Results on homotopical algebra in topoi and on diagrams of topoi

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Whitehead for simplicial sets — Theorem 3.1. An n-quasi-isomorphism of simplicial sets induces isomorphisms on homology below n and a surjection in degree n.
• Free abelian sheaves, enough points — Theorem 4.1. On a topos with enough points, the free abelian functor carries n-quasi-isomorphisms of simplicial sheaves to n-quasi-isomorphisms.
• Free abelian sheaves, homomorphisms — Theorem 7.1. On any topos, the free abelian functor carries n-quasi-isomorphisms of simplicial abelian sheaves to n-quasi-isomorphisms; no hypothesis on points.
• The Dold-Puppe bound — Proposition 2.1. A functor of degree at most k, extended to complexes through the Dold-Kan correspondence, carries complexes in degrees at most b to normalized complexes in degrees at most kb.
• Homotopy invariance — Proposition 2.3. The extension of a non-additive functor to complexes carries homotopic maps to homotopic maps.
• The Deligne-Lazard theorem — Theorem 3.2. A module over a ring of a topos is flat exactly when it is the local inductive limit of a local system of finite free modules.
• The left derived functor — Theorem 4.4. A functor of finite degree has a left derived functor on bounded-above complexes, computed by flat resolutions and compatible with local inductive limits of uniformly bounded complexes.
• Koszul sequences — Theorem 2.1. For an exact sequence of flat modules, the two Koszul complexes give exact sequences relating symmetric, exterior and divided powers.
• The shift formulas — Theorem 3.2. LS^n(E[1]) is (LΛ^n E)[n] and LΛ^n(E[1]) is (LΓ^n E)[n], canonically, for E in the bounded-above derived category; Proposition 3.4 gives the compatibility with products, without sign for the first formula and with the sign (-1)^{mn} for the second.
• Décalage is homotopically trivial — Proposition 1.1. Both décalages of a simplicial object, in any category, are homotopy equivalent to the constant object on their augmentation.
• The total décalage — Proposition 1.3. The two augmentations of the diagonal of the total décalage are homotopic; Example 1.4 shows that they are not homotopy equivalences in general.
• Nerves of internal categories — Proposition 2.4. Nine equivalent conditions characterizing nerves of internal categories in a category with fibre products, including the Segal conditions and cartesian décalage squares; Remark 2.5 shows two weaker conditions that do not suffice.
• Groupoids — Proposition 2.6. Seven equivalent conditions for an internal category to be a groupoid, in terms of its nerve and its décalages.
• Actions of monoids — Proposition 2.9. Nerves of monoid actions are exactly the simplicial objects over the nerve of the monoid that split as a product with their vertices.
• Quillen's comparison — Theorem 3.2. The first and last vertex functors from the category of simplices are weak equivalences, and so is the last-vertex map of a simplicial set; Corollary 3.4: a category and its opposite have the same homotopy type.
• Simplices of strict simplicial sets — Theorem 2.3. The nerve of the category of simplices of a strict simplicial set has the homology of its alternating face complex, naturally; Corollary 2.4 computes the truncated categories of injective ordinals.
• Deligne's theorem — Theorem 4.2. For every p, the inclusion of the ordinals with at least p+2 elements into those with at least p+1 elements, with injective maps, induces a quasi-isomorphism of chains; Corollary 4.3: these categories are weakly contractible.
• Derived colimits — Theorem 2.3. The colimit over a small category has a left derived functor on bounded-above complexes, computed by the simplicial replacement; Corollary 2.4 gives its adjunction with constant diagrams.
• Colimits over injective ordinals — Theorem 3.3. Over the category of finite ordinals and injective maps, a complex whose cohomology is eventually locally constant has the cohomology of its value at any stage from which it is locally constant.
• Exactness of the colimit over injective ordinals — Theorem 4.8. Over the category of finite ordinals and injective maps every diagram is colimit-acyclic, so the colimit is exact and the derived colimit of a bounded-above complex is its termwise colimit.
• The total topos — Theorem 4.2. The total topos of a fibred topos is the category of sheaves on an explicit total site.
• The second total topos — Theorem 4.3. For a good fibred topos whose left adjoints preserve fibre products, the second total category is a topos; Proposition 5.1 identifies it, for a diagram in a topos, with the diagrams over it.
• Direct images — Proposition 2.1. A morphism of fibred topoi induces a morphism of total topoi whose direct image is a limit over comma categories; global sections of the total topos are the limit of the global sections of the levels.
• Modules on total topoi — Proposition 4.2. Modules over a ring of a total topos are compatible families of modules; tensor products are computed levelwise, and so are internal Homs from a cartesian module.
• Diagrams of diagrams — Theorem 5.7. The second total topos of an n-diagram in a topos is the topos of diagrams of its type over it, and its first total topos is that of the diagram on the total category of its type.
• Enough levelwise acyclic sheaves — Proposition 1.4. Every abelian sheaf on a total topos is a subsheaf of an injective sheaf whose levels are totally acyclic; for flat transition morphisms, every module is a submodule of an injective module with injective levels.
• Standard resolutions — Proposition 2.1. The two standard resolutions of an abelian sheaf on a total topos are exact and become contractible on the levels.
• Cohomology of total topoi — Theorem 3.5. The derived direct image from a total topos to a topos is computed by the cosimplicial complex of the direct images of the levels, and by the complex of the direct images themselves when the levels are totally acyclic.
• Derived limits — Corollary 3.6. The derived functors of the inverse limit of abelian groups over a small category are the cohomology of its cosimplicial complex; Corollary 3.7 gives the spectral sequence from derived limits of the cohomology of the levels.
• Final objects — Corollary 3.9. Over an index category with a final object, the derived direct image of a total topos is that of the final level.
• Levelwise direct images — Proposition 4.1. Derived direct images under a morphism of fibred topoi over a fixed index category commute with restriction to the levels.
• Simplicial topoi — Theorem 2.1. The derived direct image from a simplicial topos to a topos is computed by the cosimplicial object of the direct images of the levels; Corollary 2.2 gives the spectral sequence from the higher direct images of the levels, and Theorem 2.3 the multisimplicial case.
• The associated simplicial topos — Proposition 3.2. The cohomology of the total topos of any fibred topos is that of an associated simplicial topos.
• Décalage of a simplicial topos — Proposition 4.2. The décalage of a simplicial topos has the cohomology of the level onto which it is augmented.
• Free abelian sheaves and descent — Proposition 5.2. For a simplicial object X of a topos, Rp_*p^*E is RHom(Z(X), E); Corollary 5.3 gives cohomological descent for hypercoverings in every topos, and Corollary 5.4 the invariance under morphisms inducing quasi-isomorphisms of free abelian sheaves.
• The second total topos — Theorem 2.4. For a complex whose levels stay totally acyclic under all inverse images, the derived direct image from the second total topos is that of the associated simplicial object.
• Local systems — Theorem 3.3. Cartesian objects of the two total topoi correspond, and their cohomologies agree; Corollary 3.5 computes the cohomology of simplicial sheaves over a simplicial object.
• Base change — Proposition 4.1. A criterion for derived direct images on the second total topoi to commute with restriction to the levels.
• Left derived functors of u_! — Theorem 3.2. Under an exactness condition on flat modules, the left adjoint of the inverse image of modules on total topoi has a left derived functor on bounded above complexes, computed by standard resolutions of flat complexes, with RHom(Lu_!E, F) = RHom(E, u^*F).
• Derived colimits and constant diagrams — Proposition 4.1. For diagrams of modules over a ring of a topos, the derived colimit and the constant diagram functor satisfy RHom(L colim E, F) = RHom(E, cF).
• Local inductive limits — Proposition 1.2. For a locally filtering diagram in a topos, the left adjoint of the inverse image of modules is the local inductive limit, which is exact.
• Derived left adjoints on the second total topos — Theorem 3.1. Under exactness conditions on flat modules, the left adjoint of the inverse image on the second total topos has a left derived functor, adjoint to the inverse image on RHom complexes and computed levelwise for morphisms over a fixed index category.
• Décalage for left adjoints — Proposition 4.4. The derived colimit of a simplicial module is its chain complex (Lemma 4.1), and the décalage of a simplicial topos is trivial for the derived left adjoint of its augmentation.
• The internal mixed Hom — Definition 1.1. For a diagram in a topos, the internal Hom from modules of the first total topos to modules of the second, with values in the second total topos.
• The duality formula — Theorem 2.3. RHom on the first total topos into a module pulled back from the topos is RGamma of the second total topos of the derived internal mixed Hom.
• The relative duality formula — Theorem 3.2. For a morphism of diagrams u, the canonical map from the internal mixed Hom of Lu_!E to Ru_* of the internal mixed Hom of E is an isomorphism for coefficients pulled back from the topos.
• Equivariant sheaves and their nerves — Proposition 2.2. The category BX of equivariant sheaves on a simplicial topos has colimits and finite limits computed on X_0, and its two nerves into the total topoi are fully faithful.
• Images of the nerves — Proposition 2.3. An object of the first (second) total topos is isomorphic to a nerve exactly when its maps along simplicial maps fixing the last (first) vertex are isomorphisms.
• Nerves of internal categories — Proposition 3.4. The simplicial topos of the nerve of an internal category of a topos is a good two-sided pseudo-category.
• Coinduction — Proposition 4.2. On a pseudo-category, the coinduced sheaf s_*t^*L carries a canonical equivariant structure, and coinduction is right adjoint to the forgetful functor.
• Induction — Proposition 5.1. On a good pseudo-category, the induced sheaf t_!s^*L carries a canonical equivariant structure, and induction is left adjoint to the forgetful functor.
• Equivariant modules — Proposition 6.2. Equivariant modules form an abelian category with enough injectives, the direct summands of coinduced injectives; on a good pseudo-category every module is a quotient of an induced free module.
• Homotopies of Roos complexes — Lemma 1.1. A natural transformation between two functors gives an explicit homotopy between the maps they induce on Roos complexes.
• Nerves of coinduced modules — Theorem 3.4. The nerves of a coinduced module are the direct images along the décalage of its pullback by the last vertex, and the derived direct images when the module is injective and the inverse images along last vertices preserve injectives.
• Nerves of induced modules — Theorem 4.3. On a good two-sided pseudo-category, under an exactness condition on flat modules, the nerve of an induced flat module is the derived left adjoint along the décalage of its pullback by the first vertex.
• Cohomology of BX — Theorem 4.1. If the inverse images along the last vertices preserve injective modules, the nerves compute RHom out of complexes with invertible structure; Corollary 4.2 gives the cohomology of BX and its spectral sequence from the levels.
• Derived Hom of equivariant modules — Theorem 4.3. Under an exactness condition on left adjoints as well, the second nerve, and the first for two-sided pseudo-categories, computes all RHom complexes and is fully faithful on bounded derived categories.
• Group cohomology — Example 5.3. For a group acting on a point, the complex of the levels is the inhomogeneous cochain complex of group cohomology.
• Direct images to a base — Proposition 6.1. For a pseudo-category over a topos, the derived direct image of an equivariant module to the topos is computed on either total topos.
• The mixed Hom functor — Definition 1.5. For a good fibred topos with a cartesian ring, the derived mixed Hom from modules on the second total topos to modules on the first, with values in functors on the index category, computed by complexes with injective levels.
• Mixed Hom and inverse images — Proposition 2.2. For a morphism over the index category, RHom^!(L, Rf_*M) and RHom^!(Lf^*L, M) are canonically isomorphic for L bounded above.
• Mixed Hom and derived left adjoints — Proposition 2.3. Under exactness conditions on flat modules, RHom^!(Lf_!E, F) and RHom^!(E, f^*F) are canonically isomorphic for E bounded above.
• Ext groups of equivariant modules — Theorem 3.3. Under the injectivity and exactness conditions, RHom_BX(E, F) is the derived limit over the simplex category of the mixed Hom from the second nerve of E to the nerve of F; Corollary 3.4 gives a spectral sequence from the Ext groups on the levels.
• Ext over a group ring — Example 4.1. For a group acting on a point, Ext over the group ring into a module injective over the coefficients is computed by cochains of the group with values in Hom groups.
• Reduced spectral diagrams — Proposition 2.4. Translating down by one in every coordinate is an equivalence from reduced spectral diagrams to large simplicial diagrams; reduction is its left adjoint on all spectral diagrams.
• Spectra of commutative group objects — Proposition 3.5. The spectrum of iterated nerves is fully faithful on commutative group objects, with image the spectral diagrams of iterated nerves whose structure maps are isomorphisms.
• Derived categories of spectral diagrams — Proposition 4.6. The bounded above derived category embeds fully faithfully in that of spectral diagrams, with image the objects whose cohomology consists of spectra of objects.
• The suspension theorem — Theorem 2.1. For a reduced functor and an n-reduced simplicial object, the suspension is an isomorphism in degrees below 2n and onto in degree 2n.
• The stabilization — Theorem 3.6. The homology of the stabilization of F at X in degree i is the homology of F of the n-th suspension of X in degree i+n, for every n > i.
• Stabilization and triangles — Proposition 4.3. The stabilization is additive and carries degreewise split exact sequences to distinguished triangles, through its shift isomorphism.
• Exactness of stable derived functors — Theorem 1.6. The stable derived functor, extended to bounded above complexes with its shift isomorphism, is an exact functor of triangulated categories.
• Symmetric, exterior and divided powers — Example 2.3. The stable derived functor of the n-th symmetric power is that of the exterior power shifted by n-1, which is that of the divided power shifted by n-1.
• Lifting against fibrant quasi-isomorphisms — Lemma 3.6. A fibrant quasi-isomorphism of simplicial modules in a topos has local lifts against finite monomorphisms of simplicial sets.
• Fractions for simplicial modules and algebras — Theorem 4.2. Over a simplicial ring of a topos, quasi-isomorphisms of modules up to homotopy admit left and right fractions, and quasi-isomorphisms of algebras admit right fractions.
• Suspension and loops — Proposition 1.5. On the derived category of a simplicial ring, the suspension is left adjoint to the loop functor and fully faithful, with image the modules whose homotopy vanishes in degree 0.
• Long exact Ext sequences — Theorem 3.9. Distinguished triangles of simplicial modules give long exact sequences of Ext groups in both variables.
• Constant rings — Proposition 3.10. For a ring placed in every degree, normalization carries the distinguished triangles of simplicial modules to those of the derived category, and the extension class to the connecting morphism.
• The first Ext group by extensions — Proposition 4.3. Every class in the first Ext group is the class of an extension of simplicial modules after a quasi-isomorphism, with a criterion for two extensions to give the same class.
• Flat modules and quasi-isomorphisms — Proposition 2.2. Tensoring with a degreewise flat simplicial module preserves quasi-isomorphisms, and so does tensoring a quasi-isomorphism of flat modules with any module.
• The total décalage — Corollary 1.4. For a simplicial abelian sheaf X, the maps h, v from the diagonal of the total décalage of X to X are quasi-isomorphisms.
• The derived tensor product — Proposition 3.2. The tensor product of simplicial modules over a simplicial ring has a left derived functor, computed by a flat replacement of either factor.
• Quasi-isomorphic simplicial rings — Corollary 4.3. A quasi-isomorphism of simplicial rings induces an equivalence of derived categories, through derived extension and restriction of scalars.
• The Tor spectral sequence — Theorem 2.1. The homology of a derived tensor product of simplicial modules is the abutment of a spectral sequence starting from the homology of the degreewise Tor modules.
• The Künneth spectral sequence — Theorem 4.6. A spectral sequence from Tor over the homology ring of the simplicial ring, of the homology modules of the factors, converges to the homology of their derived tensor product.
• Bounded derived categories of differential graded modules — Proposition 1.2. Over a differential graded ring in nonpositive degrees, the localizations of the categories of bounded modules are full subcategories of the derived category, cut out by bounds on cohomology.
• Truncated differential graded rings — Proposition 1.3. Modules with cohomology in an interval of length at most n over a ring in nonpositive degrees are the same as modules over its truncated ring t_n.
• Bounded resolutions of differential graded modules — Lemma 2.1. Bounded above modules have functorial good resolutions with the same bound, and bounded below modules have K-injective resolutions with the same bound.
• Derived Hom of differential graded modules — Proposition 2.3. Derived Hom and derived internal Hom are defined on all modules, compute morphisms in the derived category, and respect bounds for rings in nonpositive degrees.
• Extension of scalars for differential graded rings — Proposition 2.5. Derived extension of scalars is left adjoint to restriction, also on derived Hom, and a quasi-isomorphism of rings gives an equivalence of derived categories.
• Normalization and derived tensor products — Theorem 3.3. Normalization carries the derived tensor products and extensions of scalars of simplicial modules over a simplicial ring to those of differential graded modules over its normalized complex.
• Graded objects as equivariant sheaves — Proposition 2.3. Graded objects of a topos are the equivariant sheaves on the simplicial topos of the categories C_•(J), and C_• is their second nerve, fully faithful with an explicit image.
• Nerves of graded actions — Proposition 2.6. For a graded monoid G, the bisimplicial nerve Ner(C_•(G), C_•(X)) is a fully faithful functor of the graded G-object X, with essential image given by two isomorphism conditions.
• Base change along discrete opfibrations — Lemma 3.2. A cartesian square of diagrams in a topos with one side a discrete opfibration induces a square of topoi with invertible base change map.
• The pseudo-category of a graded action — Proposition 3.3. When the products of the grading are isomorphisms, the simplicial topos of the nerve of a graded action is a good two-sided pseudo-category whose levels are localizations of one topos.
• External tensor products of multisimplicial objects — Lemma 1.1. Normalization, reduction and translation commute with external tensor products of multisimplicial objects.
• Induced modules over rings of type C_•(J) — Proposition 1.4. Modules over a ring of type C_•(J) form a Grothendieck abelian category; induced modules are left adjoint to the forgetful functor and exact when the ring is flat.
• Large simplicial rings and spectral rings — Proposition 2.1. Translation by one is an equivalence between large simplicial rings and reduced spectral rings, and reduction is left adjoint to the inclusion of reduced spectral rings.
• Bilinear maps as maps of spectra — Proposition 3.1. Bilinear maps of abelian sheaves are the maps between their spectra of the type given by concatenation of coordinates.
• Spectra of modules — Proposition 3.4. A module over a ring is determined by its spectrum, a graded object with an action of the spectrum of the ring, with an explicit essential image.
• Modules as bisimplicial diagrams — Proposition 3.5. Modules over a ring are the bisimplicial diagrams built from products of copies of the ring and the module that satisfy four isomorphism conditions.
• Reduction and translation in derived categories — Proposition 1.1. Translation by one is fully faithful from the derived category of modules over a ring of type C_•(S_0) to that over the corresponding reduced ring of type C_•(S_1), with reduction as left adjoint.
• The diagonal comparison functor — Proposition 2.3. The normalized double diagonal is an exact functor from modules over a ring of type C_•(S_0) to differential graded modules over its diagonal ring, and it respects upper bounds.
• Derived colimits over the nerve of the injective ordinals — Proposition 3.3. The derived colimit of a coefficient system on the simplices of the nerve is the total complex of the sum of its values over the simplices of each dimension.
• The constant functor and its left adjoint — Proposition 3.4. The constant functor into objects of type C_•(S_0) is fully faithful on bounded above derived categories, with the normalized double diagonal as left adjoint.
• Objects with constant cohomology — Proposition 3.6. A bounded above object of type C_•(S_0) comes from a complex of abelian sheaves exactly when its cohomology objects are constant, and similarly for reduced spectral objects.
• The comparison theorem for diagonal rings — Theorem 4.1. For a ring of type C_•(S_0) with flat components, derived Hom into a module with constant underlying object is computed over the normalized double diagonal ring.
• Comparison for reduced spectral rings — Corollary 4.2. The comparison theorem transferred to rings of type C_•(S_1) through reduction and translation.
• Derived tensor products of rings — Proposition 1.2. The derived tensor product of two rings is a differential graded ring in degrees -1 and 0, computed from any flat presentation of one factor.
• The stable ring of a ring — Proposition 2.2. The stabilization of the free abelian group functor at a ring is a simplicial ring with flat terms, augmented to the ring by a ring homomorphism.
• Derived tensor products through the stable ring — Proposition 2.4. The truncated normalized stable ring is a flat presentation of the ring, so the derived tensor product of two rings is a truncation of a normalized simplicial ring.
• Mac Lane's theorem — Theorem 3.2. The derived tensor product of the ring with the stable module of a module, over the normalized stable ring, is the module.
• Bar resolutions — Proposition 4.1. The bar construction of a module over a ring homomorphism is contractible over the smaller ring, and a flat resolution when the ring and the module are flat over it.
• The Mac Lane resolution — Theorem 4.2. A module has a functorial resolution by sums of direct summands of the free modules on the products of copies of the ring and the module.
• Base change of the comparison theorem — Proposition 1.2. The comparison theorem for diagonal rings holds for a ring with flat components tensored with any commutative ring.
• Ext groups through spectra — Corollary 2.4. Derived Hom from the free module on the spectrum of a module into the spectrum of a complex of modules is the derived Hom of modules.
• Spectra over a derived tensor product — Theorem 3.3. Bounded complexes over the derived tensor product of a ring with a commutative ring have spectra, extending those of complexes of modules and compatible with derived Hom.
• Two consecutive degrees — Proposition 3.4. The spectrum over a derived tensor product is fully faithful on complexes with cohomology in two consecutive degrees, with an explicit essential image.
• Spectra and tensor products — Proposition 3.5. The spectrum over a derived tensor product commutes with derived tensor products over the coefficient ring when the result is bounded.
• Ext groups as cohomology of spectral nerves — Theorem 4.3. The Ext groups of modules are the cohomology of the total topos of the nerve of the action of the spectrum of the ring on the spectrum of the first module.
• Relative cohomology and cones — Proposition 4.4. Relative cohomology along a homomorphism of modules computes derived Hom out of its cone.
• Quasi-coherent modules on the big fppf site — Proposition 1.1. Quasi-coherent sheaves of a scheme and quasi-coherent modules on its big fppf site correspond, with the same cohomology.
• Adequate modules — Proposition 1.2. Adequate modules on the big site have quasi-coherent direct images on the Zariski site, no higher direct images, and direct images compatible with flat base change.
• Truncated pullbacks to the big site — Proposition 2.1. A bounded complex with quasi-coherent cohomology is recovered from the truncation of its pullback to the big fppf site.
• Derived Hom on the big site — Corollary 2.2. Derived Hom into a bounded complex with quasi-coherent cohomology is the same on the Zariski site and, after truncation, on the big fppf site.
• Flat base change for adequate complexes — Lemma 2.3. Direct images to the Zariski site of complexes with adequate cohomology commute with flat base change.
• Perfect complexes on the big site — Theorem 3.1. Complexes of perfect amplitude in [a, b] on the Zariski site and on the big fppf site of a scheme are equivalent.
• Diagrams of schemes — Lemma 4.2. The comparison holds for diagrams of schemes, and quasi-cartesian objects on the big site are those with cartesian cohomology.
• The big fpqc site — Proposition 5.3. The comparisons of quasi-coherent, adequate and perfect complexes with the Zariski site hold for the big fpqc site of a scheme, formed in a universe.
• Cotangent complex of a morphism of diagrams — Proposition 1.2. The cotangent complex of a morphism of diagrams of schemes is, at every level, the cotangent complex of the level, with the functoriality maps as structure maps.
• Transitivity triangle for diagrams — Proposition 1.3. Morphisms of diagrams X to Y to Z give a natural transitivity triangle, whose connecting map is the Kodaira-Spencer class.
• Tor-independent base change for schemes — Proposition 1.4. The cotangent complex of a morphism of schemes commutes with Tor-independent base change, so cartesian Tor-independent diagrams have quasi-cartesian cotangent complexes.
• Extensions of diagrams as ringed-topos extensions — Proposition 2.3. Square-zero extensions of a diagram of schemes by a termwise quasi-coherent ideal are the algebra extensions on its Zariski topos.
• Classification of extensions of diagrams — Theorem 2.4. Extensions of a diagram of schemes are classified by Ext^1 of its cotangent complex, with automorphisms Ext^0, also computable on the big fppf site.
• Deformations of a morphism of schemes — Lemma 3.2. A thickening of a morphism of schemes has an isomorphism on ideals exactly when its square is cartesian and a Tor_1 group vanishes.
• Deformations of squares — Corollary 3.7. Deformations of commutative squares of schemes carry cartesian squares to cartesian squares.
• Deformations of a morphism over a thickened target — Theorem 4.3. Lifting a morphism of diagrams over a square-zero thickening of its target is obstructed in Ext^2 by the product of the Kodaira-Spencer class and the class of the thickening; lifts form an Ext^1-torsor.
• Deformed diagrams of morphisms — Proposition 4.4. A deformation of a morphism over a deformation of its target deforms the diagram the morphism defines and keeps its cartesian squares cartesian.
• Nerves of action categories — Lemma 1.3. The nerve of the action category of a category scheme acting on a sheaf is the pullback of the nerve of the category scheme along its first vertex.
• Equivariant sheaves over an equivariant sheaf — Proposition 1.4. Sheaves equivariant for the action category of a G-sheaf L are the G-sheaves over L.
• The opposite action category — Proposition 1.5. The nerve functor from the bounded derived category of the opposite action category to the fppf total topos of the nerve is fully faithful, with an explicit essential image.
• Action groupoids — Proposition 1.6. For a groupoid scheme, inversion identifies the action category with its opposite.
• Tor independence and base change — Lemma 2.2. When P and R are Tor independent over Q, Tor independence of P and R' over Q is Tor independence of P x_Q R and R' over R.
• The equivariant cotangent complex — Theorem 2.4. For a morphism of G-schemes Tor independent of G, the truncated pullbacks of the cotangent complex of the nerves descend uniquely to the derived category of the opposite action category.
• Properties of the equivariant cotangent complex — Proposition 2.5. The equivariant cotangent complex has the expected underlying complex, is bounded for local complete intersections, and is the trivial pullback for trivial actions.
• Ext groups of the equivariant cotangent complex — Proposition 3.2. Ext groups of the equivariant cotangent complex into quasi-coherent equivariant modules are Ext groups of the cotangent complex of the nerves on the Zariski total topos.
• Equivariant Kodaira-Spencer class — Proposition 3.3. Morphisms X to Y to Z of G-schemes give equivariant transitivity maps, the equivariant Kodaira-Spencer class and long exact sequences of Ext groups.
• Change of the category scheme — Proposition 4.1. Restriction along an internal functor G' to G gives a base change map of equivariant cotangent complexes, an isomorphism under Tor independence.
• Closed immersions into smooth schemes — Proposition 1.2. For a closed immersion into a smooth scheme, the first truncation of the cotangent complex is isomorphic to the conormal complex, and the Kodaira-Spencer class induces the differential on H^-1.
• The sign of the transitivity triangle — Lemma 1.1. In the transitivity triangle of Z -> Z[t] -> Z, t -> 0, the third map sends the class of t in (t)/(t^2) to dt, with the triangles of short exact sequences of the Stacks project.
• Equivariant conormal complexes — Proposition 1.3. For an equivariant closed immersion into a smooth G-scheme, the truncated equivariant cotangent complex is isomorphic to the equivariant conormal complex.
• Recognizing nerves of action categories — Lemma 2.1. Simplicial objects over N(G) whose levels are the pullbacks of level 0 along the first vertex are nerves of action categories.
• Equivariant extensions and extensions of nerves — Proposition 2.3. Equivariant extensions are the extensions of the nerves whose ideal is cartesian along the faces fixing the first vertex; Tor independence makes the condition automatic.
• Classification of equivariant extensions — Theorem 2.4. Equivariant extensions of a G-scheme are classified by Ext^1 of the equivariant cotangent complex, with automorphisms Ext^0.
• Equivariant deformations and nerves — Proposition 3.2. Equivariant deformations of a morphism are the deformations of its nerve over the nerve of the thickened target.
• Equivariant deformations of a morphism — Theorem 3.3. Equivariant deformations over an equivariant extension are obstructed in Ext^2 of the equivariant cotangent complex by the product of the equivariant Kodaira-Spencer class and the class of the extension; they form an Ext^1-torsor.
• Spectral sequence of the obstruction — Proposition 4.1. A spectral sequence over the nerve computes the equivariant Ext groups, and its edge map carries the equivariant obstruction to the obstruction for the underlying morphism of schemes.
• Lifting maps to square-zero extensions — Lemma 5.1. A ring map lifts to given square-zero extensions exactly when two classes in Ext^1 of the cotangent complex agree; the lifts form a torsor under derivations.
• Morphisms between equivariant extensions — Proposition 5.2. An equivariant morphism lifts to given equivariant extensions exactly when a class in Ext^1 vanishes; the lifts form an Ext^0-torsor.
• Frobenius kernels — Lemma 1.1. The n-fold relative Frobenius of a group scheme over a field is a homomorphism and a universal homeomorphism; its kernel is a finite connected normal subgroup scheme, compatible with closed subgroups.
• The first Frobenius kernel — Lemma 1.2. The first Frobenius kernel of a group scheme locally of finite type over a field is a truncated polynomial algebra, hence a local complete intersection.
• Quotients by Frobenius kernels — Lemma 2.1. Over a perfect field, the quotient of a group scheme of finite type by a large enough Frobenius kernel is the reduced subgroup of a Frobenius twist, hence smooth.
• Group schemes over a field are local complete intersections — Theorem 2.3. Every group scheme locally of finite type over a field is a local complete intersection over the field.
• Flat group schemes are syntomic — Theorem 3.1. A group scheme flat and locally of finite presentation over a scheme is syntomic, and its cotangent complex is locally a two-term complex of finite free modules in degrees -1 and 0.
• The cotangent complex at the unit — Corollary 3.2. For a group scheme over a field, the derived pullback of the cotangent complex to the unit has H^0 dual to the Lie algebra and H^-1 of dimension dim Lie - dim G; it vanishes exactly for smooth groups.
• Trivial torsors — Lemma 1.2. A torsor with a section is isomorphic to the trivial torsor through the section.
• Torsors and Cech nerves — Proposition 1.3. A torsor under a flat group scheme locally of finite presentation is syntomic and surjective, and the nerve of the action is the Cech nerve of the torsor.
• Descent along a torsor — Proposition 2.2. Pullback is an equivalence from modules on the big fppf site of the base onto equivariant modules on the torsor, with quasi-inverse the invariant sections.
• Descent on big sites — Theorem 2.4. Pullback to the nerve of a torsor is fully faithful on bounded derived categories of the big fppf sites, with essential image the quasi-cartesian objects.
• Descent on Zariski sites — Theorem 2.5. Pullback to the nerve of a torsor is fully faithful on bounded complexes with quasi-coherent cohomology on the Zariski sites, with essential image the quasi-cartesian objects with termwise quasi-coherent cohomology.
• Direct images, sections and equivariant complexes — Corollary 2.6. A descended complex is the derived direct image of its pullback to the nerve, its pullback along a section, and the invariant direct image of its equivariant form.
• Truncating a distinguished triangle — Lemma 3.2. Truncating the first two vertices of a distinguished triangle whose third vertex lies in degrees at least n gives a distinguished triangle.
• The co-Lie complex and the Atiyah triangle — Theorem 3.3. The cotangent complex of the nerves of a torsor descends to a perfect complex of tor amplitude [-1,0] on the base, and the transitivity triangle descends to the Atiyah triangle, whose third map is the Atiyah class.
• Base change of the co-Lie complex — Proposition 3.4. The co-Lie complex of a torsor commutes with arbitrary base change of the base.
• Equivariant complexes of a torsor — Proposition 3.5. The equivariant cotangent complexes of a torsor are the pullbacks of its co-Lie complexes, and the equivariant Kodaira-Spencer class is the pullback of the Atiyah class.
• The co-Lie complex of a smooth group — Proposition 4.1. For a smooth group scheme the co-Lie complex of a torsor is the co-Lie module twisted by the torsor through the coadjoint action.
• The Atiyah sequence — Proposition 4.2. For a smooth group scheme the Atiyah triangle gives an exact sequence of invariant differential forms, whose class is the Atiyah class when the base is smooth.
• Deformations of a torsor are torsors — Lemma 1.2. Equivariant deformations of a torsor over a first-order thickening of the base are torsors, and conversely.
• The class of the thickened nerve — Lemma 2.1. The class of the thickening of the nerve of the base is the pullback of the class of the thickening.
• Deformations of torsors — Theorem 2.2. Deformations of a torsor over a first-order thickening are obstructed in Ext^2 of the co-Lie complex by the product of the class of the thickening and the Atiyah class; they form an Ext^1-torsor with automorphism groups Ext^0.
• Duality and smooth groups — Proposition 3.1. The groups classifying deformations of a torsor are cohomology groups of the dual of the co-Lie complex; torsors over affine bases always deform, and for smooth groups the coefficients are the adjoint bundle.
• The co-Lie complex of a group scheme — Proposition 1.2. The co-Lie complex of a flat group scheme locally of finite presentation is perfect of tor amplitude [-1,0], has the invariant differentials as H^0, and is the co-Lie complex of the group as a torsor.
• Base change of the co-Lie complex of a group — Proposition 1.4. The co-Lie complex of a flat group scheme commutes with arbitrary base change.
• Smoothness and the co-Lie complex — Proposition 1.5. A flat group scheme locally of finite presentation is smooth exactly when its co-Lie complex has tor amplitude [0,0].
• Exact sequences and torsors — Lemma 2.1. In an exact sequence of flat group schemes the quotient map is a torsor under the kernel.
• The triangle of an exact sequence — Proposition 2.2. An exact sequence of flat group schemes gives a distinguished triangle of co-Lie complexes, with the functoriality maps in degree 0.
• The long exact sequence of co-Lie complexes — Corollary 2.3. An exact sequence of flat group schemes gives a six-term exact sequence of the cohomology of co-Lie complexes; the map from the quotient is an isomorphism exactly when the kernel is etale.
• Descent along a torsor over a diagram — Theorem 1.4. For a torsor under a flat group scheme over a diagram of schemes, pullback to the nerve of the action is fully faithful on bounded complexes with quasi-coherent cohomology, with an explicit essential image.
• The co-Lie complex of a torsor over a diagram — Theorem 1.5. A torsor over a diagram of schemes has a quasi-cartesian co-Lie complex with perfect levels and an Atiyah triangle, restricting levelwise to those of the torsors over the schemes of the diagram.
• The adjoint co-Lie complex — Proposition 2.2. The co-Lie complex of a flat group scheme carries an action of the group, obtained by descent along the universal torsor over the nerve, as an object of the derived category of the classifying topos.
• The coadjoint action on invariant forms — Proposition 2.4. On invariant differential forms the action is the coadjoint action, and for a smooth group the adjoint co-Lie complex is this module in degree 0.
• Contravariance and base change — Proposition 2.5. A homomorphism of flat group schemes induces a map from the restricted adjoint co-Lie complex of the target to that of the source, compatibly with composition, and the formation of the adjoint co-Lie complex commutes with base change.
• The adjoint co-Lie complex through left and right translations — Proposition 2.7. The equivariant cotangent complex of G under left and right translations corresponds to the adjoint co-Lie complex under the equivalence between modules of the classifying topos of G and G x G^o-equivariant modules on G.
• Twisting by a torsor — Proposition 2.8. The co-Lie complex of a torsor under the opposite group, on the big fppf site, is the adjoint co-Lie complex twisted by the torsor.
• The cotangent complex of the nerve of a group scheme — Theorem 3.3. The cotangent complex of the nerve of a flat group scheme sits in a distinguished triangle between a copy of the co-Lie complex at each vertex and the adjoint co-Lie complex.
• Relative Ext groups of the adjoint co-Lie complex — Corollary 3.5. The Ext groups of the cotangent complex of the nerve are relative Ext groups of the adjoint co-Lie complex on the classifying topos, computed by a spectral sequence from Ext groups on the powers of the group.
• Relative cohomology with dual coefficients — Proposition 3.8. Relative Ext groups on the classifying topos out of a complex with perfect underlying object are relative cohomology groups with coefficients in its dual; so are the Ext groups of the cotangent complex of the nerve.
• The cotangent complex of the nerve of a homomorphism — Theorem 3.11. For a homomorphism F -> G, the cotangent complex of the morphism of nerves sits in a distinguished triangle between a copy of the co-Lie complex of the homomorphism at each vertex and its descent, whose classifying-topos form is a cone of the functoriality map of the adjoint co-Lie complexes.
• Relative Ext groups of a homomorphism — Corollary 3.12. The Ext groups of the cotangent complex of the morphism of nerves are relative cohomology groups of the classifying topos of F with coefficients in the cocone K of the dual functoriality map.
• Deformations of group schemes and of their nerves — Proposition 1.2. Deformations of a flat group scheme over a first-order thickening are the same as deformations of the morphism from its nerve to the base.
• Deformations of flat group schemes — Theorem 2.1. A flat group scheme deforms over a first-order thickening exactly when an obstruction in Ext^2 of the cotangent complex of its nerve vanishes; deformations then form a torsor under Ext^1, with automorphisms Ext^0.
• Smooth affine groups and Hochschild cochains — Proposition 2.4. For a smooth affine group over an affine base, the deformation groups are the cohomology of cochains of the group with values in its Lie algebra tensored with the ideal.
• Extending homomorphisms to deformations — Theorem 3.1. A homomorphism of flat group schemes extends to given deformations exactly when a class in Ext^1 vanishes; the extensions then form a torsor under Ext^0.
• Deforming a homomorphism with a given target — Theorem 3.3. Deformations of a flat group scheme together with an extension of a homomorphism to a given deformation of the target are classified by Ext groups of the relative cotangent complex of the nerves.
• Deforming a homomorphism in relative cohomology — Corollary 3.4. The obstruction to deforming a group scheme together with a homomorphism to a given deformation of the target lies in H^3 of the classifying topos relative to S with coefficients in K; solutions form a torsor under H^2, with automorphisms H^1.
• Two deformations of the target — Proposition 3.5. For two deformations of the target group, the obstructions to deforming the source together with the homomorphism differ by the image of the pulled-back difference class under the connecting map of the transitivity triangle of the nerves.
• The canonical identification — Theorem 2.3. For a triangle relating an object of the topos of the nerve to the nerve of an object of the classifying topos through the adjunction map of the level 0, the relative Ext groups on the classifying topos are canonically the Ext groups on the nerve, naturally in the triangle.
• Restriction and the canonical identification — Proposition 3.3. The canonical identifications commute with restriction along a homomorphism of group schemes and with the inverse image along the morphism of nerves.
• Deforming the target of a homomorphism — Corollary 4.4. The groups that classify deformations of the target of a homomorphism with a given deformation of the source are relative Ext groups of the adjoint co-Lie complex on the classifying topos of the target modulo that of the source.
• Equivariant deformations of the target in relative Ext groups of classifying topoi — Theorem 5.3. For morphisms of G-schemes X -> Y -> Z with g flat and an equivariant flat deformation of h = gf, the groups of obstruction, torsor and automorphisms are relative Ext groups of the truncated equivariant cotangent complex of g on the classifying topos of [G,Y]^o modulo that of [G,X]^o.
• Kodaira-Spencer classes over a fibre product — Lemma 6.1. For Y = Z_1 x_W Z_2 with Z_2 -> W flat, a morphism W -> V and X -> Y, the composites K(Z_2/W/V)[1] o K(X/Y/Z_1) and K(Z_1/W/V)[1] o K(X/Y/Z_2), through the base change isomorphisms, are opposite.
• Two deformations of the target in relative Ext groups — Corollary 6.4. For two deformations of the target of a homomorphism u: F -> G, the difference of the obstructions is minus the restricted difference class composed with the connecting map of the triangle of the co-Lie complexes, in Ext^3 of the classifying topos of F relative to S.
• Relative Ext groups as relative cohomology of the internal Hom — Theorem 7.2. The relative Ext groups of the classifying topos over S, and modulo the classifying topos of a subgroup, are the relative cohomology groups of the internal Hom, by an isomorphism of complexes natural in the first argument and compatible with restriction; for perfect K the coefficients are K^v (x) M.
• The deformation theorems for homomorphisms with dual coefficients — Corollary 7.3. The deformation groups of the target of a homomorphism are H^(j+1)(B_G/B_F, ll^v (x) eps^*I), and the difference of the obstructions for two deformations of the target is minus the dual of the connecting map applied to the restricted difference class.
• Devissage for filtered derived categories — Theorem 2.3. An exact functor of abelian categories that is fully faithful on a shift-stable class of objects of the derived category is fully faithful on the filtered objects with filtration in a fixed interval whose graded pieces lie in that class; its essential image consists of the filtered objects whose graded pieces lie in the essential image.
• The filtered transitivity complex — Proposition 3.2. For ring maps A -> B -> C of a topos, a filtered complex of flat C-modules with first step L_(B/A) (x) C, underlying object L_(C/A) and graded piece L_(C/B), whose triangle is the transitivity triangle.
• The equivariant filtered transitivity triangle — Theorem 4.1. For morphisms X -> Y -> Z of G-schemes Tor independent of G, a projective system of bounded filtered objects on the classifying category of [G,X]^o whose nerves are the truncated filtered transitivity complex of the nerves; its connecting maps give the equivariant Kodaira-Spencer class.
• Filtered descent along a torsor — Theorem 5.1. Pullback along a torsor is fully faithful on bounded filtered objects, with quasi-coherent graded pieces on Zariski sites, and on their projective systems; its essential image consists of the objects with quasi-cartesian graded pieces.
• The filtered Atiyah triangle — Theorem 5.2. The truncated filtered transitivity complex of N(G,X) -> N(G,Y) -> N(G) descends to a projective system of bounded filtered objects on Y, whose triangles are the Atiyah triangles of the torsor.
• The level filtration of a mixed Hom complex — Theorem 2.1. For equivariant modules on a good pseudo-category, the quotients of the mixed Hom complex by the columns of index greater than i are the derived limits over the strict i-truncation; for invertible structure they are the derived Hom complexes of the nerves on the truncated total topoi, the projections being the restriction maps.
• Ext groups of the truncated nerves of a group scheme — Theorem 3.1. For a flat group scheme G of finite presentation, the Ext groups of the cotangent complexes of the strict truncations of its nerve are computed by the columns 1 to i of the complex of relative Ext groups of the adjoint co-Lie complex, compatibly with the restriction maps.
• Deformations of the truncated nerves of a group — Proposition 4.2. The deformations of the strict 1-, 2- and 3-truncations of the nerve of a flat group scheme G are the deformations of G as a scheme, such deformations with a lift of the multiplication, and the deformations of G as a group scheme; an associative lift of the multiplication is a group law.
• Deformations of the truncated nerves of an action — Proposition 4.4. For an equivariant morphism of schemes with an action of a flat group scheme and an equivariant extension of the target, the deformations of the truncated nerves are the deformations of the morphism, those with a lift of the action, and the equivariant deformations.
• The graded pieces of the obstruction for a group — Theorem 5.2. The obstruction to deforming a flat group scheme has first graded piece the obstruction to deforming it as a scheme; it lies in the second and third steps of the filtration exactly when the group deforms as a scheme and when the multiplication lifts to some deformation.
• The shear isomorphism — Proposition 1.4. For a group scheme H over a diagram Y, the cotangent complex of H over Y is the pullback of the co-Lie complex of H.
• Additivity of the co-Lie complex — Lemma 1.5. The co-Lie complex of a product is the sum, and for commutative group schemes a sum of homomorphisms induces the sum of the maps of co-Lie complexes.
• The tensor formula for Lie complexes — Proposition 1.6. For a group scheme over a diagram, the internal mixed Hom from its co-Lie complex into a bounded below complex is the derived tensor product with its Lie complex.
• Torsors under a commutative group — Theorem 2.2. The co-Lie complex of a torsor under a commutative group scheme G, over a scheme or a diagram of schemes, is the pullback of the co-Lie complex of G.
• The adjoint co-Lie complex of a commutative group — Corollary 2.3. For a commutative group scheme the adjoint co-Lie complex is the co-Lie complex with the trivial action.
• The Lie complex of the nerve — Proposition 3.1. The Lie complex of the nerve of a commutative group scheme is the nerve of its Lie complex.
• The Lie complex of the spectrum — Proposition 3.5. The Lie complex of the spectrum of a commutative group scheme is the spectrum of its Lie complex.
• Second nerves on spectral pseudo-categories — Lemma 2.1. On the pseudo-category of a graded action, the second nerve is fully faithful on bounded derived categories, with image the objects whose cohomology is cartesian along the maps fixing the first vertex.
• The Lie complex of a module scheme — Proposition 3.2. The Lie complex of a module scheme over a ring scheme A has a canonical structure of a complex over the derived tensor product of A and the structure ring, determined by the spectral nerve.
• Exact sequences of module schemes — Proposition 3.5. An exact sequence of module schemes gives a distinguished triangle of their Lie complexes over the stable ring with coefficients.
• The Ext formula — Theorem 4.1. The Ext groups of the cotangent complex of the spectral nerve of a module scheme are Ext groups of modules into its Lie complex with coefficients.
• Relative Ext groups of the spectral nerve — Theorem 4.3. Relative Ext groups of the cotangent complex for a homomorphism of module schemes are Ext groups of modules out of its cone.
• Deformations of module schemes and of their spectral nerves — Proposition 1.2. Deformations of a module scheme over a first-order thickening are the same as deformations of the morphism from its spectral nerve to that of the base.
• Deformations of module schemes — Theorem 2.1. A module scheme deforms exactly when an obstruction in Ext^2 of modules into its Lie complex tensored with the ideal vanishes; deformations then form a torsor under Ext^1, with automorphisms Ext^0.
• Extending homomorphisms of module schemes — Theorem 2.2. A homomorphism of module schemes extends to given deformations exactly when a class in Ext^1 vanishes; the extensions then form a torsor under Ext^0.
• Deforming a homomorphism of module schemes with a given target — Theorem 2.3. Deformations of a module scheme together with an extension of a homomorphism to a given deformation of the target are classified by Ext groups of modules into the fibre of the map of Lie complexes.
• Two deformations of the target module scheme — Proposition 2.5. For two deformations of the target, the obstructions to extending a homomorphism differ by minus the pulled-back difference class, and the obstructions to deforming the source with the homomorphism differ by its image under the connecting map of the transitivity triangle of the spectral nerves.
• Adjoints of the degree functors on graded modules — Lemma 1.3. The degree-n component functor on graded modules has graded coinduction as right adjoint; placing a module in degree n is right adjoint to the degree-n component after reduction modulo the positive part.
• The graded cotangent complex — Theorem 3.3. The graded cotangent complex forgets to the cotangent complex; its degree-zero part is the cotangent complex of the degree-zero rings, and for B^0 = A its degree-one part is B^1.
• The graded transitivity triangle — Theorem 3.4. Maps of graded rings A -> B -> C give a distinguished triangle of graded cotangent complexes, compatible with forgetting the grading and with degree zero.
• Graded square-zero extensions — Theorem 4.5. Graded square-zero extensions are classified by Ext^1 of the graded cotangent complex in the derived category of graded modules, with automorphisms the graded derivations.
• Naturality of the Atiyah class of a module — Proposition 2.3. The Atiyah class of a module is natural in the module and in the ring map, and vanishes for modules extended from the base by a flat module.
• The Atiyah class through the graded cotangent complex — Theorem 2.4. The Atiyah class of a module M is the connecting map of the degree-one part of the graded transitivity triangle of the dual numbers B + M.
• Principal parts and connections — Proposition 3.2. Connections on a module correspond to splittings of its principal parts extension, whose class is the image of the Atiyah class.
• Deformations of modules over a square-zero thickening — Theorem 2.3. An extension of modules with prescribed action of the ideal exists exactly when an obstruction in Ext^2 vanishes; the extensions then form a torsor under Ext^1, with automorphisms Hom.
• The obstruction as the product of the Atiyah class and the class of the thickening — Theorem 3.2. The canonical obstruction class of a module over a square-zero thickening is the composite of its Atiyah class and the class of the thickening.
• The universal first-order thickening — Corollary 3.3. For the thickening A/K^2 of A/K the canonical class of a module is the class of the two-term truncation of its derived tensor product with A/K over A.
• The Atiyah class of an invertible module — Proposition 3.8. On a smooth scheme the Atiyah class of an invertible module is its first Chern class in Hodge cohomology, the image of its class under dlog.
• Extending vector bundles to the first infinitesimal neighbourhood — Theorem 3.10. For a closed immersion of smooth schemes the obstruction to extending a vector bundle to the first infinitesimal neighbourhood is minus the boundary of its Atiyah class in the conormal sequence; on the quadric surface in P^3 over Z neither ruling line bundle extends (Example 3.11).
• Deforming a morphism of modules with given source — Proposition 4.2. Deforming the target of a morphism of modules together with the morphism is governed by the obstruction of the target and a class in a cokernel of Ext^1 groups.
• The single obstruction for deforming a morphism with given source — Theorem 5.3. Deforming the target of a morphism f_o of modules with given source is obstructed by one class in Ext^2(C(f_o), K), C(f_o) the cone of f_o; the solutions form a torsor under Ext^1(C(f_o), K) and their automorphisms are Hom(coker f_o, K).
• The canonical obstruction for morphisms of modules — Proposition 5.5. The obstruction for deforming a morphism with given source is the image of a canonical class with coefficients in a pushout, which is the composite of an Atiyah class of f_o with the class of the thickening.
• The graded obstruction for modules and the derived tensor product — Proposition 5.6. For the deformations of a module, the obstruction obtained from the graded transitivity triangle of the dual numbers is the canonical class of the derived tensor product; the two identifications of the torsor are inverse to each other.
• The filtered derived category — Proposition 2.2. Complexes with finite filtrations up to filtered homotopy, localized at the maps that are quasi-isomorphisms on all graded pieces, form a triangulated category; a morphism is an isomorphism when its graded pieces are.
• Filtered objects and their Yoneda extensions — Theorem 3.6. Filtered objects form the full subcategory of objects whose graded pieces have cohomology in degree 0, and their filtered Ext groups are the Yoneda extensions with exact graded pieces.
• The filtered derived Hom — Proposition 4.6. With enough injectives, the filtered Hom complex into a bounded below complex of injective type computes the filtered derived Hom, and its H^n F^0 is the filtered Ext group.
• The spectral sequence of a filtered derived Hom — Theorem 5.2. The filtration of the derived Hom gives a convergent spectral sequence from the Ext groups of the graded pieces, whose first differential is composition with the extension classes, with explicit signs.
• Graded pieces of a tensor product — Lemma 1.4. If one filtered complex has flat graded pieces, the graded pieces of the tensor product are the tensor products of the graded pieces.
• The filtered derived category is symmetric monoidal — Theorem 3.3. The derived tensor product makes the filtered derived category of a ringed topos a symmetric monoidal category, and the underlying complex and the associated graded are symmetric monoidal functors.
• Cartan's isomorphism for filtered complexes — Theorem 3.1. RHom(L (x) M, N) = RHom(L, RHom(M, N)) on the filtered derived category, compatibly with the associated graded; the filtered derived category is closed symmetric monoidal.
• Derived pullback and pushforward of filtered complexes — Proposition 4.1. Derived pullback is left adjoint to derived pushforward on filtered derived categories, with the internal form RHom(L, Rf_*M) = Rf_* RHom(Lf^*L, M), compatibly with the associated graded.
• Composition and product of internal Homs — Proposition 3.5. In the filtered derived category, the composition of internal Homs is compatible with their product: the product of two composites is the composite of the products, after the symmetry.
• Global composition and global product — Proposition 4.3. The global composition of derived Hom complexes is compatible with the global product.
• Duality for perfect filtered complexes — Theorem 3.3. For filtered perfect L, the biduality morphism, the morphism from the dual tensor M to RHom(L, M) and the product of internal Homs are isomorphisms, with no boundedness condition.
• Dualizable objects of the filtered derived category — Theorem 4.2. An object of the filtered derived category has a dual exactly when its graded pieces are perfect; the dual is RHom(L, A) with evaluation and an explicit coevaluation.
• Additivity of traces — Theorem 2.3. For a filtered perfect complex E and a filtered morphism E (x) K -> E (x) L with K, L unfiltered, the trace on E is the sum of the traces on the graded pieces.
• Multiplicativity of traces — Proposition 2.5. The trace of the tensor product of two morphisms of perfect filtered complexes is the tensor product of their traces, on morphisms and internally.
• Cup products and traces — Proposition 2.7. The cup product of u: E (x) K1 -> F (x) L1 and v: F (x) K2 -> E (x) L2 is additive on graded pieces and multiplicative, and Tr(uv) = Tr(vu) = <u, v> for endomorphism-type pairs.
• Graded pieces of powers of a filtered module — Proposition 1.6. If the graded pieces of a filtered module are flat, the graded pieces of its symmetric, exterior and divided powers are the corresponding powers of its graded module.
• Derived powers of filtered and graded complexes — Theorem 2.6. Symmetric, exterior and divided powers derive to the bounded above filtered and graded derived categories; the graded pieces of LT^n(L) are sums of derived tensor products of the LT^m(gr^i L).
• Amplitude of derived powers — Proposition 2.8. If L has tor or perfect amplitude in an interval [a,b] containing 0, then LT^n(L) has tor or perfect amplitude in [na,nb].
• Filtered Koszul sequences — Proposition 1.2. For a sequence of filtered modules with exact flat graded pieces, the Koszul sequences are exact on every filtration step, with graded pieces the Koszul sequences of the graded pieces.
• Shift isomorphisms on filtered derived categories — Theorem 2.2. LS^n(M[1]) = LΛ^n(M)[n] and LΛ^n(M[1]) = LΓ^n(M)[n] on the bounded above filtered and graded derived categories, with the shift formulas of complexes as underlying isomorphisms.
• Graded pieces of the shift isomorphisms — Proposition 2.3. On graded pieces the shift isomorphisms are sums of tensor products of the shift isomorphisms of the graded pieces, composed with the isomorphisms of shifted tensor products, with the sign (-1)^(sum over i<j of n_i n_j) for the second one.
• Diagonal maps and Koszul sequences — Proposition 1.4. The diagonal maps carry the Koszul sequence of an exact sequence tensored with M to the Koszul sequence tensored with S^n M.
• Derived diagonal maps — Proposition 2.2. For filtered complexes concentrated in degrees at most 0: LS^n(L (x) M) -> LS^n L (x) LS^n M and LΛ^n(L (x) M) -> LΛ^n L (x) LS^n M, natural and compatible with graded pieces.
• Diagonal maps and shift isomorphisms — Proposition 2.4. The diagonal maps commute with the shift isomorphisms of L (x) M and of L, for L and M concentrated in degrees at most 0.
• Shift isomorphisms as ring isomorphisms — Proposition 1.3. The shift isomorphisms identify Hom(A, LS(M[1])) with Hom(A, LΛ(M)[1]) as rings of series, and, after the signs (-1)^(n(n-1)/2), Hom(A, LΛ(M[1])) with Hom(A, LΓ(M)[1]).
• Trace of a shifted morphism — Lemma 2.1. For E perfect and g: E (x) K -> E (x) L, Tr(g[a]) = (-1)^a Tr(g).
• L-series on filtrations and gradings — Theorem 3.3. The L-series of a filtered morphism E -> E (x) M is that of its graded pieces; for a graded morphism it is the product of the L-series of the components.
• L-series of a shifted morphism — Corollary 3.4. L_t(f[1]) = L_t(f)^(-1) and L_t(f[2]) = L_t(f) for f: E -> E (x) M with E, M concentrated in degrees at most 0.
• Characteristic series through exterior powers — Corollary 3.7. c_t(f) = sum over n of (-1)^n Tr(psi^n LΛ^n(f)) t^n for E and M concentrated in degrees at most 0.
• Characteristic series of maps of complexes — Theorem 4.2. For a map of complexes phi of a strictly perfect complex, c_t of its class is the product over i of det(1 - t phi^i)^((-1)^i), which therefore depends only on the class.
• Filtered Dold-Puppe — Proposition 2.1. Normalization identifies filtered simplicial objects of an abelian category with filtered complexes in degrees at most 0, and their localization with D^{<=0}F.
• Filtered complexes and chains — Theorem 2.4. The filtered derived category of objects with filtration in [0,n] is equivalent to the derived category of chains of n+1 modules, with graded pieces the cones of the maps of the chain.
• Filtered Atiyah class — Theorem 3.1. A filtered complex in D^{<=0}F has an Atiyah class in DF whose underlying morphism is the Atiyah class and whose graded pieces are the Atiyah classes of the graded pieces.
• Atiyah class and shifts — Corollary 3.2. at(E[1]) = at(E)[1] for the Atiyah class written E -> E (x) L[1].
• Atiyah class and principal parts — Proposition 4.2. The image of the Atiyah class in E (x) Omega[1] is the class of the sequence of principal parts of B over A tensored with E, also for filtered complexes.
• Multiplicativity of Chern classes — Theorem 1.3. The total Chern class of a perfect complex is multiplicative on filtrations, gradings and distinguished triangles.
• Base change of Chern classes — Proposition 1.5. Chern classes are compatible with base change along squares of rings, morphisms of topoi and morphisms of ringed topoi.
• The integral rank — Proposition 1.8. Over a local ring of a topos, perfect complexes have an integral rank in Gamma(T, Z), additive on triangles; with the total Chern class it defines a homomorphism on the Grothendieck group.
• Chern classes in Hodge cohomology — Proposition 2.3. The Hodge-cohomology Chern classes are computed from the principal parts class and satisfy the Whitney formula and functoriality.
• First Chern class of an invertible module — Proposition 2.4. For an invertible module, c'_t = 1 + c'^1 t, with c'^1 the negative of the class of the principal parts extension twisted by the dual.
• The first Chern class of an invertible module on a smooth scheme — Corollary 2.5. For an invertible module on a smooth scheme, c'^1 is minus the first Chern class in Hodge cohomology defined through dlog.
• Cohomology on the topos of an open subset and its complement — Proposition 1.3. On Top^o(i), R Gamma of (i^*F -> F), (G -> 0) and (0 -> F) is R Gamma(X, F), R Gamma(U, G) and R Gamma_Y(X, F), compatibly with the triangle of cohomology with supports.
• Partial connections as Leibniz maps — Lemma 2.4. Partial connections relative to 0 -> L -> Omega|_U -> M -> 0 correspond to additive maps nabla: E|_U -> M (x) E|_U with nabla(bm) = b nabla(m) + v(db) (x) m.
• The localized class of a partial connection — Proposition 2.7. The localized class at(E, D) on Top^o(i), with coefficients in (L -> Omega), restricts on X to the principal parts class of E.
• Chern classes with supports — Theorem 3.3. The Chern classes of (E, D) lift those of E; Chern polynomials of weight larger than r, with L locally a direct summand of B^r, lie in H^n_Y(X, Omega^n) and map to the Chern polynomials of E.
• Bott's partial connection — Proposition 4.1. For a locally free involutive submodule F of the tangent module, locally a direct summand on U, nabla(bar s)(xi) = bar [xi, s] is a partial connection on the normal module.
• Chern polynomials of the normal module of a foliation — Corollary 4.2. Chern polynomials of weight larger than the rank of the normal module on U come from cohomology with supports in Y, and vanish when U is the whole space.
• Principal parts of complex spaces — Proposition 5.4. For a morphism of complex spaces, the principal parts of order one on the diagonal of the fibre product are the principal parts of the analytic differential; for a smooth morphism the differentials are free on the dz_k.
• Chern classes of complex spaces — Proposition 5.5. The Chern classes of a complex of bounded perfect amplitude on a complex space, in analytic Hodge cohomology, come from the analytic principal parts and satisfy the Whitney formula, functoriality and the dlog formula.
• Localized classes for smooth morphisms of complex spaces — Corollary 5.6. For a smooth morphism of complex spaces, a partial connection on an open subset localizes Chern polynomials of large weight in cohomology with supports in the complement.
• Strict graded commutativity — Lemma 1.1. In the homotopy of a simplicial ring, classes of odd degree have square zero, and a linear map M -> pi_1(R) extends uniquely to an algebra map from the exterior algebra of M.
• The Tor algebra — Proposition 2.2. Tor^A_*(B, B) = pi_*(P (x)_A B) is a graded ring, independent of the flat simplicial resolution, and functorial in squares of rings.
• Weakly regular ideals — Definition 2.4. For B = A/I, theta: Lambda_B(I/I^2) -> Tor^A_*(B, B) extends the inverse of the conormal isomorphism; I is weakly regular if I/I^2 is flat and theta is an isomorphism.
• The Koszul complex maps to Tor of the associated graded ring — Proposition 3.2. theta extends to a homomorphism of bigraded algebras Lambda(I/I^2) (x) S(I/I^2) -> Tor^A(B, gr_I A) carrying the Koszul differential to the connecting maps.
• Associated graded ring of a weakly regular ideal — Theorem 3.3. For I weakly regular, Lambda^q(I/I^2) (x) S^n(I/I^2) = Tor^A_q(B, I^n/I^(n+1)), so S^n(I/I^2) = I^n/I^(n+1) is flat, and Tor_q(B, A/I^(k+1)) -> Tor_q(B, A/I^k) is zero for q >= 1.
• Tor of the quotients by the powers of a weakly regular ideal — Corollary 3.4. For I weakly regular and q >= 1, the Koszul terms Lambda^(q+j)(I/I^2) (x) S^(n-j)(I/I^2) resolve Tor^A_q(B, A/I^(n+1)), naturally in (A, I).
• The ideal of the variables is weakly regular — Proposition 4.1. For a flat B-module F, the ideal S^+_B(F) of S_B(F) is weakly regular.
• Koszul-regular ideals are weakly regular — Proposition 4.2. An ideal generated locally by Koszul-regular sequences is weakly regular, and I/I^2 is free on the classes of the generators.
• Connectivity after base change to pi_0 — Lemma 1.3. A module with flat terms over a simplicial ring B is k-connective if its base change to pi_0(B) is.
• Connectivity of Koszul terms — Proposition 1.4. For a module V with flat terms over a simplicial ring and pi_0(V) = 0, Lambda^a V (x) S^b V is (a + b)-connective.
• Quillen's convergence theorem — Theorem 2.1. If J is an ideal of a simplicial ring R with pi_0(J) = 0 and every J_n weakly regular in R_n, then J^m is m-connective for every m.
• Convergence for the standard resolution — Corollary 2.2. If B (x)_A B = B, P is the standard resolution and Q = P (x)_A B, then J^m is m-connective and pi_n(Q) = pi_n(Q/J^m) for m >= n + 1.
• Low-degree stable homology of the free abelian group — Theorem 2.7. For an abelian sheaf X, the stabilization of the free abelian group functor has H_2 = X/2X and H_3 = the kernel of multiplication by 2 on X, and its homology in positive degrees is torsion.
• Spectra over the derived tensor product — Proposition 2.8. Over the integers, the extension of spectra to complexes over the derived tensor product is not fully faithful: for the ring R of that lesson, Hom(R, R[4]) = 0, while the corresponding group of spectra is Z/2.
• The augmentation ideal of the Tor resolution — Proposition 1.1. For B (x)_A B = B and Q = P (x)_A B, J/J^2 = L_{B/A} and J^q/J^(q+1) = S^q(L_{B/A}) as simplicial modules; pi_0(L_{B/A}) = 0, and for B = A/I, H_1(L_{B/A}) = I/I^2.
• The two conormal isomorphisms agree — Proposition 1.2. For B = A/I, the isomorphism H_1(L_{B/A}) = I/I^2 through Tor^A_1(B, B) and the augmentation ideal of the Tor resolution is the one given by the naive cotangent complex.
• Symmetric powers of 1-connective modules — Proposition 2.3. For L with flat terms and pi_0(L) = 0 over a ring, S^k L is k-connective and products of classes of degree one give Lambda^k(pi_1 L) = pi_k(S^k L).
• Comparison with the suspension of pi_1 — Proposition 2.4. If moreover pi_1(L) is flat and pi_i(L) = 0 for 2 <= i <= n, then pi_j(S^k L) = 0 for j <= n + k - 2, j != k (k >= 2); if pi_i(L) = 0 for all i >= 2, then pi_j(S^k L) = 0 for all j != k.
• Multiplicative spectral sequences — Lemma 3.1. The spectral sequence of a finite multiplicative filtration of a differential graded ring is a spectral sequence of bigraded rings, converging to the graded ring of the homology.
• The fundamental spectral sequence — Theorem 3.2. For B (x)_A B = B: E^2_(p,q) = H_(p+q)(S^q L_{B/A}) => Tor^A_(p+q)(B, B), a convergent first-quadrant spectral sequence of bigraded algebras.
• Weakly regular ideals and the cotangent complex — Theorem 4.2. For B = A/I: I is weakly regular iff D_q(B/A, M) = 0 for all M and q >= 2 iff I/I^2 is flat and L_{B/A} = (I/I^2)[1].
• Cotangent complex of a Koszul-regular quotient — Corollary 4.3. If I is Koszul-regular, then I/I^2 is locally free and L_{B/A} = (I/I^2)[1].
• Tor amplitude — Lemma 1.1. Tor amplitude in a ringed topos: connective objects, base change, faithfully flat descent and extensions in triangles.
• Examples of weak complete intersections — Proposition 2.2. Algebras with L = Omega flat (polynomial algebras), quotients by weakly regular ideals, and local complete intersection homomorphisms of rings and morphisms of schemes have L of tor amplitude in [-1, 0].
• Composition, cancellation and descent — Proposition 3.1. Weak complete intersections compose; cancel against a factor with L = Omega flat; and descend along faithfully flat C when C is one over A and over B.
• Quotients of algebras with flat differentials — Theorem 4.2. For P -> B surjective with kernel I and L_{P/A} = Omega_{P/A} flat, B is a weak complete intersection over A iff I is weakly regular; then L_{B/A} = (I/I^2 -> Omega_{P/A} (x) B).
• Tor-independent base change — Proposition 5.1. In a topos with enough points, L_{B/A} (x)^L B' = L_{B'/A'} for Tor-independent base change, and weak complete intersections are stable under it.
• Tor-independent base change in every topos — Corollary 5.3. In every topos, L_{B/A} (x)^L B' = L_{B'/A'} for Tor-independent base change, and weak complete intersections are stable under it.
• Products in degree two — Proposition 1.1. For B = A/I in a topos, H_3(L_{B/A}) -> Lambda^2(I/I^2) -> Tor_2^A(B, B) -> H_2(L_{B/A}) -> 0 is exact, the middle map being theta.
• Koszul homology and the Tor algebra — Proposition 2.2. For generators f of I, Lambda^2(B^r) -> Tor_2 -> H_1(f; A) -> B^r -> Tor_1 -> 0 is exact; if the classes of f are a basis of I/I^2 and theta_2 is surjective, f is H_1-regular.
• The Ext criterion — Proposition 3.3. I/I^2 is projective and I weakly regular iff Ext^q_B(L_{B/A}, M) = 0 for all M and q >= 2.
• Regular sequences over noetherian local rings — Proposition 3.4. For A noetherian local and I in the maximal ideal: I is generated by a regular sequence iff I is weakly regular iff D_2(B/A, k) = 0; then every minimal set of generators is regular.
• Weakly regular ideals of noetherian rings — Theorem 3.5. For A noetherian: weakly regular = regular ideal = Koszul-regular ideal = I/I^2 projective with H_2(L) = 0 = vanishing of D_2 or Ext^2 for finite modules or of D_2 at the residue fields.
• Local complete intersections of finite type — Theorem 4.1. For B of finite type over a noetherian A: lci iff weak complete intersection iff L_{B/A} perfect of tor amplitude [-1, 0] iff D_2(B/A, -) vanishes on finite modules.
• Local complete intersection morphisms — Corollary 4.3. A morphism of schemes locally of finite type over a locally noetherian base is lci iff it is a weak complete intersection.
• Regular immersions — Corollary 4.4. A closed subscheme of a locally noetherian scheme is regularly immersed iff its ideal sheaf is weakly regular.
• Regular local rings by the cotangent complex — Theorem 1.2. A noetherian local ring is regular iff its maximal ideal is generated by a regular sequence iff D_2(k/A) = 0 iff L_{k/A} = (m/m^2)[1].
• Presentations by regular local rings — Proposition 2.4. For P -> C surjective with P regular local and kernel I: D_q(k/C) = D_{q-1}(C/P, k) for q >= 3, and 0 -> D_2(k/C) -> I/m_P I -> m_P/m_P^2 -> m_C/m_C^2 -> 0.
• Complete intersections by D_3 — Theorem 2.6. A noetherian local ring is a complete intersection iff D_3(k/A) = 0 iff D_q(k/A) = 0 for q >= 3; then every presentation of the completion by a regular local ring has kernel generated by a regular sequence.
• Minimal presentations — Proposition 2.7. dim D_2(k/A) = mu(I) - edim P + edim A; presentations with I in m_P^2 exist, and for them dim D_2(k/A) is the minimal number of relations.
• Cohomology of the arrow topos — Proposition 2.2. For a morphism of ringed topoi f: X -> Y and C = (E <- F) in D^+(XY), restriction to Y is an isomorphism RGamma(XY, C) -> RGamma(Y, F).
• Derived Hom out of objects over Y — Proposition 2.8. For M in D^-(Y) and C = (E <- F) in D^+(XY): i_X^* RHom((0 <- M), C) = 0, and there are distinguished triangles i_Y^* RHom((0 <- M), C) -> RHom(M, F) -> RHom(M, Rf_*E) and RHom((0 <- M), C) -> RHom(M, F) -> RHom(Lf^*M, E).
• Relative Ext groups — Proposition 3.2. For M in D^-(Y) and F in D^+(Y) with Lf^*F in D^+(X), the relative Ext groups sit in a long exact sequence ... -> Ext^(i-1)(Lf^*M, Lf^*F) -> Ext^i(Y/X; M, F) -> Ext^i(M, F) -> Ext^i(Lf^*M, Lf^*F) -> ...
• The cotangent complex of a morphism of arrow topoi — Proposition 4.2. For a square q o f = g o p of ringed topoi and (p, q): XY -> ZT: i_X^* L_(XY/ZT) = L_(X/Z) and i_Y^* L_(XY/ZT) = L_(Y/T); likewise for (p, q): XY -> Z when q o f = p.
• Deforming a morphism with its target: the general problem — Theorem 3.1. Obstruction in Ext^2_T((0 <- L_(Y_o/Z_o)), (I <- J)) on the arrow topos of f_o, torsor under Ext^1, automorphisms Ext^0, with the long exact sequence through Ext(L_(Y_o/Z_o), J) and Ext(Lf_o^* L_(Y_o/Z_o), I).
• Flat deformations of a target with an extension of the morphism — Theorem 3.2. For g_o flat and X a flat deformation of X_o over Z: obstruction in Ext^2(Y_o/X_o; L_(Y_o/Z_o), g_o^* K), torsor under Ext^1, automorphisms the derivations over Z_o into g_o^* K vanishing on X_o.
• The obstruction as a product — Theorem 4.1. The obstruction is the Kodaira-Spencer class of X_oY_o -> X_oZ_o -> S composed with the pullback of the class of the thickening XZ of X_oZ_o.
• Deforming the target first — Proposition 5.1. The restriction of the obstruction to Y_o is the obstruction to deforming Y_o over Z; if Y_o deforms, the obstruction comes from Ext^1(Lf_o^* L_(Y_o/Z_o), h_o^* K).
• Extending a morphism into a given deformation of the target — Proposition 5.2. For a flat deformation Y of Y_o over Z: the obstruction o(Y) in Ext^1(Lf_o^* L_(Y_o/Z_o), h_o^* K) to extending f_o to X -> Y over Z, o(Y') - o(Y) is minus the pullback of the difference of the classes of Y' and Y, and the connecting map carries o(Y) to the obstruction of Theorem 3.2.
• Extending first-order operators to antiderivations — Proposition 2.2. K-linear d_0: A -> E and d_1: E -> Lambda^2 E with the Leibniz rules extend uniquely to a K-antiderivation of degree 1 of Lambda_A E; d^2 = 0 if and only if it vanishes on A and E.
• De Rham complexes on quotients of Omega^1 — Proposition 2.4. For a surjection pi: Omega^1_(A/K) -> E with kernel N there is at most one de Rham complex with d|_A = pi o d; it exists if and only if Lambda^2 pi(d w) = 0 for w in N.
• The Lie algebroid of a de Rham complex — Proposition 4.4. theta_x = [d, i_x] and [theta_x, i_y] = i_([x,y]) make E^vee a Lie algebroid over A/K when d^2 = 0; Chevalley-Eilenberg formula for d.
• De Rham complexes and Lie algebroids on locally free modules — Theorem 5.3. For E locally free of finite type, Jacobi holds if and only if d^2 = 0 (Theorem 5.2), and de Rham complexes are anti-equivalent to Lie algebroids, the inverse given by the Chevalley-Eilenberg differential.
• Invariant forms on torsors and the Atiyah sequence — Theorem 6.5. For a torsor X -> Y under a flat group scheme over S: Omega^1_(Y/S) -> omega^1_(X/S) -> omega^1_(X/Y) -> 0 exact, the dual sequence ends with the anchor, its kernel is the Lie algebra of G_Y twisted by X through the adjoint action; exact on both sides for G smooth.
• A formal category that is not strict — Proposition 1.9. The formal additive group over a field of characteristic 0 is not isomorphic to a projective system of affine categories; in characteristic p it is one.
• Divided powers extend along tensor products of augmented algebras — Lemma 2.4. For augmented A-algebras B, C, divided powers on I_B extend to B (x)_A C without flatness; with divided powers on I_C the augmentation ideal of B (x)_A C gets a unique structure making both coprojections PD morphisms.
• Divided powers on the nerve of an affine category — Proposition 2.6. Divided powers on the augmentation ideal I of the arrows extend uniquely to every degree of the nerve; they are compatible with all cofaces exactly when the composition is a PD morphism.
• PD completion of a formal category — Proposition 2.7. The completion of the nerve of an affine PD formal category along the divided powers of its augmentation ideal is the nerve of a PD-adic formal category with arrows the system of the P/I^[r], the composition induced through P/I^[2r-1]; it need not be strict.
• Divided powers on arbitrary formal categories — Proposition 2.9. For an arbitrary formal category with divided powers on the augmentation ideal of its arrows, the augmentation ideal of the nerve carries divided powers, unique up to unique isomorphism, making the factor inclusions PD morphisms; they make it a PD formal category iff the composition is a PD morphism, and its PD completion is PD-adic.
• Envelopes of tensor products of augmented algebras — Proposition 3.3. D_(B (x)_A C)(I_B (x) C + B (x) I_C) = D_B(I_B) (x)_A D_C(I_C) for augmented A-algebras; hence the envelope of the nerve of an affine category is the nerve of an affine category (Corollary 3.4), and the Čech–Alexander complex is defined.
• The Alexander-Whitney algebra of a cosimplicial algebra — Proposition 1.2. The cochains of a cosimplicial algebra with d = sum (-1)^i d^i and the front-back product form a differential graded algebra; the normalized cochains form a sub-algebra.
• The tensor algebra of the augmentation ideal and the normalized nerve — Proposition 2.4. T_A(I) with the antiderivation extending d in degrees 0, 1 maps to the normalized nerve of an affine category, isomorphically in degrees <= 2, and everywhere when d^0 = d^1 on A.
• The exterior derivative of a PD formal category — Proposition 3.4. A unique antiderivation of Lambda_A(I/I^[2]) compatible with the tensor algebra; a de Rham complex when d^0 = d^1, when Omega^1 is generated by the d^0(a) - d^1(a), or when Omega^1 is flat.
• The Čech–Alexander complex gives the de Rham complex — Example 3.5. For the Čech–Alexander complex of A/K, Omega^1 = Omega^1_(A/K) and the exterior derivative is the de Rham differential.
• The graded ring of a flat PD formal category — Theorem 2.4. If Omega^1_R = I/I^[2] is flat over A, Gamma_A(Omega^1_R) -> gr P is an isomorphism and the graded de Rham complex of the shifted category is the Koszul complex of the identity of Omega^1_R.
• The formal Poincaré lemma — Corollary 3.1. For flat Omega^1_R the augmented de Rham complex A -> P (x) Omega^bullet_R of the shifted category is acyclic modulo every step of the divided power filtration, hence levelwise for PD-adic R.
• Cohomology of flat PD-adic formal categories — Theorem 3.5. For a PD-adic formal category with flat Omega^1_R, affine, strict or the PD completion of one, p: N(J^[m]) -> F^m Omega_R is a quasi-isomorphism for every m: its cohomology is de Rham cohomology, with the divided power and Hodge filtrations.
• Invariance of the cotangent complex of simplicial rings — Theorem 3.1. For a commutative square of simplicial rings A -> B, A' -> B' whose maps A -> A' and B -> B' are quasi-isomorphisms, L^Delta_(B/A) (x)_B B' -> L^Delta_(B'/A') is a quasi-isomorphism.
• Degreewise free resolutions compute the cotangent complex — Corollary 3.5. For a map of rings A -> B and a quasi-isomorphism P -> B of simplicial A-algebras with P degreewise free, Omega^1_(P/A) (x)_P B is canonically isomorphic to L_(B/A) in D(B).
• Tor-independent base change — Proposition 3.6. For maps of rings A -> B, A -> A' of a topos with Tor^A_i(B, A') = 0 for i > 0, P_A(B) (x)_A A' resolves B' = B (x)_A A' by free A'-algebras and L_{B/A} (x)^L_B B' -> L_{B'/A'} is an isomorphism.
• The cotangent complex corepresents derivations — Proposition 4.2. Hom_(D(B))(L^Delta_(B/A), M) = Hom_(D(A-Alg/B))(B, B + M), naturally in the B-module M.
• The diagonal graded cotangent complex — Theorem 2.4. For a map of graded simplicial rings, the diagonal graded cotangent complex is invariant under quasi-isomorphisms, compares with the ungraded one through theta, and its parts of degree 0 and 1 are the cotangent complex of the degree 0 rings and the corresponding degree 1 complex.
• The graded transitivity triangle — Theorem 4.1. A composite of maps of graded simplicial rings gives a graded transitivity sequence and a distinguished triangle of graded cotangent complexes, compatible with the parts of degree 0.
• The Atiyah class of a simplicial module — Theorem 5.1. The Atiyah class of a simplicial module M over B is the third map of the degree one part of the graded transitivity triangle of A -> B -> B + M.
• Flat epimorphisms have vanishing cotangent complex — Proposition 2.1. For a map of rings A -> B of a topos which is flat and an epimorphism of rings, in particular a localization morphism, L_(B/A) = 0 in D(B).
• The cotangent complex on affine schemes — Proposition 3.2. For a map of rings A -> B, the cotangent complex of Spec B -> Spec A is quasi-isomorphic to the quasi-coherent module associated with L_(B/A).
• Tor-independent base change for ringed topoi — Theorem 4.2. For a square of ringed topoi whose structure ring of X' is a localization of u^-1 O_X (x) f'^-1 O_S', with vanishing higher Tor, u^*L_(X/S) = L_(X'/S') and L_(X'/S) = u^*L_(X/S) + f'^*L_(S'/S).
• The derived de Rham complex does not depend on the free resolution — Theorem 2.3. For every degreewise free resolution P' -> B over A, a canonical chain of filtered quasi-isomorphisms connects L Omega_(B/A) with the realization of Omega_(P'/A), with the Hodge filtrations; gr^m = Lambda^m L_(B/A)[-m].
• Derived de Rham complexes over rings containing Q — Theorem 3.2. If every nonzero integer is invertible in A, then A -> L Omega_(B/A) is a quasi-isomorphism for every A-algebra B.
• Derived exterior powers of a two-term complex — Proposition 5.2. For a map u: N -> M of flat modules, L Lambda^n(N -> M) is isomorphic to the Koszul complex of divided powers Gamma^(n-q) N (x) Lambda^q M, shifted by n.
• The Cartier isomorphism for free algebras — Theorem 2.1. Over a ring with p = 0, the inverse Cartier map Omega_(B^(p)/A) -> H(Omega_(B/A)) is an isomorphism of graded algebras for B free, or a symmetric algebra of a flat module.
• The derived Cartier isomorphism — Theorem 3.2. The conjugate filtration of L Omega_(B/A) has graded pieces Lambda^i L_(B^(p)/A)[-i] when Tor_j^A(B, A_F) = 0 for j > 0; in general the twisted free resolution computes them.
• Derived de Rham complexes of smooth morphisms in characteristic p — Corollary 4.1. For a smooth morphism of schemes in characteristic p, L Omega_(X/S) -> Omega_(X/S) is a quasi-isomorphism carrying the conjugate filtration to the canonical filtration.
• The first graded piece of the completed Čech nerve — Proposition 2.3. For a presentation B = Q/I with Omega^1_(Q/A) flat, the normalized cochains of K/K^2 are I/I^2 -> Omega^1_(Q/A) (x)_Q B in degrees 0, 1.
• The first graded piece for an arbitrary affine category — Proposition 2.5. For an affine category with a surjection onto a constant algebra and flat module of differentials, the normalized first graded piece of the kernel is the two-term complex I/I^2 -> U/U^2 (x) B.
• Hodge-completed derived de Rham complexes of complete intersections in characteristic zero — Theorem 4.1. Over a ring containing Q, for B = Q/I with Q weakly smooth and I weakly regular, the Hodge-completed derived de Rham complex is filtered quasi-isomorphic to the normalized Čech nerve of Q completed along the kernel to B.
• Gluing monomorphic quasi-isomorphisms of simplicial sets — Proposition 2.5. A pushout of a monomorphism of simplicial sets that is a quasi-isomorphism is a quasi-isomorphism, and so is its product with a zigzag interval relative to the two ends.
• Pushouts along monomorphisms preserve quasi-isomorphisms — Proposition 2.6. A map of pushout diagrams of simplicial sets, with one leg a monomorphism in both, that is a quasi-isomorphism at each vertex induces a quasi-isomorphism of pushouts.
• Left fractions in the homotopy category of simplicial sheaves — Theorem 5.1. If the topos has enough points, the quasi-isomorphisms of simplicial sheaves form a left multiplicative system in the homotopy category.
• Kan's functor Ex^∞ — Theorem 3.6. For a simplicial set X, Ex^∞ X is a Kan complex and X -> Ex^∞ X is a monomorphic quasi-isomorphism; Ex^∞ preserves finite limits and Kan fibrations.
• Kan fibrations that are quasi-isomorphisms — Theorem 3.7. A Kan fibration of simplicial sets, over any base, is a quasi-isomorphism if and only if it lifts against every monomorphism.
• Anodyne extensions are the monomorphic quasi-isomorphisms — Corollary 3.8. A monomorphism of simplicial sets is anodyne if and only if it is a quasi-isomorphism.
• Kan's functor for simplicial sheaves — Proposition 4.1. For a simplicial sheaf X, X -> Ex^∞(X) is a monomorphic quasi-isomorphism into a Kan simplicial sheaf; Ex^∞ commutes with finite limits and inverse images and preserves fibrations.
• Moore's theorem for simplicial groups of a topos — Proposition 1.2. For a simplicial group G of a topos, G is Kan, pi_0 G = G_0/d_0(N_1 G) and pi_n(G) is the n-th homology sheaf of the normalized complex, abelian for n >= 1.
• Fibrations of simplicial groups — Proposition 1.4. A homomorphism of simplicial groups of a topos is a fibration iff it is an epimorphism on the normalized complexes in positive degrees, iff G -> H x_(pi_0 H) pi_0 G is an epimorphism in every degree.
• Right fractions among Kan simplicial sheaves — Theorem 4.1. If the topos has enough points, quasi-isomorphisms between Kan simplicial sheaves form a right multiplicative system in the homotopy category.
• The Kan derived category is the simplicial derived category — Theorem 4.3. For every topos, localizing the homotopy category of Kan simplicial sheaves at quasi-isomorphisms gives a category equivalent to the simplicial derived category, through Ex^∞.
• Free simplicial modules on derived categories — Proposition 1.3. If the topos has enough points, X -> Lambda^(X) induces a left adjoint of the forgetful functor D(Lambda) -> D(T) on simplicial derived categories; so Ext^p(Lambda^(X), F) = Hom_D(T)(X, sigma^p F).
• Ext groups through the Hom complex — Proposition 2.4. For a simplicial sheaf Y and a simplicial Lambda-module M, the p-th cohomology of the Hom complex of C(Lambda^(Y)) and C(M) is Hot(Y, sigma^p M).
• Ext groups of free simplicial modules as colimits — Theorem 3.2. If the topos has enough points and X is a Kan simplicial sheaf, Ext^p(Lambda^(X), F) is the colimit of H^p Hom(X', F) over the fibrant quasi-isomorphisms X' -> X.
• Verdier's hypercovering theorem — Corollary 3.5. If the topos has enough points, H^p(T, F) is the colimit of the Čech cohomology of the hypercoverings; hypercoverings are the fibrant quasi-isomorphisms to the final object (Lemma 3.4).
• Giraud's theorem — Theorem 3.1. A locally small category with finite limits, disjoint coproducts stable under pullback, effective equivalence relations with quotients stable under pullback and a small set of generators is equivalent to the sheaves on a small full subcategory of generators closed under fibre products, with the epimorphic families as coverings.
• Topoi satisfy Giraud's conditions — Proposition 3.2. The category of sheaves on a small site satisfies Giraud's conditions; so the conditions characterize topoi up to equivalence.
• Second total topoi are topoi — Corollary 4.1. For every good topos X fibred over a small category, the second total category Top^o(X) is a topos, and the level functors are inverse images of morphisms of topoi.
• Classifying topoi of pseudo-categories — Corollary 4.2. For a good pseudo-category X, the category BX of equivariant sheaves is a topos, and the nerves are inverse images of morphisms of topoi into it.
• Deforming the target of a morphism of diagrams — Theorem 1.4. For diagrams of schemes with g_o flat and a flat deformation X of X_o over a thickening Z of Z_o: obstruction in Ext^2(Y_o/X_o; L_(Y_o/Z_o), g_o^* K) to a flat deformation of Y_o with an extension of f_o, torsor under Ext^1, automorphisms Ext^0.
• Equivariant deformations of the target — Theorem 2.2. For morphisms of G-schemes X -> Y -> Z with g flat and an equivariant flat deformation X' -> Z' of h = gf: obstruction, torsor and automorphisms in the relative Ext groups of the nerves N(G,X) -> N(G,Y).
• Deforming a group scheme with a homomorphism from a deformed source — Theorem 3.1. For a homomorphism u: F -> G of flat group schemes and a deformation F' of F: obstruction in Ext^2(N(G)/N(F); L_(N(G)/S), f_G^* I) to a deformation G' of G with an extension F' -> G' of u, torsor under Ext^1, automorphisms Ext^0.
• Deforming a module scheme with a homomorphism from a deformed source — Theorem 4.1. For a homomorphism u: F -> G of A-module schemes and a deformation F' of F: obstruction, torsor and automorphisms in the relative Ext groups of the spectral nerves N(A,F) -> N(A,G).
• A given deformation of the target module scheme — Proposition 4.2. If G has a deformation G', the obstruction is the image under the connecting map of the obstruction to extending u to F' -> G', which changes by minus the restriction of the difference of two deformations.
• Ext groups over the ring scheme — Corollary 4.4. For a homomorphism u: F -> G of module schemes over A_S and a deformation of F, the groups of obstruction, torsor and automorphisms are Ext groups over A_S of the cone of u with coefficients in the Lie complex of G tensored with the ideal.

Reading order

Eighty-seven lessons, to be read in order. The first three develop homotopical algebra in a ringed topos; the fourth to sixth treat simplicial objects, nerves and colimits over categories of ordinals; the seventh and eighth construct the total topoi of fibred topoi and their modules, the ninth to eleventh compute their cohomology, the twelfth and thirteenth construct the left derived functors of the left adjoints of inverse images of modules, on both total topoi, the fourteenth proves a duality formula for diagrams in a topos, the fifteenth treats equivariant sheaves on simplicial topoi, the sixteenth proves Giraud's characterization of topoi, the seventeenth to nineteenth treat the nerves of induced and coinduced modules and the cohomology and Ext groups of equivariant modules, the twentieth to twenty-second construct spectral diagrams, the stabilization of non-additive functors and stable derived functors, the twenty-third to twenty-sixth develop the homotopy theory of simplicial modules over a simplicial ring: the calculus of fractions, triangles and Ext groups, derived tensor products and their spectral sequences, the twenty-seventh treats differential graded modules in a topos and compares them with simplicial modules, the twenty-eighth and twenty-ninth treat graded monoids and the nerves of their actions, graded rings and the spectra of modules, the thirtieth proves the adjunction formulas for diagonal rings, the thirty-first constructs the stable ring of a ring and the Mac Lane resolution, the thirty-second expresses Ext groups of modules as cohomology of spectral nerves, the thirty-third compares the Zariski and the big fppf sites of schemes and of diagrams of schemes for quasi-coherent and perfect complexes, the thirty-fourth classifies extensions and deformations of diagrams of schemes and of their morphisms, the thirty-fifth constructs equivariant cotangent complexes, the thirty-sixth classifies equivariant extensions and deformations, the thirty-seventh proves that flat group schemes locally of finite presentation are local complete intersections, the thirty-eighth treats torsors, descent along them, co-Lie complexes and the Atiyah class, the thirty-ninth classifies deformations of torsors, the fortieth treats the co-Lie complex of a group scheme, the forty-first constructs its adjoint form and computes the cotangent complex of the nerve of a group scheme, the forty-second classifies deformations of flat group schemes and of their homomorphisms, the forty-third computes the Lie complexes of commutative group schemes, of their torsors, nerves and spectra, the forty-fourth constructs the Lie complexes of module schemes and computes the Ext groups of their spectral nerves, the forty-fifth classifies deformations of module schemes and of their homomorphisms, the forty-sixth develops graded modules and the graded cotangent complex, the forty-seventh constructs the Atiyah class of a module, the forty-eighth classifies deformations of modules and of their morphisms and expresses the obstruction through the Atiyah class, the forty-ninth to fifty-second develop filtered derived categories, filtered tensor products, the filtered internal Hom and the duality of perfect filtered complexes, the fifty-third proves the additivity and multiplicativity of traces, the fifty-fourth to fifty-sixth develop filtered and graded powers of complexes, the filtered shift isomorphisms and the diagonal maps of symmetric and exterior powers, the fifty-seventh to fifty-ninth define the L-series of endomorphisms of perfect complexes, the Atiyah class of filtered complexes and the Chern classes of perfect complexes, the sixtieth constructs Chern classes with supports from partial connections, the sixty-first constructs the cotangent complex of a map of simplicial rings, with Tor-independent base change, the sixty-second extends it to graded simplicial rings, with graded derivations and the graded transitivity triangle, the sixty-third treats localization morphisms and the cotangent complex, the sixty-fourth treats weakly regular ideals, the sixty-fifth proves Quillen's convergence theorem, the sixty-sixth constructs the fundamental spectral sequence and characterizes weakly regular ideals by the cotangent complex, the sixty-seventh treats weak complete intersections, the sixty-eighth weakly regular ideals of noetherian rings, the sixty-ninth regular local rings and complete intersections, the seventieth the arrow topos and relative Ext groups, the seventy-first deforming a morphism together with its target, the seventy-second de Rham complexes and Lie algebroids, the seventy-third formal categories with divided powers and the Čech–Alexander complex, the seventy-fourth the de Rham complex of a formal category with divided powers, the seventy-fifth décalage and the formal Poincaré lemma, the seventy-sixth cohomology of flat formal categories with divided powers, the seventy-seventh the derived de Rham complex, the seventy-eighth derived de Rham complexes in characteristic p, the seventy-ninth the completed derived de Rham complex of a complete intersection, the eightieth the simplicial derived category of a topos, the eighty-first local fibrations and Kan's subdivision functor, the eighty-second simplicial groups and right fractions, the eighty-third hypercoverings and Ext groups of free simplicial modules, the eighty-fourth deforming the target of a morphism of diagrams, equivariantly and for group and module schemes, and the eighty-fifth relative cohomology of classifying topoi, with the relative Ext groups of these deformation problems for group schemes and for equivariant morphisms, the eighty-sixth filtered transitivity triangles and the filtered Atiyah triangle, and the eighty-seventh truncated nerves and the level filtration. The simplicial homotopy theory and Quillen's Theorem A are taken from the course Algebraic spaces and stacks; sites, modules and derived categories from the Stacks project.