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.

Theorems and assumptions · Reader and editable sources · Course record

Cotangent complexes and deformations in ringed topoi

Prerequisites