EGA III
The items of EGA III 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.
§1
- (1.1.1): 17.24 Koszul complexes (“more general”), 15.29 The Koszul complex (“(f)”)
- (1.1.5): 15.29 The Koszul complex
- (1.1.7): 15.29 The Koszul complex (“more general”)
- (1.1.10): 51.2 Generalities (“principal ideal, Noeth.”)
- (1.2.2): 28.18 Sections over principal opens (“special case”), 26.5 Affine schemes (“special case”)
- (1.2.5): 20.10 Čech cohomology as a functor on presheaves (“partial”)
- (1.3.1): 30.2 Čech cohomology of quasi-coherent sheaves (“more general”), 20.37 Derived limits (“more general”), 21.23 Derived and homotopy limits (“more general”)
- (1.3.2): 30.5 Cohomology and base change, I
- (1.3.3): 30.2 Čech cohomology of quasi-coherent sheaves
- (1.3.4): 30.2 Čech cohomology of quasi-coherent sheaves (“g to a point”)
- (1.4.1): 30.2 Čech cohomology of quasi-coherent sheaves
- (1.4.2): 30.2 Čech cohomology of quasi-coherent sheaves (“more general”)
- (1.4.3): 51.2 Generalities (“(1.4.3.2), special case”), 51.8 Finiteness of pushforwards, I (“(1.4.3.2), special case”; “special case”)
- (1.4.5): 28.18 Sections over principal opens (“H⁰, L = 𝒪_X, qcqs”)
- (1.4.6): 28.18 Sections over principal opens (“degree zero, F = O_X”)
- (1.4.7): 30.3 Vanishing of cohomology (“(ii), special case”)
- (1.4.9): 30.7 Cohomology and base change, II (“affine diagonal”)
- (1.4.10): 30.4 Quasi-coherence of higher direct images (“more general”), 26.24 Functoriality for quasi-coherent modules (“q = 0”)
- (1.4.12): 30.4 Quasi-coherence of higher direct images (“existence of r”)
- (1.4.13): 28.18 Sections over principal opens (“q = 0 and F = O_X”)
- (1.4.14): 30.4 Quasi-coherence of higher direct images (“affine base, Z=point”)
- (1.4.15): 26.17 Fibre products of schemes
- (1.4.15.5): 28.20 Flat modules (“case A = B, φ = id”)
- (1.4.17): 26.7 Quasi-coherent sheaves on affines (“affine case”)
§2
- (2.1.1): 17.25 Invertible modules (“(2.1.1.1)”)
- (2.1.2): 17.25 Invertible modules (“partial”), 36.9 Koszul complexes (“special case”), 28.18 Sections over principal opens (“special case”)
- (2.1.13): 30.8 Cohomology of projective space (“more general”)
- (2.1.14): 30.8 Cohomology of projective space (“more general”)
- (2.1.15): 30.8 Cohomology of projective space (“partial”)
- (2.1.16): 30.8 Cohomology of projective space (“(ii)”)
- (2.2.2): 30.14 Coherent sheaves on Proj, I (“(ii), P^n case”; “clause (i), special case”)
- (2.2.5): 30.19 Higher direct images of coherent sheaves (“partial”)
- (2.3.3): 30.14 Coherent sheaves on Proj, I (“(F), special case”)
- (2.3.4): 31.33 Blowing up (“(i), special case”)
- (2.4.1): 32.19 Base change in top degree (“Clause (i), special case”)
- (2.4.2): 30.20 The theorem on formal functions (“special case”)
- (2.5.1): 33.33 Euler characteristics (“field base”)
- (2.5.2): 33.33 Euler characteristics (“special case”)
- (2.6.1): 30.3 Vanishing of cohomology (“c) => a), absolute”)
- (2.6.2.1): 10.36 Finite and integral ring extensions (“special case”), 29.45 Integral and finite morphisms (“more general”), 65.4 Lists of useful properties of morphisms of schemes (“more general”)
- (2.6.2.3): 28.27 Ample invertible sheaves (“special case”)
§3
- (3.2.1): 36.11 Derived category of coherent modules (“Noetherian target”), 69.20 Higher direct images of coherent sheaves (“more general”), 75.7 Being proper over a base (“more general”)
- (3.2.2): 30.9 Coherent sheaves on locally Noetherian schemes (“finite case”)
- (3.2.3): 30.19 Higher direct images of coherent sheaves
- (3.2.4): 36.11 Derived category of coherent modules (“special case”)
- (3.3.1): 30.20 The theorem on formal functions (“special case”), 27.8 Proj of a graded ring (“special case”)
- (3.3.2): 30.20 The theorem on formal functions
§4
- (4.1.3): 26.10 Immersions of schemes (“canonical immersion i_k”)
- (4.1.5): 69.22 The theorem on formal functions (“partial”)
- (4.1.7): 10.86 Mittag-Leffler systems, 30.20 The theorem on formal functions, 12.31 Inverse systems, 69.22 The theorem on formal functions (“special case”)
- (4.1.7.12): 30.20 The theorem on formal functions (“more general”)
- (4.1.7.21): 10.86 Mittag-Leffler systems (“more general”), 10.87 Inverse systems (“more general”), 20.37 Derived limits (“H^i(U”), 12.31 Inverse systems (“more general”)
- (4.1.10): 36.11 Derived category of coherent modules (“more general”), 75.7 Being proper over a base (“more general”)
- (4.2.1): 26.18 Base change in algebraic geometry (“special case”)
- (4.2.2): 30.20 The theorem on formal functions (“partial”)
- (4.2.4): 30.20 The theorem on formal functions (“Degree p = 0 case”)
- (4.3.1): 27.4 Relative spectrum as a functor, 37.53 Stein factorization
- (4.3.4): 10.159 Constructing flat ring maps (“partial”), 42.15 Push and pull (“more general”), 53.8 The genus of a curve (“more general”), 37.28 Connected components of fibres (“geom. # of components”), 37.53 Stein factorization (“more general”; “partial”; “special case”), 29.42 Proper morphisms (“more general”), 65.4 Lists of useful properties of morphisms of schemes (“more general”), 76.30 Connected components of fibres (“more general”), 76.36 Stein factorization (“more general”), 33.7 Geometrically connected schemes (“special case”), 33.9 Geometrically integral schemes (“special case”)
- (4.3.5): 10.36 Finite and integral ring extensions (“more general”)
- (4.3.6): 15.108 Local irreducibility (“unibranch local ring”), 28.16 Local irreducibility (“normal ⇒ unibranch”; “unibranch point”)
- (4.3.8): 37.53 Stein factorization (“special case”)
- (4.3.11): 29.24 Open morphisms
- (4.3.13): 10.57 Proj of a graded ring, 10.59 Noetherian local rings, 27.8 Proj of a graded ring
- (4.4.1): 29.57 Zariski's Main Theorem (algebraic version) (“more general”)
- (4.4.2): 37.44 Applications of Zariski's Main Theorem, I (“more general”), 29.45 Integral and finite morphisms (“more general”), 76.35 Applications of Zariski's Main Theorem, I (“a) <=> c)”; “a) => c)”; “c) => a)”), 67.45 Integral and finite morphisms (“a) <=> b)”)
- (4.4.3): 10.123 Zariski's Main Theorem (“affine case, X' open”; “affine case, X=X'”), 37.43 Zariski's Main Theorem (“more general”), 29.57 Zariski's Main Theorem (algebraic version) (“special case”)
- (4.4.5): 37.43 Zariski's Main Theorem (“special case”)
- (4.4.6): 10.123 Zariski's Main Theorem (“special case”)
- (4.4.7): 10.5 Finite modules and finitely presented modules, 10.9 Localization, 11.2 Noncommutative algebras, 9.7 Finite extensions
- (4.4.9): 33.17 Generically finite morphisms (“proper case”)
- (4.4.10): 10.123 Zariski's Main Theorem (“affine case”)
- (4.4.11): 37.44 Applications of Zariski's Main Theorem, I (“more general”), 76.35 Applications of Zariski's Main Theorem, I (“proper q-finite ⇒ finite”)
- (4.4.12): 37.43 Zariski's Main Theorem (“(i) quasi-affine”; “Quasi-finite case”), 37.49 Quasi-projective schemes (“more general”), 29.57 Zariski's Main Theorem (algebraic version) (“partial”), 67.34 Morphisms and dimensions of fibres (“partial”), 67.52 Zariski's Main Theorem (representable case) (“more general”)
- (4.5.1): 10.71 Ext groups (“special case”)
- (4.6.2): 10.112 Homomorphisms and dimension (“special case”), 26.18 Base change in algebraic geometry (“partial”)
- (4.6.7): 57.16 No deformations (“(ii)”), 26.18 Base change in algebraic geometry (“partial”)
- (4.6.7.1): 10.99 Criteria for flatness
- (4.8.5): 29.45 Integral and finite morphisms (“ordinary schemes”)
- (4.8.6): 30.9 Coherent sheaves on locally Noetherian schemes (“scheme case”)
- (4.8.11): 30.21 Applications of the theorem on formal functions (“ordinary schemes”)
§5
- (5.1.1): 87.14 Completion along a closed subset (“Spf(A) ≅ Ŷ”), 87.38 Completion along a closed subspace (“Spf(A) ≅ Ŷ”)
- (5.2.3): 30.17 Ample invertible sheaves and cohomology (“(ii), special case”)
- (5.2.5): 30.27 Grothendieck's existence theorem, III (“more general”), 28.27 Ample invertible sheaves (“Ampleness of L'_0 = L/JL”)
- (5.2.6): 30.27 Grothendieck's existence theorem, III (“more general”)
- (5.3.5): 26.10 Immersions of schemes, 26.12 Reduced schemes (“partial”)
- (5.4.6): 10.32 Locally nilpotent ideals (“more general”)
§6
- (6.1.1): 10.39 Flat modules and flat ring maps (“special case”), 41.9 Flat morphisms (“special case”), 26.17 Fibre products of schemes
- (6.1.4): 12.13 Complexes (“special case”)
- (6.2.1): 42.2 Periodic complexes and Herbrand quotients, 20.29 Cohomology of filtered complexes (“absolute case, Y=point”), 12.13 Complexes (“f = id_X”), 64.7 Filtered derived category
- (6.2.3): 30.4 Quasi-coherence of higher direct images (“special case”)
- (6.2.5): 36.11 Derived category of coherent modules (“special case”)
- (6.3.2): 15.63 Spectral sequences for Tor (“Q• = module M, K bdd”), 15.65 Künneth spectral sequence (“(6.3.2.2)”)
- (6.3.5): 10.75 Tor groups and flatness (“(i), special case”)
- (6.3.6): 10.76 Functorialities for Tor (“special case”)
- (6.3.8): 10.71 Ext groups (“partial”)
- (6.3.12): 10.76 Functorialities for Tor (“special case”)
- (6.4.1): 26.7 Quasi-coherent sheaves on affines (“special case”)
- (6.5.2): 26.17 Fibre products of schemes (“Affineness clause”; “more general”)
- (6.5.4): 15.65 Künneth spectral sequence (“(6.5.4.2)”)
- (6.5.5): 10.75 Tor groups and flatness (“S = X = Y = Spec R”)
- (6.5.6): 10.12 Tensor products (“special case”), 10.76 Functorialities for Tor (“special case”)
- (6.5.10): 10.39 Flat modules and flat ring maps (“special case”)
- (6.5.13): 10.75 Tor groups and flatness (“special case”)
- (6.6.4): 26.17 Fibre products of schemes (“partial”), 26.21 Separation axioms (“binary intersections”)
- (6.6.7): 20.25 Čech cohomology of complexes (“partial”)
- (6.7.1): 10.17 The spectrum of a ring (“special case”)
- (6.7.9): 10.75 Tor groups and flatness (“special case”)
- (6.7.13): 10.75 Tor groups and flatness (“special case”)
- (6.8.2): 15.65 Künneth spectral sequence (“m=2, modules over a ring”)
- (6.9.1): 56.5 Functors between categories of quasi-coherent modules (“partial”), 29.15 Morphisms of finite type (“partial”), 29.21 Quasi-finite morphisms (“partial”), 26.17 Fibre products of schemes (“partial”), 26.19 Quasi-compact morphisms (“partial”), 26.21 Separation axioms (“separatedness of f′_i”), 65.4 Lists of useful properties of morphisms of schemes (“partial”; “separatedness of f′_i”)
- (6.9.3): 15.63 Spectral sequences for Tor (“special case”)
- (6.9.3.2): 15.63 Spectral sequences for Tor (“special case”)
- (6.9.3.3): 15.63 Spectral sequences for Tor (“special case”)
- (6.9.4): 56.5 Functors between categories of quasi-coherent modules (“more general”), 29.15 Morphisms of finite type (“more general”), 29.21 Quasi-finite morphisms (“more general”), 26.19 Quasi-compact morphisms (“more general”), 65.4 Lists of useful properties of morphisms of schemes (“more general”)
- (6.9.8): 15.63 Spectral sequences for Tor (“(6.9.8.1)”)
- (6.9.10): 26.18 Base change in algebraic geometry
- (6.10.4): 26.17 Fibre products of schemes (“partial”)
§7
- (7.1.2): 22.24 Linear categories, 90.11 Tangent spaces of functors
- (7.1.3): 7.5 Functoriality of categories of presheaves (“partial”), 7.19 More functoriality of presheaves (“partial”), 7.44 Sheaves of algebraic structures (“partial”)
- (7.2.1): 10.12 Tensor products (“special case”), 22.12 Tensor product
- (7.2.2): 22.33 Derived tensor product (“more general”)
- (7.2.5): 56.3 Functors between categories of modules (“(a) <=> (c)”)
- (7.2.6): 10.12 Tensor products (“partial”; “special case”), 10.14 Base change, 22.24 Linear categories, 90.11 Tangent spaces of functors, 56.2 Functors on module categories, 56.3 Functors between categories of modules (“(1) <=> (2)”; “N ↦ T_N”; “commutative rings”)
- (7.3.1): 56.3 Functors between categories of modules (“(a) <=> (e)”)
- (7.3.4): 10.78 Finite projective modules (“(1) => (8)”), 28.21 Locally free modules
- (7.3.5): 10.101 Flatness criteria over Artinian rings (“commutative case”)
- (7.4.1): 10.39 Flat modules and flat ring maps (“partial”)
- (7.4.2): 12.3 Preadditive and additive categories (“more general”)
- (7.4.3): 10.120 Factorization (“special case”)
- (7.4.6): 10.5 Finite modules and finitely presented modules, 11.2 Noncommutative algebras, 9.7 Finite extensions
- (7.4.7): 10.75 Tor groups and flatness (“(i), special case”)
- (7.6.4): 15.63 Spectral sequences for Tor (“(i) spectral sequence”), 15.65 Künneth spectral sequence (“(i) short exact sequence”)
- (7.6.11): 42.2 Periodic complexes and Herbrand quotients, 20.9 The Čech complex and Čech cohomology
- (7.7.2): 10.39 Flat modules and flat ring maps (“special case”), 42.15 Push and pull (“special case”), 56.5 Functors between categories of quasi-coherent modules (“quasi-compactness of f'”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“special case”), 29.15 Morphisms of finite type (“quasi-compactness of f'”), 29.21 Quasi-finite morphisms (“quasi-compactness of f'”), 29.26 Flat morphisms (“special case”), 26.17 Fibre products of schemes, 26.19 Quasi-compact morphisms (“quasi-compactness of f'”), 26.21 Separation axioms (“separatedness of f'”), 65.4 Lists of useful properties of morphisms of schemes (“quasi-compactness of f'”; “separatedness of f'”; “special case”)
- (7.7.9): 36.36 The resolution property (“(ii)”), 6.26 Morphisms of ringed spaces and modules (“more general”), 18.23 Local types of modules (“partial”)
- (7.7.12): 10.83 Descent for finite projective modules (“more general”)
- (7.8.6): 10.143 Étale ring maps
- (7.8.6.1): 10.21 Open and closed subsets of spectra (“affine case Z = Spec R”; “special case”)
- (7.8.10): 37.53 Stein factorization (“more general”; “partial”), 58.15 Homotopy exact sequence (“partial”), 76.36 Stein factorization (“partial”)
- (7.9.1): 10.5 Finite modules and finitely presented modules, 10.78 Finite projective modules, 11.2 Noncommutative algebras, 9.7 Finite extensions, 17.14 Locally free sheaves, 28.21 Locally free modules (“M proj f.t. <=> M~ l.f.”)
- (7.9.3): 36.31 Perfect complexes (“more general”)
- (7.9.4): 36.32 Applications (“special case”)
- (7.9.5): 36.32 Applications (“special case”)
- (7.9.7): 10.39 Flat modules and flat ring maps (“special case”), 42.15 Push and pull (“special case”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“special case”), 29.26 Flat morphisms (“special case”), 36.32 Applications (“special case”), 65.4 Lists of useful properties of morphisms of schemes (“special case”), 33.33 Euler characteristics (“special case”)
- (7.9.10): 36.32 Applications (“partial”)
- (7.9.10.1): 37.72 Contracting rational curves (“partial”)
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