  85. Relative cohomology of classifying topoi
  86. Filtered transitivity triangles and the filtered Atiyah triangle
  87. Truncated nerves and the level filtration

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites