SGA 6

The items of SGA 6 that a declaration of an article carries, numbered as printed, each with the articles it is in and the attribute's comments where there are any.

Exposé 0

(2.1): 10.131 Differentials (“special case”), 29.33 Sheaf of differentials of a morphism, 26.10 Immersions of schemes (“special case”), 76.5 Conormal sheaf of an immersion (“more general”)
(2.3.Corollary2): 11.2 Noncommutative algebras (“affine clause”), 9.7 Finite extensions (“affine clause”)
(2.5): 35.8 Quasi-coherent sheaves and topologies, I (“partial”), 17.10 Quasi-coherent modules, 26.7 Quasi-coherent sheaves on affines (“partial”)
(2.9): 10.39 Flat modules and flat ring maps (“affine case”), 42.15 Push and pull (“special case”), 41.9 Flat morphisms (“affine case”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“special case”), 29.26 Flat morphisms (“special case”), 65.4 Lists of useful properties of morphisms of schemes (“special case”), 45.13 Exterior powers and K-groups (“partial”)
(2.11): 10.9 Localization (“affine step”)
(3.1): 30.22 Cohomology and base change, III (“affine-target flat case”), 12.11 K-groups (“more general”), 15.121 Perfect complexes and K-groups (“special case”), 37.61 Perfect morphisms (“partial”; “special case”), 36.11 Derived category of coherent modules (“partial”), 36.27 Producing perfect complexes (“more general”), 36.38 K-groups (“more general”; “special case”), 21.47 Perfect complexes (“partial”)
(3.2): 10.135 Local complete intersections (“special case”)
(4.1): 30.22 Cohomology and base change, III (“special case”), 37.61 Perfect morphisms, 36.27 Producing perfect complexes (“more general”)
(4.2): 20.49 Perfect complexes (“more general”)
(4.3): 10.134 The naive cotangent complex (“partial”), 98.22 Obstruction theories (“partial”), 92.14 The cotangent complex of a local complete intersection (“special case”), 49.10 Quasi-finite syntomic morphisms (“special case”), 49.11 Finite syntomic morphisms (“special case”)
(4.4): 10.134 The naive cotangent complex (“special case”), 10.138 Formally smooth maps (“partial”), 98.22 Obstruction theories (“special case”), 91.2 Deformations of rings and the naive cotangent complex (“special case”), 15.87 The naive cotangent complex (“affine case”), 88.11 Finite type morphisms (“partial”), 18.35 The naive cotangent complex (“NL_{B/A} construction”)
(6.Lemma3): 38.36 Blow up squares and the ph topology (“partial”)

Exposé I

(1.1.3): 13.3 The definition of a triangulated category (“partial”), 22.24 Linear categories (“partial”), 90.11 Tangent spaces of functors (“partial”)
(1.1.5): 42.2 Periodic complexes and Herbrand quotients, 13.6 Quotients of triangulated categories (“more general”), 13.10 Distinguished triangles in the homotopy category (“K(C_X) is triangulated”; “K^-, K^b triangulated”; “partial”), 13.11 Derived categories (“D^+ degree condition”; “D^- degree condition”; “D^b degree condition”; “the derived fiber D(C_X)”; “the fiber D⁻(C_X)”), 13.17 Triangulated subcategories of the derived category (“partial”), 12.5 Abelian categories (“partial”), 12.13 Complexes (“partial”), 64.7 Filtered derived category (“partial”; “the fiber K⁺(C_X)”)
(1.2): 10.5 Finite modules and finitely presented modules (“special case”), 11.2 Noncommutative algebras (“special case”), 9.7 Finite extensions (“special case”), 12.28 Projectives
(1.3): 10.5 Finite modules and finitely presented modules (“more general”; “special case”), 10.51 More Noetherian rings (“more general”), 11.2 Noncommutative algebras (“special case”), 9.4 Vector spaces (“more general”), 9.7 Finite extensions (“special case”), 19.13 Additional remarks on Grothendieck abelian categories (“partial”), 17.3 The abelian category of sheaves of modules (“partial”), 17.9 Modules of finite type, 6.10 Sheaves of modules (“partial”), 18.10 Sheaves of modules (“partial”), 18.14 The abelian category of sheaves of modules (“partial”), 18.23 Local types of modules
(1.3.1): 10.78 Finite projective modules (“affine case; (i)⇔(iii)”), 12.28 Projectives (“(i) ⇔ (ii), special case”)
(2.2): 20.47 Pseudo-coherent modules (“a)”; “b) (ii')=>(ii)”), 21.45 Pseudo-coherent modules (“a)”)
(2.5): 20.47 Pseudo-coherent modules (“b), ringed space”), 15.66 Pseudo-coherent modules, I (“b), modules over a ring”; “b), pseudo-coherent case”), 21.45 Pseudo-coherent modules (“b)”)
(2.6): 15.66 Pseudo-coherent modules, I (“partial”; “special case”), 37.59 Relative pseudo-coherence (“row 1 (n+1, n, n)”; “row 2 (n, n, n)”; “row 3 (n, n, n-1)”)
(2.8): 10.5 Finite modules and finitely presented modules (“special case”), 11.2 Noncommutative algebras (“special case”), 9.7 Finite extensions (“special case”)
(2.9): 15.66 Pseudo-coherent modules, I (“special case”)
(2.11): 15.66 Pseudo-coherent modules, I (“Part a), special case”; “Part b), special case”; “b), H^i=0 for i>m+1”)
(2.12): 20.47 Pseudo-coherent modules (“n-pc: sum ⇒ F'”; “n-pc: sum ⇒ F''”; “n-pc: sum ⇒ F',F''”), 15.66 Pseudo-coherent modules, I (“necessity”), 21.45 Pseudo-coherent modules (“n-pc: sum ⇒ F'”; “n-pc: sum ⇒ F''”; “pc: sum ⇒ F'”)
(2.13): 15.66 Pseudo-coherent modules, I (“special case”), 75.13 Pseudo-coherent and perfect complexes (“special case”)
(2.14): 15.83 Relatively pseudo-coherent modules (“(ii), affine case”), 37.59 Relative pseudo-coherence (“(ii)”)
(2.15.1): 10.78 Finite projective modules (“affine case, (3)⇒(7)”), 17.14 Locally free sheaves (“affine case”; “ringed space case”)
(2.16): 20.47 Pseudo-coherent modules (“special case”)
(2.16.1): 43.14 Intersection multiplicities using Tor formula (“b), special case”)
(2.16.2): 15.77 Lifting complexes (“(i), punctual”), 37.71 Relatively perfect objects (“(i), special case”)
(3.1): 10.90 Coherent rings (“special case”)
(3.2): 110.33 A finite flat module which is not projective (“special case”), 55.4 The Picard group of a numerical type (“special case”)
(3.3): 10.90 Coherent rings (“special case”), 17.9 Modules of finite type (“c) finite type”)
(3.4): 10.90 Coherent rings (“partial”; “special case”), 103.17 Coherent sheaves on locally Noetherian stacks (“special case”)
(3.5): 10.90 Coherent rings (“a) (i)⇔(ii)”), 17.12 Coherent modules (“a) (ii)⇒(i)”), 15.66 Pseudo-coherent modules, I (“b), coherent ring”), 36.10 Pseudo-coherent and perfect complexes (“b) last claim, Noeth X”; “b), Noetherian scheme”)
(3.6): 10.90 Coherent rings (“special case”)
(3.7): 10.39 Flat modules and flat ring maps (“special case”), 10.71 Ext groups (“Clause b), special case”), 48.2 Dualizing complexes on schemes (“Clause b), special case”), 41.9 Flat morphisms (“special case”), 15.67 Pseudo-coherent modules, II (“affine case, modules”)
(4.1): 20.46 Strictly perfect complexes (“ringed space”; “special case”), 13.19 Projective resolutions (“special case”), 21.44 Strictly perfect complexes (“special case”)
(4.2): 20.46 Strictly perfect complexes (“ringed space case”), 21.44 Strictly perfect complexes
(4.8): 15.76 Perfect complexes (“affine (ring) case”; “affine case, module”), 6.10 Sheaves of modules (“ambient S-category C”), 18.10 Sheaves of modules (“ambient S-category C”)
(4.10): 64.10 Perfectness (“Clause (i), special case”)
(4.12.2): 64.10 Perfectness (“Case C = R-modules”)
(4.15): 15.76 Perfect complexes (“special case”)
(4.19): 20.49 Perfect complexes (“single ringed space”), 15.76 Perfect complexes (“D(R), R a ring”)
(4.19.2): 36.39 Determinants of complexes (“special case”)
(5.1): 20.48 Tor dimension (“ringed space case”), 15.68 Tor dimension (“affine case”; “module over a ring”), 21.46 Tor dimension
(5.1.1): 10.39 Flat modules and flat ring maps (“special case”), 20.26 Flat resolutions (“special case”)
(5.2): 20.48 Tor dimension (“Tor-amp(F) ⊂ [m,n]”; “finite flat amplitude”), 15.68 Tor dimension (“Tor-amp(F) ⊂ [m,n]”; “finite flat amplitude”), 21.46 Tor dimension (“Tor-amp(F) ⊂ [m,n]”; “finite flat amplitude”)
(5.3): 20.48 Tor dimension (“partial”)
(5.4): 15.68 Tor dimension (“commutative ring”)
(5.6.2): 20.48 Tor dimension (“ringed space case”), 21.46 Tor dimension (“converse, enough points”; “fiber direction”)
(5.7): 10.39 Flat modules and flat ring maps (“a), special case”), 15.56 Injective modules (“(ii) ⇒ (i)”), 15.71 Injective dimension (“(i) definition”)
(5.8): 20.49 Perfect complexes (“(ii) ⟹ (i)”), 15.76 Perfect complexes (“special case”), 21.47 Perfect complexes (“(ii) ⟹ (i)”)
(5.8.1): 20.49 Perfect complexes (“necessity”; “sufficiency”), 15.76 Perfect complexes (“module over a ring”), 21.47 Perfect complexes
(5.8.3): 10.78 Finite projective modules (“commutative ring case”), 10.83 Descent for finite projective modules (“(ii) ⇒ (i)”)
(5.8.4): 42.56 An Adams operator (“5.8.4 a), special case”), 57.8 Fourier-Mukai functors (“5.8.4 b), special case”), 36.37 The resolution property and perfect complexes (“special case”), 36.38 K-groups (“5.8.4 a), special case”; “special case”), 76.52 Relatively perfect objects (“5.8.4 a), special case”)
(5.9): 15.68 Tor dimension (“special case”)
(6.0): 10.55 K-groups (“(ii), special case”), 10.78 Finite projective modules (“special case”), 17.14 Locally free sheaves (“existence of rg_U(L)”)
(6.3): 13.28 K-groups (“triangulated case”)
(6.10): 17.14 Locally free sheaves (“special case”)
(6.11): 11.2 Noncommutative algebras (“special case”), 9.7 Finite extensions (“special case”)
(7.1.1): 15.75 Derived hom (“partial”)
(7.2): 20.50 Duals (“special case”)
(7.3): 15.76 Perfect complexes (“affine (ring) case”; “finite Tor-dim stability”), 21.48 Duals (“perfect ⇒ dual perfect”)
(7.4): 20.42 Internal hom in the derived category (“first isomorphism”), 15.75 Derived hom (“affine case, 1st iso”), 21.35 Internal hom in the derived category (“first isomorphism”; “functoriality, 1st iso”)
(7.6): 10.78 Finite projective modules (“special case”), 15.100 Some evaluation maps (“affine case, E perfect”)
(7.7): 20.50 Duals (“special case”), 15.76 Perfect complexes (“affine case”)
(8.1): 21.35 Internal hom in the derived category (“partial”)
(8.5): 20.49 Perfect complexes (“special case”), 21.47 Perfect complexes (“partial”)
(8.6): 64.15 Preliminaries and sorites (“Punctual (affine) case”)

Exposé IX

(1.1): 6.32 Closed immersions and (pre)sheaves (“Support clause”)
(1.5): 10.110 Regular rings and global dimension (“partial”)
(1.7): 10.55 K-groups (“Lemma 1.7, case A = k[T]”; “special case”), 15.22 Torsion free modules (“more general”)
(3.4): 10.55 K-groups (“Case S = Spec k”)

Exposé V

(2.2.1): 12.28 Projectives (“special case”), 45.13 Exterior powers and K-groups (“Second conjunct only”)
(6.4): 9.6 Field extensions

Exposé VI

(1.2): 12.16 Graded objects (“partial”)
(1.5.2): 10.83 Descent for finite projective modules (“Case B = A, n = 0”), 10.114 Dimension of finite type algebras over fields (“more general”)
(1.6): 10.71 Ext groups (“partial”), 15.26 Flatness and finiteness conditions (“partial”)
(2.2): 6.10 Sheaves of modules (“more general”), 18.10 Sheaves of modules (“more general”)
(2.3.1): 6.26 Morphisms of ringed spaces and modules (“more general”; “partial”)
(2.3.2): 27.8 Proj of a graded ring (“(f), special case”; “affine base”; “affineness of D₊(f)”)
(2.7): 24.4 Sheaves of graded modules (“partial”)
(4.1): 27.22 Grassmannians (“Grassmannian, free case”)
(4.9): 27.22 Grassmannians (“G_S^(n, special case”; “special case”)
(6.1): 10.32 Locally nilpotent ideals

Exposé VII

(1.1): 17.24 Koszul complexes (“special case”), 15.29 The Koszul complex (“more general”)
(1.4): 31.21 Regular ideal sheaves (“1.4.2”; “regular ideal J”; “special case”), 31.22 Regular immersions (“regular immersion”)
(1.5): 35.23 Properties of morphisms local in the fpqc topology on the target (“special case”), 31.22 Regular immersions (“flat base change, H1”; “flat base change, Koszul”; “flat base change, quasi”), 15.31 Koszul regular sequences (“special case”), 15.33 Regular ideals (“converse, affine case”)
(1.6): 10.139 Smoothness and differentials (“special case”), 29.32 Conormal sheaf of an immersion
(1.7): 31.22 Regular immersions
(1.8): 31.33 Blowing up
(1.9): 37.61 Perfect morphisms (“more general”)
(1.10): 42.60 Gysin maps for diagonals (“special case”), 31.23 Relative regular immersions (“more general”; “section case”), 37.62 Local complete intersection morphisms (“special case”)
(2.1): 22.3 Differential graded algebras, 15.64 Products and Tor (“partial”)
(2.6): 36.38 K-groups (“special case”)
(3.7): 38.36 Blow up squares and the ph topology (“more general”), 26.17 Fibre products of schemes
(4.2): 31.22 Regular immersions (“partial”), 31.23 Relative regular immersions (“more general”)
(4.6): 10.14 Base change (“more general”)

Exposé VIII

(1.1): 31.22 Regular immersions (“special case”), 31.23 Relative regular immersions (“more general”), 37.62 Local complete intersection morphisms (“at a point”)
(1.2): 42.59 Gysin maps for local complete intersection morphisms, 37.62 Local complete intersection morphisms
(1.3): 42.60 Gysin maps for diagonals (“special case”), 31.22 Regular immersions (“clause (i), special case”), 31.23 Relative regular immersions (“more general”), 41.10 Topological properties of flat morphisms (“more general”), 59.26 Étale morphisms (“more general”), 111.33 Schemes, 15.31 Koszul regular sequences (“VII 1.5, special case”), 29.3 Immersions (“(iv)”), 29.26 Flat morphisms (“more general”), 29.35 Smooth morphisms (“more general”), 29.37 Étale morphisms (“more general”), 26.17 Fibre products of schemes, 26.21 Separation axioms (“(iv)”)
(1.4): 49.10 Quasi-finite syntomic morphisms (“special case”), 49.11 Finite syntomic morphisms (“special case”)
(1.5): 42.59 Gysin maps for local complete intersection morphisms (“special case”)
(1.6): 10.126 Algebras and modules of finite presentation (“special case”), 10.137 Smooth ring maps (“special case”), 37.62 Local complete intersection morphisms (“flat base change stable”), 29.35 Smooth morphisms (“partial”), 26.17 Fibre products of schemes (“partial”), 26.18 Base change in algebraic geometry (“partial”), 65.4 Lists of useful properties of morphisms of schemes (“partial”)
(1.7): 15.84 Pseudo-coherent and perfect ring maps (“affine case”), 37.61 Perfect morphisms (“special case”), 37.62 Local complete intersection morphisms (“partial”)
(1.8): 10.137 Smooth ring maps (“affine case”), 31.23 Relative regular immersions (“partial, special case”), 29.3 Immersions (“partial”), 29.35 Smooth morphisms (“partial”), 26.21 Separation axioms (“partial”), 65.4 Lists of useful properties of morphisms of schemes (“partial”)
(2.1): 10.131 Differentials (“special case”), 10.134 The naive cotangent complex (“affine case”; “affine case, R[S]”), 91.2 Deformations of rings and the naive cotangent complex (“affine case”)
(2.2): 10.131 Differentials (“partial”), 10.134 The naive cotangent complex (“special case”), 37.7 Universal first order thickenings (“partial”), 37.13 The naive cotangent complex (“more general”), 29.32 Conormal sheaf of an immersion (“special case”), 29.33 Sheaf of differentials of a morphism (“(b)”), 76.5 Conormal sheaf of an immersion (“partial”), 76.44 Regular immersions (“partial”)
(2.4): 10.131 Differentials, 10.139 Smoothness and differentials (“special case”), 92.14 The cotangent complex of a local complete intersection (“special case”), 49.10 Quasi-finite syntomic morphisms (“partial”), 49.11 Finite syntomic morphisms (“Forward direction”), 31.23 Relative regular immersions (“special case”), 37.62 Local complete intersection morphisms (“more general”; “partial”)
(2.6): 10.131 Differentials (“partial”), 10.138 Formally smooth maps (“partial”), 10.139 Smoothness and differentials (“partial”), 37.7 Universal first order thickenings (“partial”; “right exactness”), 37.13 The naive cotangent complex (“more general”), 37.62 Local complete intersection morphisms (“partial”)
(3.4): 42.56 An Adams operator (“more general”)
(3.5): 17.25 Invertible modules (“partial”), 18.32 Invertible modules (“partial”)
(5.2): 10.131 Differentials (“partial”; “special case”), 27.6 Vector bundles, 50.11 de Rham cohomology of projective space (“special case”)

Exposé X

(1.1): 36.38 K-groups
(1.1.2): 10.62 Support and dimension of modules (“special case”), 42.10 Cycle associated to a coherent sheaf (“partial”)
(1.1.8): 36.38 K-groups
(1.2): 45.13 Exterior powers and K-groups (“clause (1.2.1)”)
(1.5): 10.105 Catenary rings (“Affine ambient case”), 42.7 Setup (“CM S, (1.5.1) on S”; “special case”), 68.25 Catenary algebraic spaces (“(1.5.2) is a dim. fn”), 29.17 Universally catenary schemes (“more general”), 29.53 The dimension formula (“special case”)
(1.6): 28.11 Catenary schemes (“more general”)
(2.3.1): 72.18 Numerical intersections (“more general”; “special case”)
(2.4): 44.4 The Picard functor (“definition of Pic_{X/S}”), 99.11 The Picard functor (“definition of Pic_{X/B}”)
(4.4): 30.20 The theorem on formal functions (“Stalk form of (4.4.5)”), 72.18 Numerical intersections (“defn, algebraic spaces”; “partial”), 33.44 Degrees on curves (“(4.4.3), special case”; “special case”), 33.45 Numerical intersections (“defn of (L_1..L_d;F)”)
(5.1): 17.25 Invertible modules (“Pic(X) definition”; “special case”), 17.26 Rank and determinant (“special case”), 15.120 Determinants (“affine case”), 18.32 Invertible modules (“Pic(X) definition”)
(5.2.2): 31.15 Effective Cartier divisors and invertible sheaves (“more general”)
(6.1): 10.13 Tensor algebra (“special case”), 10.78 Finite projective modules (“partial”), 17.21 Symmetric and exterior powers (“special case”)
(6.2.7): 17.26 Rank and determinant (“(6.2.9), special case”; “special case”)
(7.1): 10.139 Smoothness and differentials (“special case”), 37.61 Perfect morphisms (“partial”), 37.62 Local complete intersection morphisms (“partial”)
(7.5): 42.15 Push and pull (“partial”), 53.8 The genus of a curve (“partial”), 29.42 Proper morphisms (“partial”), 65.4 Lists of useful properties of morphisms of schemes (“partial”)
(7.7): 10.106 Regular local rings (“special case”), 10.163 Ascending properties (“special case”)
(7.8): 55.9 The geometry of a regular model (“O_X(X_s) ≅ O_X”; “partial, special case”), 15.123 A regular local ring is a UFD (“more general”; “special case”)
(7.12): 55.9 The geometry of a regular model (“special case”)
(7.13.3): 10.137 Smooth ring maps (“special case”), 29.35 Smooth morphisms (“partial”), 65.4 Lists of useful properties of morphisms of schemes (“partial”)
(7.15): 10.31 Noetherian rings (“special case”), 41.8 Examples of unramified morphisms (“special case”)
(7.17): 10.31 Noetherian rings (“special case”)
(7.18.1): 10.12 Tensor products, 10.39 Flat modules and flat ring maps, 41.9 Flat morphisms

Exposé XII

(1.2): 10.49 Geometrically integral algebras (“special case”), 33.9 Geometrically integral schemes (“partial”)
(1.6): 37.20 Normal morphisms, 29.45 Integral and finite morphisms (“special case”)
(2.3): 10.39 Flat modules and flat ring maps (“(a)”), 42.15 Push and pull (“(a)”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“(a)”), 29.26 Flat morphisms (“(a)”), 65.4 Lists of useful properties of morphisms of schemes (“(a)”)
(2.4): 99.4 The Isom functor (“partial”)
(2.5): 37.53 Stein factorization (“partial”)
(4.1): 42.15 Push and pull (“(1)”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“(1)”), 29.26 Flat morphisms (“(1)”), 65.4 Lists of useful properties of morphisms of schemes (“(1)”)
(4.5): 38.22 Flattening stratification over an Artinian ring (“special case”)
(4.6): 77.9 Flattening a map (“partial”)

Exposé XIII

(1.3): 33.35 Coherent sheaves on projective space (“(ii), special case”; “(iii), special case”)
(1.13): 37.54 Generic flatness stratification (“special case”)
(2.1): 33.45 Numerical intersections (“all L_i equal”; “coeff of n_1⋯n_s”)
(2.2): 28.3 Integral, irreducible, and reduced schemes
(3.1): 44.4 The Picard functor, 99.11 The Picard functor
(4.1): 39.7 Properties of group schemes over a field (“G^0 over a field”)
(4.6): 53.5 Riemann-Roch (“more general”)
(5.3): 45.7 Classical Weil cohomology theories (“partial”)
(6.1): 33.35 Coherent sheaves on projective space (“partial”)
(7.1.4): 31.33 Blowing up (“more general”)

Exposé XIV

(2.1): 92.14 The cotangent complex of a local complete intersection (“special case”), 37.62 Local complete intersection morphisms (“(1) ⇒ (3), special case”; “special case”)
(2.2): 30.22 Cohomology and base change, III (“special case”), 37.61 Perfect morphisms (“special case”), 36.27 Producing perfect complexes (“special case”)
(2.3): 36.11 Derived category of coherent modules (“special case”)
(4.1): 42.19 Rational equivalence (“partial”)
(4.2): 33.25 Schemes smooth over fields (“Clause”)

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