EGA I
The items of EGA I 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): 10.9 Localization (“more general”), 10.17 The spectrum of a ring (“more general”), 10.18 Local rings, 10.112 Homomorphisms and dimension (“A_x is a local ring”)
- (1.1.2): 10.17 The spectrum of a ring (“(iii): V(⋃E_λ)=⋂V(E_λ)”; “(v) for ideals”), 47.9 Local cohomology (“(iii), special case”)
- (1.1.5): 10.17 The spectrum of a ring (“more general”; “partial”), 47.9 Local cohomology (“special case”)
- (1.1.6): 10.31 Noetherian rings
- (1.1.8): 10.17 The spectrum of a ring (“more general”), 10.26 Irreducible components of spectra (“more general”), 26.11 Zariski topology of schemes (“more general”)
- (1.1.9): 10.17 The spectrum of a ring (“last clause”; “more general”), 26.5 Affine schemes (“partial”)
- (1.1.10): 10.17 The spectrum of a ring (“more general”), 10.26 Irreducible components of spectra (“more general”), 29.52 Generically finite morphisms (“more general”), 28.25 Sections with support in a closed subset (“more general”)
- (1.2.1): 10.17 The spectrum of a ring
- (1.2.2): 10.17 The spectrum of a ring (“more general”)
- (1.2.4): 10.17 The spectrum of a ring (“more general”), 10.29 Images of ring maps of finite presentation (“more general”), 61.5 Constructing w-local affine schemes (“localization R → M⁻¹R”)
- (1.2.6): 10.17 The spectrum of a ring (“partial”), 10.29 Images of ring maps of finite presentation (“special case”), 110.22 Underlying space Noetherian not Noetherian (“partial”)
- (1.2.7): 10.30 More on images (“special case”)
- (1.3.1): 10.9 Localization (“more general”)
- (1.3.2): 26.5 Affine schemes (“(a) ⇔ (f)”)
- (1.3.5): 26.5 Affine schemes
- (1.3.6): 10.17 The spectrum of a ring (“partial”), 10.29 Images of ring maps of finite presentation (“partial”)
- (1.3.7): 26.5 Affine schemes (“Case M = A”)
- (1.3.12): 26.7 Quasi-coherent sheaves on affines (“(i)”; “(ii)”)
- (1.3.13): 6.20 Sheafification of presheaves of modules (“more general”), 18.11 Sheafification of presheaves of modules (“more general”)
- (1.4.1): 28.18 Sections over principal opens (“special case”), 26.7 Quasi-coherent sheaves on affines (“a ⇒ c for V = Spec A”; “c ⇒ a for V = Spec A”)
- (1.4.2): 28.23 Extending quasi-coherent sheaves
- (1.4.3): 26.7 Quasi-coherent sheaves on affines (“special case”)
- (1.5.1): 30.9 Coherent sheaves on locally Noetherian schemes (“(a) <=> (b)”), 51.8 Finiteness of pushforwards, I (“(a) => (c)”)
- (1.5.2): 51.8 Finiteness of pushforwards, I (“special case”)
- (1.5.4): 30.10 Coherent sheaves on Noetherian schemes (“colimit”; “filtered system”)
- (1.6.1): 10.17 The spectrum of a ring (“more general”)
- (1.6.2): 10.17 The spectrum of a ring (“partial”; “special case”), 10.29 Images of ring maps of finite presentation (“special case”), 61.5 Constructing w-local affine schemes (“partial”)
- (1.6.4): 29.4 Closed immersions and quasi-coherent sheaves (“closed immersion”)
- (1.7.1): 33.26 Types of varieties
- (1.7.2): 10.17 The spectrum of a ring (“partial”)
- (1.7.3): 26.6 The category of affine schemes (“more general”)
- (1.8.2): 26.2 Locally ringed spaces
- (1.8.6): 26.7 Quasi-coherent sheaves on affines (“u = id on Spec R”)
§2
- (2.1.2): 26.9 Schemes
- (2.1.3): 26.11 Zariski topology of schemes
- (2.1.4): 26.11 Zariski topology of schemes
- (2.1.5): 26.11 Zariski topology of schemes (“partial”)
- (2.1.7): 111.33 Schemes, 26.9 Schemes, 26.10 Immersions of schemes
- (2.1.8): 28.3 Integral, irreducible, and reduced schemes, 5.7 Connected components
- (2.2.1): 26.2 Locally ringed spaces (“more general”), 26.9 Schemes
- (2.2.2): 26.9 Schemes
- (2.2.3): 41.14 Topological properties of étale morphisms (“more general”), 111.33 Schemes (“more general”), 26.3 Open immersions of locally ringed spaces (“more general”), 26.9 Schemes, 26.10 Immersions of schemes
- (2.2.4): 66.33 Morphisms towards schemes
- (2.2.6): 29.8 Dominant morphisms, 29.9 Surjective morphisms, 29.24 Open morphisms, 67.18 Dominant morphisms (“dominant”), 5.17 Characterizing proper maps
- (2.2.7): 29.9 Surjective morphisms, 29.24 Open morphisms, 65.4 Lists of useful properties of morphisms of schemes
- (2.2.8): 35.24 Properties of morphisms local in the fppf topology on the target (“more general”)
- (2.2.9): 68.22 Birational morphisms (“more general”), 29.51 Birational morphisms (“more general”)
- (2.2.10): 111.33 Schemes, 26.9 Schemes, 26.10 Immersions of schemes
- (2.3.1): 26.14 Glueing schemes
- (2.3.2): 26.14 Glueing schemes (“X not affine”; “gluing construction”; “partial”; “Γ(X,O_X) ≅ K”)
- (2.4.1): 10.18 Local rings (“more general”)
- (2.4.2): 61.5 Constructing w-local affine schemes (“topological embedding”), 26.13 Points of schemes, 26.23 Monomorphisms (“partial”), 5.19 Specialization
- (2.4.3): 66.11 Generic points
- (2.4.4): 26.13 Points of schemes
- (2.4.5): 10.18 Local rings
- (2.4.6): 26.13 Points of schemes (“factorization via κ(x)”)
- (2.4.7): 26.13 Points of schemes (“zero ideal a_y = 0”)
- (2.4.8): 31.29 More on invertible modules (“more general”)
- (2.5.1): 90.6 Prorepresentable functors and predeformation categories, 26.18 Base change in algebraic geometry
- (2.5.2): 26.18 Base change in algebraic geometry
- (2.5.3): 61.12 The pro-étale site, 73.7 Fppf topology, 73.8 The ph topology, 34.3 The Zariski topology, 34.4 The étale topology, 34.7 The fppf topology, 34.8 The ph topology
- (2.5.4): 61.12 The pro-étale site (“more general”), 73.7 Fppf topology (“more general”), 73.8 The ph topology (“more general”), 34.3 The Zariski topology (“more general”), 34.4 The étale topology (“more general”), 34.7 The fppf topology (“more general”), 34.8 The ph topology (“more general”)
- (2.5.5): 26.18 Base change in algebraic geometry (“more general”)
§3
- (3.2.1): 26.17 Fibre products of schemes (“more general”)
- (3.2.2): 26.17 Fibre products of schemes (“partial”)
- (3.2.6): 26.16 Existence of fibre products of schemes (“more general”)
- (3.3.1): 90.6 Prorepresentable functors and predeformation categories
- (3.3.2): 14.9 Products of cosimplicial objects (“more general”)
- (3.3.5): 26.16 Existence of fibre products of schemes (“existence for n = 2”; “more general”), 26.17 Fibre products of schemes
- (3.3.6): 26.17 Fibre products of schemes (“more general”), 26.18 Base change in algebraic geometry
- (3.3.7): 61.12 The pro-étale site (“more general”), 26.18 Base change in algebraic geometry (“more general”), 73.7 Fppf topology (“more general”), 73.8 The ph topology (“more general”), 34.3 The Zariski topology (“more general”), 34.4 The étale topology (“more general”), 34.7 The fppf topology (“more general”), 34.8 The ph topology (“more general”), 33.36 Frobenii (“more general”)
- (3.3.12): 39.3 Equivalence relations (“more general”), 26.23 Monomorphisms (“more general”), 65.4 Lists of useful properties of morphisms of schemes (“more general”), 78.4 Equivalence relations (“more general”), 67.10 Monomorphisms (“more general”)
- (3.3.13): 26.17 Fibre products of schemes (“Affineness of X ×_S S′”)
- (3.3.14): 26.18 Base change in algebraic geometry (“more general”)
- (3.3.15): 26.15 A representability criterion (“partial”), 26.16 Existence of fibre products of schemes
- (3.4.1): 59.9 Presheaves (“more general”)
- (3.4.3): 59.9 Presheaves, 14.8 Pushouts of simplicial objects (“more general”)
- (3.4.5): 26.13 Points of schemes (“location map”; “partial, special case”)
- (3.4.7): 29.9 Surjective morphisms (“partial”)
- (3.4.9): 26.17 Fibre products of schemes (“partial”)
- (3.5.1): 26.18 Base change in algebraic geometry (“more general”)
- (3.5.2): 29.9 Surjective morphisms, 65.4 Lists of useful properties of morphisms of schemes
- (3.5.4): 29.10 Radicial and universally injective morphisms
- (3.5.6): 29.10 Radicial and universally injective morphisms, 65.4 Lists of useful properties of morphisms of schemes
- (3.5.8): 10.45 Perfect fields (“special case”), 29.10 Radicial and universally injective morphisms
- (3.5.10): 26.18 Base change in algebraic geometry (“special case”)
- (3.5.11): 76.3 Radicial morphisms (“radicial => univ. inj.”)
- (3.6.1): 26.18 Base change in algebraic geometry (“special case”)
- (3.6.2): 26.18 Base change in algebraic geometry
- (3.7.2): 26.18 Base change in algebraic geometry (“more general”)
- (3.7.3): 32.15 Noetherian valuative criterion (“more general”)
§4
- (4.1.1): 27.4 Relative spectrum as a functor (“more general”), 17.10 Quasi-coherent modules (“direct sums (binary)”), 26.7 Quasi-coherent sheaves on affines (“affine case”; “affine, cokernel”; “affine, kernel”)
- (4.1.2): 10.40 Supports and annihilators (“affine case”), 17.5 Supports of modules and sections (“more general”), 26.10 Immersions of schemes
- (4.1.3): 111.33 Schemes (“special case”), 26.9 Schemes (“special case”), 26.10 Immersions of schemes (“special case”)
- (4.1.6): 29.2 Closed immersions, 26.24 Functoriality for quasi-coherent modules, 65.4 Lists of useful properties of morphisms of schemes, 65.12 Immersions and Zariski coverings of algebraic spaces (“closed subprescheme”; “subprescheme case”)
- (4.1.7): 111.33 Schemes (“special case”), 26.3 Open immersions of locally ringed spaces (“special case”), 26.10 Immersions of schemes (“special case”)
- (4.1.8): 29.27 Flat closed immersions (“special case”), 26.12 Reduced schemes (“special case”)
- (4.2.1): 111.33 Schemes (“partial”), 26.3 Open immersions of locally ringed spaces (“more general”), 26.10 Immersions of schemes
- (4.2.2): 29.2 Closed immersions (“(b) closed case”)
- (4.2.3): 30.2 Čech cohomology of quasi-coherent sheaves (“more general”), 29.2 Closed immersions (“more general”), 29.11 Affine morphisms (“more general”)
- (4.2.4): 29.3 Immersions (“(a), special case”), 67.12 Immersions (“(a) immersion case”; “(b)”)
- (4.2.5): 87.27 Closed immersions (“closed immersions”), 29.2 Closed immersions, 26.24 Functoriality for quasi-coherent modules, 65.4 Lists of useful properties of morphisms of schemes, 65.12 Immersions and Zariski coverings of algebraic spaces (“closed immersions”; “immersions”; “open immersions”), 100.9 Immersions of algebraic stacks (“closed immersions”; “immersions”; “open immersions”)
- (4.3.1): 29.2 Closed immersions (“special case”), 26.17 Fibre products of schemes (“more general”; “special case”), 26.18 Base change in algebraic geometry (“special case”), 65.4 Lists of useful properties of morphisms of schemes (“special case”)
- (4.3.2): 37.2 Thickenings (“thickenings”), 29.2 Closed immersions, 29.14 Types of morphisms defined by properties of ring maps (“affine open immersions”), 26.17 Fibre products of schemes, 26.18 Base change in algebraic geometry, 65.4 Lists of useful properties of morphisms of schemes, 65.12 Immersions and Zariski coverings of algebraic spaces (“closed immersion”; “immersion”; “open immersion”)
- (4.4.1): 42.29 Gysin homomorphisms (“closed case”)
- (4.5.5): 29.2 Closed immersions (“(i), special case”; “(iii), special case”), 29.14 Types of morphisms defined by properties of ring maps (“(iii), special case”), 61.3 Local isomorphisms (“(i), special case”; “(iii), special case”), 26.17 Fibre products of schemes (“(iii), special case”), 26.18 Base change in algebraic geometry (“(iii), special case”), 26.24 Functoriality for quasi-coherent modules (“(i), special case”), 65.4 Lists of useful properties of morphisms of schemes (“(i), special case”; “(iii), special case”)
§5
- (5.1.2): 26.12 Reduced schemes (“uniqueness”)
- (5.1.3): 26.12 Reduced schemes
- (5.1.4): 28.3 Integral, irreducible, and reduced schemes (“partial”), 26.12 Reduced schemes
- (5.1.5): 26.12 Reduced schemes (“reduced source factors”)
- (5.1.9): 26.10 Immersions of schemes (“more general”)
- (5.1.9.2): 30.2 Čech cohomology of quasi-coherent sheaves (“more general”)
- (5.2.1): 26.12 Reduced schemes (“closed case, uniqueness”)
- (5.2.2): 26.12 Reduced schemes (“more general”)
- (5.2.3): 51.8 Finiteness of pushforwards, I (“special case”), 29.3 Immersions (“factorization only”)
- (5.3.1): 83.5 Quotients as orbit spaces
- (5.3.8): 26.23 Monomorphisms (“more general”), 67.10 Monomorphisms
- (5.3.10): 67.4 Separation axioms
- (5.3.11): 45.3 Correspondences (“separated case”)
- (5.3.13): 29.3 Immersions, 26.21 Separation axioms
- (5.3.14): 29.3 Immersions (“more general”), 26.21 Separation axioms (“more general”)
- (5.4.1): 39.4 Group schemes, 26.21 Separation axioms, 67.4 Separation axioms
- (5.4.2): 26.21 Separation axioms (“special case”), 67.4 Separation axioms, 45.3 Correspondences (“special case”)
- (5.4.3): 26.21 Separation axioms, 45.3 Correspondences
- (5.4.4): 41.6 Topological properties of unramified morphisms (“special case”), 29.2 Closed immersions (“special case”), 29.3 Immersions, 26.21 Separation axioms, 67.12 Immersions
- (5.4.5): 26.21 Separation axioms (“j = id_X”)
- (5.4.6): 41.6 Topological properties of unramified morphisms, 29.3 Immersions (“more general”), 26.21 Separation axioms (“more general”)
- (5.5.1): 29.10 Radicial and universally injective morphisms (“more general”), 26.21 Separation axioms (“(v), special case”), 26.23 Monomorphisms, 65.4 Lists of useful properties of morphisms of schemes, 67.10 Monomorphisms (“(i) immersions”), 67.12 Immersions (“(v)”)
- (5.5.3): 26.21 Separation axioms (“more general”), 65.4 Lists of useful properties of morphisms of schemes (“more general”)
- (5.5.4): 26.12 Reduced schemes
- (5.5.6): 26.21 Separation axioms (“necessity”; “necessity of generation”; “sufficiency”)
- (5.5.7): 29.11 Affine morphisms (“more general”), 26.21 Separation axioms (“more general”)
- (5.5.8): 26.21 Separation axioms (“sufficiency”)
- (5.5.9): 29.11 Affine morphisms (“special case”), 26.21 Separation axioms (“special case”)
- (5.5.10): 26.21 Separation axioms (“special case”)
- (5.5.11): 26.14 Glueing schemes (“doubled origin constr.”; “more general”; “morphism to Spec k[t]”; “two origins over 0”), 79.12 The finite part of a morphism (“partial”)
- (5.5.12): 29.2 Closed immersions (“(v), special case”), 29.3 Immersions (“(i), special case”; “special case”), 29.45 Integral and finite morphisms (“(i), special case”), 26.18 Base change in algebraic geometry, 26.21 Separation axioms (“(i), special case”; “special case”)
§6
- (6.1.1): 28.5 Noetherian schemes (“more general”)
- (6.1.3): 28.5 Noetherian schemes (“more general”)
- (6.1.4): 28.5 Noetherian schemes (“special case”)
- (6.1.6): 10.32 Locally nilpotent ideals (“affine case”)
- (6.1.9): 5.9 Noetherian topological spaces
- (6.1.12): 28.3 Integral, irreducible, and reduced schemes (“necessity”)
- (6.2.2): 10.53 Artinian rings (“affine, special case”), 10.60 Dimension (“affine case”)
- (6.3.1): 10.6 Ring maps of finite type and of finite presentation (“(P)”), 29.15 Morphisms of finite type (“partial”), 29.21 Quasi-finite morphisms (“partial”), 33.20 Algebraic schemes (“partial”)
- (6.3.4): 10.6 Ring maps of finite type and of finite presentation (“clause (ii), affine case”; “clause (v), affine case”), 10.14 Base change (“(iv), affine case”), 42.15 Push and pull (“clause (iv)”), 56.5 Functors between categories of quasi-coherent modules (“clause (iv)”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“clause (iv)”), 29.3 Immersions (“(v)”), 29.15 Morphisms of finite type (“(i)”; “clause (i)”; “clause (ii)”; “clause (iv)”), 29.21 Quasi-finite morphisms (“(i)”; “clause (iv)”; “clause (v)”), 29.45 Integral and finite morphisms (“more general”), 26.19 Quasi-compact morphisms (“clause (iv)”), 26.21 Separation axioms (“(v)”), 65.4 Lists of useful properties of morphisms of schemes (“clause (ii)”; “clause (iv)”)
- (6.3.5): 29.15 Morphisms of finite type (“partial”), 29.21 Quasi-finite morphisms (“partial”)
- (6.3.6): 10.6 Ring maps of finite type and of finite presentation (“affine case”), 29.21 Quasi-finite morphisms (“local finite type of f”)
- (6.3.7): 29.15 Morphisms of finite type (“more general”), 67.23 Morphisms of finite type (“locally Noetherian”)
- (6.3.8): 10.31 Noetherian rings (“affine case”)
- (6.3.10): 67.24 Points and geometric points (“more general”)
- (6.4.2): 29.21 Quasi-finite morphisms (“sufficiency”)
- (6.4.4): 10.61 Applications of dimension theory (“affine case”)
- (6.4.5): 11.2 Noncommutative algebras, 9.7 Finite extensions, 29.45 Integral and finite morphisms (“more general”)
- (6.4.6): 29.45 Integral and finite morphisms (“more general”), 65.4 Lists of useful properties of morphisms of schemes (“more general”)
- (6.4.7): 9.14 Purely inseparable extensions
- (6.4.11): 10.54 Homomorphisms essentially of finite type (“special case”), 9.6 Field extensions
- (6.4.12): 29.45 Integral and finite morphisms (“more general”), 65.4 Lists of useful properties of morphisms of schemes (“more general”)
- (6.5.1): 29.43 Valuative criteria (“(i)”; “(ii)”; “(ii), X reduced case”)
- (6.5.4): 29.51 Birational morphisms (“(ii), special case”)
- (6.5.5): 29.51 Birational morphisms
- (6.6.1): 29.15 Morphisms of finite type, 29.21 Quasi-finite morphisms, 26.19 Quasi-compact morphisms, 33.20 Algebraic schemes
- (6.6.2): 29.15 Morphisms of finite type, 29.21 Quasi-finite morphisms, 67.23 Morphisms of finite type, 33.20 Algebraic schemes
- (6.6.3): 37.71 Relatively perfect objects (“special case”)
- (6.6.4): 56.5 Functors between categories of quasi-coherent modules, 29.3 Immersions (“more general”), 29.15 Morphisms of finite type, 29.21 Quasi-finite morphisms, 29.41 Quasi-projective morphisms (“(ii), special case”), 26.19 Quasi-compact morphisms, 26.21 Separation axioms (“more general”), 65.4 Lists of useful properties of morphisms of schemes
- (6.6.5): 29.8 Dominant morphisms
- (6.6.6): 10.6 Ring maps of finite type and of finite presentation (“(ii), affine schemes”; “(v), affine schemes”), 42.15 Push and pull, 63.5 Weightings and trace maps for locally quasi-finite morphisms, 29.15 Morphisms of finite type (“(i), closed immersions”; “(i), for immersions”), 29.21 Quasi-finite morphisms (“(i), for immersions”), 65.4 Lists of useful properties of morphisms of schemes, 67.12 Immersions (“(v)”), 67.23 Morphisms of finite type (“(v)”)
- (6.6.7): 10.31 Noetherian rings (“affine case”)
§7
- (7.1.1): 29.50 Rational maps, 67.47 Rational maps (“equivalence definition”; “is an equivalence rel.”)
- (7.1.2): 29.50 Rational maps, 67.47 Rational maps (“rational map”)
- (7.1.3): 31.24 Meromorphic functions and sections (“special case”), 31.25 Meromorphic functions and sections; Noetherian case (“special case”), 31.26 Meromorphic functions and sections; reduced case (“special case”)
- (7.1.4): 5.15 Constructible sets (“partial”)
- (7.1.5): 29.50 Rational maps (“field when integral”; “integral case”; “more general”; “κ(η) when integral”)
- (7.1.8): 29.50 Rational maps (“partial”; “special case”)
- (7.1.11): 29.50 Rational maps (“bijectivity”; “bijectivity, X reduced”; “injectivity”)
- (7.1.13): 29.50 Rational maps (“more general”)
- (7.2.1): 29.50 Rational maps, 67.47 Rational maps (“defined at a point”; “domain of definition”)
- (7.2.2): 29.50 Rational maps (“special case”), 67.47 Rational maps
- (7.2.8): 28.3 Integral, irreducible, and reduced schemes (“partial”)
- (7.3.3): 29.50 Rational maps (“partial”; “special case”)
- (7.3.6): 31.26 Meromorphic functions and sections; reduced case (“F = O_X, integral”; “integral case”), 28.3 Integral, irreducible, and reduced schemes (“partial”)
- (7.4.1): 31.11 Torsion free modules (“quasi-coherence of T”; “special case”; “torsion free”; “torsion section”), 15.22 Torsion free modules (“special case”)
- (7.4.7): 15.22 Torsion free modules (“special case”)
§8
- (8.1.1): 10.50 Valuation rings (“(i) ⇔ (ii)”)
- (8.1.2): 33.37 Glueing dimension one rings (“partial”)
- (8.2.3): 29.50 Rational maps (“closed point, separated”; “sufficiency direction”)
- (8.2.4): 29.50 Rational maps (“closed point case”)
§9
- (9.1.1): 30.9 Coherent sheaves on locally Noetherian schemes (“coherent tensor”; “more general”), 35.10 Quasi-coherent sheaves and topologies, II (“special case”), 17.16 Tensor product (“coherent tensor”), 18.26 Tensor product (“coherent tensor”; “more general”), 103.10 Quasi-coherent modules (“more general”), 96.12 Locally quasi-coherent modules (“more general”), 96.15 Quasi-coherent sheaves on algebraic stacks (“more general”)
- (9.1.12): 6.26 Morphisms of ringed spaces and modules (“case Y = S”)
- (9.2.2): 27.4 Relative spectrum as a functor (“special case”)
- (9.3.1): 28.18 Sections over principal opens (“(i)”)
- (9.3.2): 28.18 Sections over principal opens (“Case L = O”), 26.5 Affine schemes (“special case”)
- (9.3.3): 28.18 Sections over principal opens (“special case”), 26.5 Affine schemes (“special case”)
- (9.3.4): 10.62 Support and dimension of modules (“affine case”), 30.10 Coherent sheaves on Noetherian schemes, 69.13 Coherent sheaves on Noetherian spaces (“more general”)
- (9.4.2): 6.31 Open immersions and (pre)sheaves (“(i)”)
- (9.4.3): 28.23 Extending quasi-coherent sheaves (“qcqs scheme”)
- (9.4.5): 28.23 Extending quasi-coherent sheaves
- (9.4.6): 28.23 Extending quasi-coherent sheaves (“qcqs case”)
- (9.4.7): 28.23 Extending quasi-coherent sheaves (“(b)”)
- (9.4.8): 28.23 Extending quasi-coherent sheaves (“special case”)
- (9.4.9): 30.10 Coherent sheaves on Noetherian schemes (“Noetherian scheme”), 28.23 Extending quasi-coherent sheaves (“quasi-compact scheme”)
- (9.5.1): 29.6 Scheme theoretic image (“factorization clause”), 29.7 Scheme theoretic closure and density
- (9.5.2): 29.6 Scheme theoretic image, 29.7 Scheme theoretic closure and density
- (9.5.3): 29.6 Scheme theoretic image, 29.7 Scheme theoretic closure and density, 101.38 Scheme theoretic image
- (9.5.4): 29.6 Scheme theoretic image (“partial”)
- (9.5.8): 29.6 Scheme theoretic image (“identification over V”)
- (9.5.10): 29.7 Scheme theoretic closure and density (“Y open in closure”; “topological density”)
- (9.6.1): 29.11 Affine morphisms
- (9.6.4): 30.9 Coherent sheaves on locally Noetherian schemes (“quotient, B = O_X”; “submodule, B = O_X”)
- (9.6.6): 28.23 Extending quasi-coherent sheaves (“partial”)
§10
- (10.1.3): 26.5 Affine schemes (“special case”)
- (10.2.1): 10.17 The spectrum of a ring (“special case”)
- (10.2.2): 87.2 Formal schemes à la EGA
- (10.3.7): 87.13 Colimits of algebraic spaces along thickenings (“more general”)
- (10.4.2): 28.5 Noetherian schemes (“special case”), 26.9 Schemes (“special case”)
- (10.4.3): 26.11 Zariski topology of schemes (“special case”)
- (10.4.4): 111.33 Schemes (“special case”), 28.5 Noetherian schemes (“special case”), 26.9 Schemes (“special case”), 26.10 Immersions of schemes (“special case”)
- (10.4.5): 26.2 Locally ringed spaces (“special case”), 26.9 Schemes (“special case”)
- (10.4.6): 87.2 Formal schemes à la EGA (“Affine case X = Spf B”), 87.9 Affine formal algebraic spaces (“X affine formal”)
- (10.4.7): 90.6 Prorepresentable functors and predeformation categories
- (10.6.4): 10.97 Completion for Noetherian rings (“special case”), 10.160 The Cohen structure theorem (“special case”)
- (10.8.1): 26.12 Reduced schemes (“more general”)
- (10.8.5): 87.14 Completion along a closed subset (“affine case Spf(Â)”), 87.38 Completion along a closed subspace (“affine case Spf(Â)”)
- (10.8.6): 10.97 Completion for Noetherian rings (“special case”), 87.37 Recompletion (“more general”; “special case”), 87.38 Completion along a closed subspace (“special case”)
- (10.9.9): 87.37 Recompletion
- (10.10.1): 87.14 Completion along a closed subset (“Spf(A) = X_{/X}”), 87.38 Completion along a closed subspace (“Spf(A) = X_{/X}”), 26.7 Quasi-coherent sheaves on affines (“Representation clause”)
- (10.10.2): 26.5 Affine schemes (“(i), special case”)
- (10.12.2): 85.27 Descent in terms of simplicial schemes (“more general”)
- (10.13.1): 88.11 Finite type morphisms (“affine case”)
- (10.13.2): 87.19 Morphisms representable by algebraic spaces (“special case”)
- (10.13.3): 87.24 Morphisms of finite type (“representable morphisms”)
- (10.13.4): 88.11 Finite type morphisms
- (10.13.5): 10.6 Ring maps of finite type and of finite presentation (“(i), special case”), 10.31 Noetherian rings (“(ii)”), 42.15 Push and pull (“(ii)”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“(ii)”), 29.15 Morphisms of finite type (“(i), special case”; “(ii)”), 29.21 Quasi-finite morphisms (“(ii)”), 65.4 Lists of useful properties of morphisms of schemes (“(i), special case”; “(ii)”)
- (10.14.2): 26.10 Immersions of schemes (“partial”)
- (10.14.3): 29.45 Integral and finite morphisms (“special case”)
- (10.14.5): 29.2 Closed immersions (“(i), ordinary schemes”; “(ii), ordinary schemes”), 26.17 Fibre products of schemes (“(ii), ordinary schemes”), 26.18 Base change in algebraic geometry (“(ii), ordinary schemes”), 26.24 Functoriality for quasi-coherent modules (“(i), ordinary schemes”), 65.4 Lists of useful properties of morphisms of schemes (“(i), ordinary schemes”; “(ii), ordinary schemes”)
- (10.15.1): 87.30 Separation axioms for morphisms (“separated morphism”), 39.4 Group schemes (“Separated morphism X → S”), 83.5 Quotients as orbit spaces, 26.21 Separation axioms (“Separated morphism X → S”; “special case”)
- (10.15.3): 26.21 Separation axioms (“(i), ordinary preschemes”; “(iii), special case”; “(iv), special case”), 65.4 Lists of useful properties of morphisms of schemes (“(i), ordinary preschemes”; “(iii), special case”)
- (10.15.4): 45.3 Correspondences (“ordinary schemes”)
- (10.15.7): 87.16 Separation axioms for formal algebraic spaces (“more general”; “partial”)
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