EGA IV

The items of EGA IV 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): 29.15 Morphisms of finite type, 29.21 Quasi-finite morphisms, 26.19 Quasi-compact morphisms, 33.20 Algebraic schemes
(1.1.2): 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, 26.19 Quasi-compact morphisms, 26.21 Separation axioms (“more general”), 65.4 Lists of useful properties of morphisms of schemes
(1.1.3): 29.9 Surjective morphisms, 67.8 Quasi-compact morphisms
(1.1.4): 35.23 Properties of morphisms local in the fpqc topology on the target (“special case”)
(1.1.5): 29.8 Dominant morphisms (“a) <=> b)”)
(1.2.1): 26.21 Separation axioms, 67.4 Separation axioms
(1.2.2): 56.5 Functors between categories of quasi-coherent modules (“(v), special case”), 29.22 Morphisms of finite presentation (“(ii), special case”), 26.21 Separation axioms, 26.23 Monomorphisms (“more general”), 65.4 Lists of useful properties of morphisms of schemes
(1.2.3): 56.5 Functors between categories of quasi-coherent modules, 29.22 Morphisms of finite presentation (“(ii), necessity”), 26.21 Separation axioms
(1.2.4): 56.5 Functors between categories of quasi-coherent modules (“Case Z = Spec ℤ”), 29.3 Immersions, 26.21 Separation axioms, 67.8 Quasi-compact morphisms, 101.7 Quasi-compact morphisms
(1.2.6): 56.5 Functors between categories of quasi-coherent modules (“partial”), 29.22 Morphisms of finite presentation (“partial”), 26.21 Separation axioms (“special case”)
(1.2.7): 26.21 Separation axioms, 5.12 Quasi-compact spaces and maps (“partial”)
(1.3.1): 10.6 Ring maps of finite type and of finite presentation, 10.14 Base change, 35.4 Descent for universally injective morphisms (“special case”), 15.82 Relatively finitely presented modules
(1.3.2): 29.15 Morphisms of finite type (“finite type at x”), 29.21 Quasi-finite morphisms, 33.20 Algebraic schemes
(1.3.3): 98.25 Limit preserving functors on Noetherian schemes (“special case”), 29.15 Morphisms of finite type, 28.5 Noetherian schemes (“open immersions”; “special case”), 67.23 Morphisms of finite type
(1.3.4): 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), immersions”), 29.21 Quasi-finite morphisms (“(i), immersions”), 65.4 Lists of useful properties of morphisms of schemes, 67.12 Immersions (“(v)”), 67.23 Morphisms of finite type (“(v)”)
(1.3.5): 10.31 Noetherian rings (“affine schemes”), 29.15 Morphisms of finite type (“identity base change”)
(1.3.7): 67.24 Points and geometric points (“more general”)
(1.3.8): 10.54 Homomorphisms essentially of finite type, 9.6 Field extensions
(1.3.9): 10.54 Homomorphisms essentially of finite type
(1.3.10): 10.9 Localization
(1.4.1): 10.6 Ring maps of finite type and of finite presentation, 10.14 Base change, 10.31 Noetherian rings (“more general”), 10.126 Algebras and modules of finite presentation, 15.82 Relatively finitely presented modules (“more general”)
(1.4.2): 29.22 Morphisms of finite presentation (“fin. presentation at x”)
(1.4.3): 10.6 Ring maps of finite type and of finite presentation (“(ii), affine schemes”; “(v), affine schemes”), 10.14 Base change (“(iii), special case”), 10.126 Algebras and modules of finite presentation (“(iii), special case”), 41.14 Topological properties of étale morphisms (“(i), open immersions”), 63.5 Weightings and trace maps for locally quasi-finite morphisms, 29.22 Morphisms of finite presentation (“(i), open immersions”), 65.4 Lists of useful properties of morphisms of schemes
(1.4.3.1): 29.22 Morphisms of finite presentation (“special case”), 67.28 Morphisms of finite presentation (“more general”)
(1.4.4): 10.6 Ring maps of finite type and of finite presentation (“necessity”)
(1.4.5): 29.22 Morphisms of finite presentation (“closed immersion case”), 71.5 Fitting ideals (“the closed subspaces”), 67.28 Morphisms of finite presentation (“closed immersion case”)
(1.4.7): 10.7 Finite ring maps (“necessity”)
(1.5.1): 37.71 Relatively perfect objects (“(a) ⇒ (c), special case”)
(1.5.3): 10.31 Noetherian rings (“affine case”)
(1.5.4): 10.6 Ring maps of finite type and of finite presentation (“(ii) for affine schemes”; “special case”), 10.14 Base change (“(iii), special case”), 29.3 Immersions (“(v)”), 29.15 Morphisms of finite type (“(i)”), 29.21 Quasi-finite morphisms (“(i)”), 29.45 Integral and finite morphisms (“more general”), 26.21 Separation axioms (“(v)”)
(1.5.5): 10.31 Noetherian rings (“affine case”)
(1.5.6): 29.21 Quasi-finite morphisms (“partial”)
(1.6.2): 10.6 Ring maps of finite type and of finite presentation (“(ii), affine case”; “(v), affine case”), 10.14 Base change (“(iii), special case”), 10.126 Algebras and modules of finite presentation (“(iii), special case”), 29.22 Morphisms of finite presentation
(1.7.9): 28.23 Extending quasi-coherent sheaves (“I.9.6.5 clause”)
(1.7.17): 30.3 Vanishing of cohomology (“(d') => affine”)
(1.7.21): 30.4 Quasi-coherence of higher direct images (“base change vanishing”; “vanishing bound”)
(1.8.1): 59.15 The fpqc topology (“more general”), 28.2 Constructible sets (“partial”; “retrocompact criterion”), 28.25 Sections with support in a closed subset (“qc open ⟺ finite ∪ aff”), 26.19 Quasi-compact morphisms (“more general”), 5.12 Quasi-compact spaces and maps (“special case”), 5.15 Constructible sets (“partial”), 5.16 Constructible sets and Noetherian spaces (“special case”)
(1.8.2): 29.23 Constructible sets (“locally constructible”), 5.16 Constructible sets and Noetherian spaces (“special case”)
(1.8.4): 10.29 Images of ring maps of finite presentation (“affine case”), 67.29 Constructible sets
(1.8.4.1): 10.29 Images of ring maps of finite presentation (“affine source”), 67.29 Constructible sets
(1.8.4.2): 10.168 Colimits and maps of finite presentation, II
(1.8.5): 10.29 Images of ring maps of finite presentation (“affine case”)
(1.8.6): 10.112 Homomorphisms and dimension (“affine case”), 37.24 Generic fibres (“first assertion”), 37.25 Relative assassins (“second assertion”)
(1.8.7): 26.18 Base change in algebraic geometry (“partial”)
(1.8.7.1): 29.10 Radicial and universally injective morphisms (“more general”)
(1.9.3.1): 10.127 Colimits and maps of finite presentation (“more general”)
(1.9.5): 10.29 Images of ring maps of finite presentation (“(viii), special case”)
(1.9.6): 26.21 Separation axioms (“special case”)
(1.9.14): 5.23 Spectral spaces (“(iii)”; “iii, special case”; “iv, special case”)
(1.9.15): 5.23 Spectral spaces (“(ii)(iii) suff, spectral”)
(1.10.3): 29.24 Open morphisms (“a) ⇒ b)”), 5.19 Specialization (“a) ⇒ b)”)
(1.10.4): 41.10 Topological properties of flat morphisms (“b) ⇒ a), special case”), 59.26 Étale morphisms (“b) ⇒ a), special case”), 29.24 Open morphisms (“b) ⇒ a)”; “b) ⇒ a), special case”), 29.35 Smooth morphisms (“b) ⇒ a), special case”), 29.37 Étale morphisms (“b) ⇒ a), special case”), 5.19 Specialization (“a) ⇒ b), special case”)

§2

(2.1.1): 41.9 Flat morphisms, 39.4 Group schemes, 17.20 Flat morphisms of ringed spaces (“partial”), 29.26 Flat morphisms
(2.1.2): 28.20 Flat modules (“A = B, φ = id”)
(2.1.4): 10.39 Flat modules and flat ring maps (“special case”), 42.15 Push and pull, 63.5 Weightings and trace maps for locally quasi-finite morphisms, 29.26 Flat morphisms (“global module case”; “pointwise statement”), 65.4 Lists of useful properties of morphisms of schemes
(2.1.5): 10.39 Flat modules and flat ring maps (“special case”), 42.14 Flat pullback (“global case Y' = Y”), 29.26 Flat morphisms (“global case Y' = Y”), 65.4 Lists of useful properties of morphisms of schemes (“global case Y' = Y”)
(2.1.6): 10.39 Flat modules and flat ring maps (“special case”), 42.14 Flat pullback, 29.26 Flat morphisms, 65.4 Lists of useful properties of morphisms of schemes
(2.1.8): 10.39 Flat modules and flat ring maps (“(i), affine case”; “(ii)”), 18.39 Pullbacks of flat modules (“(i), G' = O_{Y'}”)
(2.1.10): 10.39 Flat modules and flat ring maps (“Affine case”; “special case”), 17.17 Flat modules (“Absolute case f = id”), 18.28 Flat modules (“Absolute case f = id”; “special case”)
(2.1.11): 10.40 Supports and annihilators (“affine case”)
(2.1.12): 10.12 Tensor products, 10.39 Flat modules and flat ring maps, 10.78 Finite projective modules (“affine, flat everywhere”), 41.9 Flat morphisms, 17.14 Locally free sheaves, 28.21 Locally free modules (“special case”)
(2.1.13): 10.164 Descending properties (“special case”)
(2.2.1): 10.39 Flat modules and flat ring maps (“affine case, a)<=>b)”), 10.112 Homomorphisms and dimension (“special case”)
(2.2.2): 10.39 Flat modules and flat ring maps (“a)<=>b'), affine F=O_X”; “partial”)
(2.2.3): 10.39 Flat modules and flat ring maps (“M = B with A → B flat”)
(2.2.4): 10.39 Flat modules and flat ring maps (“special case”), 41.9 Flat morphisms (“special case”)
(2.2.5): 10.39 Flat modules and flat ring maps (“special case”)
(2.2.6): 10.39 Flat modules and flat ring maps (“special case”)
(2.2.7): 10.39 Flat modules and flat ring maps (“partial”; “special case”), 10.82 Universally injective module maps (“partial”), 35.4 Descent for universally injective morphisms (“special case”), 41.9 Flat morphisms (“partial”; “special case”)
(2.2.8): 10.82 Universally injective module maps (“affine case”)
(2.2.10): 10.39 Flat modules and flat ring maps (“special case”)
(2.2.11): 10.39 Flat modules and flat ring maps (“(i), affine case”; “(iii), special case”; “special case”), 42.14 Flat pullback (“(iii), special case”), 29.26 Flat morphisms (“(iii), special case”), 65.4 Lists of useful properties of morphisms of schemes (“(iii), special case”)
(2.2.13): 10.39 Flat modules and flat ring maps (“(i) for affine schemes”; “(iii)”), 42.14 Flat pullback (“(iii)”), 42.15 Push and pull (“(i)”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“(i)”), 29.26 Flat morphisms (“(i)”; “(iii)”), 65.4 Lists of useful properties of morphisms of schemes (“(i)”; “(iii)”)
(2.2.14): 10.164 Descending properties (“affine source and target”)
(2.2.16): 59.45 Topological invariance of the small étale site (“special case”)
(2.2.17): 10.101 Flatness criteria over Artinian rings (“special case”)
(2.3.2): 29.6 Scheme theoretic image, 29.7 Scheme theoretic closure and density, 29.26 Flat morphisms, 67.30 Flat morphisms, 101.38 Scheme theoretic image (“more general”)
(2.3.4): 42.13 Preparation for flat pullback (“(iii), special case”), 68.4 Finiteness conditions and points (“(i), special case”), 29.26 Flat morphisms (“(i), special case”), 5.19 Specialization
(2.3.5): 10.47 Geometrically irreducible algebras (“(iii), special case”), 29.24 Open morphisms (“partial”; “special case”), 29.26 Flat morphisms (“special case”), 5.19 Specialization (“(i), special case”)
(2.3.6): 5.19 Specialization (“(i), special case”)
(2.3.7): 56.5 Functors between categories of quasi-coherent modules (“(i)”), 29.15 Morphisms of finite type (“(i)”), 29.21 Quasi-finite morphisms (“(i)”), 26.19 Quasi-compact morphisms (“(i)”), 65.4 Lists of useful properties of morphisms of schemes (“(i)”)
(2.3.12): 41.10 Topological properties of flat morphisms (“special case”), 67.30 Flat morphisms (“closed case”; “open case”)
(2.4.2): 29.24 Open morphisms, 29.46 Universal homeomorphisms (“universal homeomorphism”)
(2.4.3): 35.23 Properties of morphisms local in the fpqc topology on the target (“more general”), 29.24 Open morphisms (“(ii), universally open”), 29.42 Proper morphisms (“(ii)”; “(iii)”; “(v), universally closed”), 26.19 Quasi-compact morphisms, 65.4 Lists of useful properties of morphisms of schemes (“(ii)”; “(ii), universally open”; “(iii)”; “(iii), universally open”; “more general”)
(2.4.5): 10.46 Universal homeomorphisms (“(i), affine case”), 29.46 Universal homeomorphisms (“(i)”)
(2.4.6): 41.10 Topological properties of flat morphisms, 59.26 Étale morphisms, 29.26 Flat morphisms, 29.35 Smooth morphisms, 29.37 Étale morphisms, 67.30 Flat morphisms (“in particular clause”), 67.31 Flat modules
(2.4.7): 41.10 Topological properties of flat morphisms (“more general”), 59.26 Étale morphisms (“more general”), 29.35 Smooth morphisms (“more general”), 29.37 Étale morphisms (“more general”)
(2.4.8): 29.46 Universal homeomorphisms (“(ii) univ. homeo”), 67.53 Universal homeomorphisms (“(ii) univ. homeo”)
(2.4.9): 67.6 Open morphisms (“Y = Spec k”)
(2.4.10): 67.6 Open morphisms
(2.4.11): 5.6 Submersive maps (“more general”), 33.24 Closures of products (“more general”)
(2.5.1): 10.39 Flat modules and flat ring maps (“flat part, affine”; “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”), 18.19 Localization of ringed sites (“partial”), 18.21 Localization of ringed topoi (“partial”), 65.4 Lists of useful properties of morphisms of schemes (“special case”)
(2.5.2): 10.83 Descent for finite projective modules (“(ii), descent, affine”), 35.4 Descent for universally injective morphisms (“(i), affine case”; “(ii), affine case”)
(2.5.3): 10.5 Finite modules and finitely presented modules, 10.83 Descent for finite projective modules (“fin. pres., descent”), 11.2 Noncommutative algebras, 9.7 Finite extensions
(2.5.4): 35.4 Descent for universally injective morphisms (“(i), affine case”; “(ii), affine case”)
(2.5.4.1): 10.82 Universally injective module maps (“more general”)
(2.5.5): 10.112 Homomorphisms and dimension (“affine fiber module”)
(2.5.5.1): 10.52 Length (“F = structure sheaf”; “special case”)
(2.5.5.2): 10.52 Length (“case N = B”; “special case”)
(2.5.5.3): 10.52 Length (“in particular”; “special case”)
(2.5.5.4): 10.39 Flat modules and flat ring maps (“special case”)
(2.5.8): 10.14 Base change (“partial”)
(2.5.8.8): 10.20 Nakayama's lemma (“partial, special case”), 10.39 Flat modules and flat ring maps (“partial”), 10.53 Artinian rings (“special case”), 11.4 Lemmas on algebras (“special case”)
(2.5.8.9): 11.4 Lemmas on algebras (“more general”; “partial”; “special case”)
(2.6.2): 35.23 Properties of morphisms local in the fpqc topology on the target (“(iv), special case”), 65.4 Lists of useful properties of morphisms of schemes (“(iv), special case”)
(2.6.4): 35.23 Properties of morphisms local in the fpqc topology on the target (“more general”), 56.5 Functors between categories of quasi-coherent modules (“necessity in (v)”), 29.15 Morphisms of finite type (“necessity in (v)”), 29.21 Quasi-finite morphisms (“necessity in (v)”), 29.42 Proper morphisms (“(ii)”), 26.19 Quasi-compact morphisms (“necessity in (v)”), 65.4 Lists of useful properties of morphisms of schemes (“(ii)”; “more general”; “necessity in (i)”; “necessity in (v)”; “partial”)
(2.7.1): 10.126 Algebras and modules of finite presentation (“special case”), 42.15 Push and pull (“(iii)”; “implication”), 53.8 The genus of a curve (“implication”), 35.23 Properties of morphisms local in the fpqc topology on the target (“(iv)”; “(xii)”; “(xv)”; “(xvii)”; “descent direction of (x)”; “more general”), 56.5 Functors between categories of quasi-coherent modules (“(ii)”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“(iii)”; “(iv)”), 29.2 Closed immersions (“implication”), 29.13 Quasi-affine morphisms (“(xiv)”), 29.15 Morphisms of finite type (“(iii)”), 29.21 Quasi-finite morphisms (“(iii)”), 29.22 Morphisms of finite presentation (“(iv)”), 29.42 Proper morphisms (“implication”), 29.45 Integral and finite morphisms (“implication”), 26.17 Fibre products of schemes (“(x)”; “implication”), 26.18 Base change in algebraic geometry (“(x)”; “implication”), 26.21 Separation axioms (“(ii)”; “implication”), 26.23 Monomorphisms (“(ix)”), 65.4 Lists of useful properties of morphisms of schemes (“(ii)”; “(iii)”; “(iv)”; “(ix)”; “(x)”; “(xiii)”; “implication”; “more general”), 74.11 Descending properties of morphisms in the fpqc topology (“more general”), 67.10 Monomorphisms (“(ix)”)
(2.7.1.1): 10.14 Base change (“partial”), 10.126 Algebras and modules of finite presentation (“partial”)
(2.8.1): 108.7 Properties of the Hilbert functor (“special case”)
(2.8.1.1): 108.5 Properties of Quot (“special case”)

§3

(3.1.2): 31.2 Associated points (“more general”)
(3.1.3): 31.4 Embedded points (“special case”)
(3.1.6): 20.24 Alternative view of the Čech complex (“more general”), 5.28 Partitions and stratifications (“more general”)
(3.1.10.1): 10.63 Associated primes (“Contraction inclusion”), 10.66 Weakly associated primes (“Lifting inclusion”; “Quotient maps A → A/I”)
(3.1.10.2): 27.4 Relative spectrum as a functor (“special case”)
(3.2.1): 10.157 Serre's criterion for normality (“a), special case”)
(3.3.1): 10.65 Relative assassin (“affine case”), 10.112 Homomorphisms and dimension (“special case”), 31.3 Morphisms and associated points (“equality (Noetherian)”; “inclusion (3.3.1.1)”)
(3.3.5): 10.163 Ascending properties (“affine case”)
(3.3.6): 10.65 Relative assassin (“affine case, X = S”), 31.3 Morphisms and associated points (“one factor = base”; “⊇ only, one factor =”)
(3.4.1): 10.18 Local rings, 10.50 Valuation rings, 10.52 Length, 26.12 Reduced schemes
(3.4.6): 31.16 Effective Cartier divisors on Noetherian schemes (“partial”)

§10

(10.1.1): 5.16 Constructible sets and Noetherian spaces (“global comparison”)
(10.1.2): 5.18 Jacobson spaces (“a′ ⇒ c and c′, a′ ⇒ c″”)
(10.2.7): 5.18 Jacobson spaces (“special case”)
(10.2.8): 18.42 Constant sheaves (“special case”)
(10.3.1): 28.6 Jacobson schemes, 5.18 Jacobson spaces (“Forward direction”)
(10.3.2): 28.6 Jacobson schemes (“special case”), 5.18 Jacobson spaces (“closed subspace case”; “locally closed case”; “more general”; “special case”)
(10.3.3): 28.6 Jacobson schemes (“necessity”; “sufficiency”), 5.18 Jacobson spaces (“necessity”)
(10.4.1): 10.35 Jacobson rings (“ring: Spec(A) Jacobson”), 28.6 Jacobson schemes (“ring: Spec(A) Jacobson”)
(10.4.2): 10.35 Jacobson rings (“special case”), 28.6 Jacobson schemes (“special case”)
(10.4.3): 10.35 Jacobson rings (“example ℤ”; “principal ideal ring”; “special case”), 10.60 Dimension (“partial”), 10.61 Applications of dimension theory (“Artinian, PID clauses”), 28.6 Jacobson schemes (“Open subpreschemes”), 5.18 Jacobson spaces (“Closed subpreschemes”; “Open subpreschemes”; “more general”)
(10.4.4): 10.9 Localization, 10.17 The spectrum of a ring
(10.4.5): 10.34 Hilbert Nullstellensatz (“a⇒b′”; “a⇒b′ over a field”), 10.35 Jacobson rings (“a) => b)”; “a⇒b′”; “b) => a)”)
(10.4.5.1): 5.18 Jacobson spaces (“sufficiency”)
(10.4.6): 10.34 Hilbert Nullstellensatz (“partial”), 10.35 Jacobson rings (“in particular”; “preimage of max ideal”), 29.16 Points of finite type and Jacobson schemes (“over Z”)
(10.4.7): 10.35 Jacobson rings (“special case”), 68.23 Jacobson spaces (“Y Jacobson”), 29.16 Points of finite type and Jacobson schemes (“(1) => (3)”; “1-dim Noeth domain base”; “base = Spec Z”; “punctured spectrum base”), 28.6 Jacobson schemes (“open immersions”), 5.18 Jacobson spaces (“closed subschemes”; “open immersions”; “special case”)
(10.4.9): 29.16 Points of finite type and Jacobson schemes (“more general”; “over Z”)
(10.4.10): 10.162 Nagata rings (“more general”), 5.18 Jacobson spaces (“partial”; “special case”)
(10.5.1): 10.17 The spectrum of a ring, 10.60 Dimension, 10.61 Applications of dimension theory (“e) ⇒ f) for local B”)
(10.5.2): 10.35 Jacobson rings (“special case”), 10.61 Applications of dimension theory (“sufficiency”)
(10.5.5): 10.61 Applications of dimension theory (“(ii) sufficiency, affine”)
(10.5.7): 29.16 Points of finite type and Jacobson schemes (“special case”), 28.6 Jacobson schemes (“Noetherian local ring”; “affine case”)
(10.5.8): 15.10 Zariski pairs, 28.6 Jacobson schemes (“Spec(A)-V(R) Jacobson”; “local case”)
(10.5.9): 28.6 Jacobson schemes (“Jacobson part”; “Y is Jacobson”)
(10.6.1): 10.105 Catenary rings (“(i)”), 42.7 Setup (“(i)”), 29.16 Points of finite type and Jacobson schemes (“(i), condition 1° for X”), 29.17 Universally catenary schemes (“(i), condition 2° for X”; “(i), special case”), 33.20 Algebraic schemes (“(iii), special case”)
(10.6.3): 33.20 Algebraic schemes (“special case”; “sup_i dim(Y_i)”)
(10.7.1): 10.35 Jacobson rings (“partial”), 29.16 Points of finite type and Jacobson schemes (“1° over Z”; “1°, partial”; “special case”), 29.17 Universally catenary schemes (“2° over Z”; “2° over a field”; “more general”)
(10.7.2): 10.105 Catenary rings (“more general”)
(10.7.3): 10.17 The spectrum of a ring (“partial”), 10.105 Catenary rings (“partial”), 110.21 Dimension in Noetherian Jacobson rings (“n = 2 case”), 110.22 Underlying space Noetherian not Noetherian (“partial”), 61.5 Constructing w-local affine schemes (“partial”)
(10.9.1): 5.18 Jacobson spaces (“more general”)
(10.9.2): 10.35 Jacobson rings (“more general”), 29.16 Points of finite type and Jacobson schemes (“more general”), 5.18 Jacobson spaces (“partial”; “special case”)
(10.10.3): 10.6 Ring maps of finite type and of finite presentation (“more general”)
(10.10.4): 28.3 Integral, irreducible, and reduced schemes (“forward direction”), 26.12 Reduced schemes (“forward direction”)

§11

(11.1.1): 41.10 Topological properties of flat morphisms (“F = O_X case”)
(11.1.6): 10.129 Openness of the flat locus (“case C = B”), 41.10 Topological properties of flat morphisms (“M = C = B case”), 38.13 Flat finite type modules, Part II (“case C = B”)
(11.2.5): 10.99 Criteria for flatness (“case b), local rings”; “special case”), 10.101 Flatness criteria over Artinian rings (“special case”)
(11.2.6): 10.168 Colimits and maps of finite presentation, II (“affine case, (ii)”), 32.10 Descending relative objects (“(ii)”), 70.6 Descending properties of morphisms (“F = O_X case”)
(11.2.6.1): 10.39 Flat modules and flat ring maps (“(ii), sufficiency”), 10.168 Colimits and maps of finite presentation, II (“(ii) necessity”), 77.7 Flattening functors (“(ii)”)
(11.2.8): 26.18 Base change in algebraic geometry
(11.2.9.4): 15.26 Flatness and finiteness conditions (“Module clause”; “special case”)
(11.3.1): 10.129 Openness of the flat locus (“U open, affine case”), 41.10 Topological properties of flat morphisms (“Noetherian, F = O_X”; “special case”), 59.26 Étale morphisms (“special case”), 37.15 Openness of the flat locus (“U open”; “more general”), 29.26 Flat morphisms (“f|U open”), 29.35 Smooth morphisms (“special case”), 29.37 Étale morphisms (“special case”), 67.31 Flat modules (“f|U open”)
(11.3.3): 10.129 Openness of the flat locus (“case C = B”), 38.13 Flat finite type modules, Part II (“WeakAss finite”; “wk assoc pts loc finite”)
(11.3.7): 10.79 Open loci defined by module maps (“affine case, b)=>a)”)
(11.3.8): 10.99 Criteria for flatness (“(c) ⇒ (a), special case”), 10.128 More flatness criteria (“(c) ⇒ (a), special case”; “c) ⇒ a)”), 37.23 Slicing Cohen-Macaulay morphisms (“c) ⇒ b') for F = O_X”)
(11.3.10): 10.99 Criteria for flatness (“special case”), 10.128 More flatness criteria (“special case”), 37.16 Critère de platitude par fibres (“1°) a) ⇔ b)”; “2°) a) ⇔ b)”; “2°) open locus”)
(11.3.10.1): 10.99 Criteria for flatness (“(a) ⇒ (b)”)
(11.3.10.2): 10.39 Flat modules and flat ring maps (“1°”; “1°, special case”)
(11.3.11): 10.101 Flatness criteria over Artinian rings (“(a) ⇒ (b), special case”), 10.128 More flatness criteria (“(a) ⇒ (b), special case”)
(11.3.12): 10.78 Finite projective modules (“more general”), 10.99 Criteria for flatness (“b⇒a when J is nilpotent”; “b⇒a, special case”; “partial”), 10.101 Flatness criteria over Artinian rings (“special case”), 10.128 More flatness criteria (“b⇒a, b⇒a for arbitrary A”)
(11.3.13): 10.163 Ascending properties (“(ii) reduced, Noetherian”), 10.164 Descending properties (“(i), reduced case”)
(11.3.16): 35.14 Descent of finiteness and smoothness properties of morphisms (“affine, fin. pres. case”; “affine, finite type case”; “loc. fin. presentation”; “locally of finite type”), 74.8 Descent of finiteness and smoothness properties of morphisms (“loc. fin. presentation”; “locally of finite type”)
(11.3.17): 35.14 Descent of finiteness and smoothness properties of morphisms (“finite presentation case”; “finite type case”)
(11.4.1): 10.101 Flatness criteria over Artinian rings (“special case”)
(11.4.4): 15.17 Flattening over an Artinian ring (“flat clause, Artinian”)
(11.4.6): 10.118 Generic flatness (“special case”)
(11.4.7): 10.64 Symbolic powers
(11.4.9): 10.101 Flatness criteria over Artinian rings (“sufficiency (flat case)”)
(11.5.4): 42.15 Push and pull (“necessity”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“necessity”), 29.26 Flat morphisms (“necessity”), 65.4 Lists of useful properties of morphisms of schemes (“necessity”)
(11.5.5): 15.21 Descent of flatness along integral maps (“finite injective case”)
(11.9.5): 26.17 Fibre products of schemes (“Geometric clause”), 26.18 Base change in algebraic geometry (“Geometric clause”), 65.4 Lists of useful properties of morphisms of schemes (“Geometric clause”)
(11.9.14): 10.39 Flat modules and flat ring maps (“affine case”), 10.82 Universally injective module maps (“special case”), 67.19 Universally injective morphisms (“base change stability”), 101.14 Universally injective morphisms (“base change stability”; “fppf descent clause”)
(11.9.16): 18.19 Localization of ringed sites (“more general”), 18.21 Localization of ringed topoi (“more general”)
(11.9.18): 10.82 Universally injective module maps (“special case”)
(11.10.2): 67.17 Scheme theoretic closure and density (“single open subspace”)
(11.10.3): 27.4 Relative spectrum as a functor

§12

(12.0.2): 5.19 Specialization (“more general”)
(12.1.1): 10.130 Openness of Cohen-Macaulay loci (“(vi), affine, F=O_X”), 76.26 Cohen-Macaulay morphisms (“(vi), F = O_X”)
(12.1.1.4): 10.112 Homomorphisms and dimension (“special case”), 37.18 Flat modules and relative assassins (“equidimensional Z'→Y'”; “fibre dimension equality”; “z ∈ Ass(F_y)”)
(12.1.1.5): 37.18 Flat modules and relative assassins (“dim(Z'_y)=dim(Z'_{y'})”; “equidimensionality”; “z ∈ Ass(F_y)”)
(12.1.5): 10.163 Ascending properties (“(i), F=O_X, G=O_Y”)
(12.1.7): 10.130 Openness of Cohen-Macaulay loci (“(iii)”; “(iii), affine flat”; “(iii), over a field”), 37.22 Cohen-Macaulay morphisms (“(iii), flat case”)
(12.1.8): 10.157 Serre's criterion for normality (“more general”)
(12.2.1.1): 37.22 Cohen-Macaulay morphisms (“partial”)
(12.2.4): 76.29 Reduced fibres (“more general”)
(12.3.1): 10.83 Descent for finite projective modules (“a ⇒ b”; “b ⇒ a”)
(12.3.2): 10.71 Ext groups (“(i), special case”)

§13

(13.1.3): 29.29 Morphisms and dimensions of fibres
(13.1.4): 10.123 Zariski's Main Theorem (“affine case”)
(13.1.5): 37.30 Dimension of fibres (“universally closed”)
(13.2.6): 67.35 The dimension formula (“more general”)
(13.2.12): 37.30 Dimension of fibres (“(ii), special case”)
(13.3.1.1): 33.18 Variants of Noether normalization (“quasi-finite factoriz.”)
(13.3.4): 10.113 The dimension formula (“partial”; “special case”), 29.53 The dimension formula (“Inequality dim O_{X_j”; “partial”), 33.20 Algebraic schemes (“special case”)

§14

(14.1.6): 29.24 Open morphisms (“partial”), 5.19 Specialization (“special case”)
(14.2.4): 37.30 Dimension of fibres
(14.2.5): 37.30 Dimension of fibres (“flat fin. pres. case”), 76.31 Dimension of fibres (“flat fin. pres. case”)
(14.3.1): 29.24 Open morphisms
(14.3.4): 29.24 Open morphisms (“(i), special case”), 65.4 Lists of useful properties of morphisms of schemes (“(i), special case”; “(ii), global form”; “(ii), special case”)
(14.3.10): 10.125 Dimension of fibres (“affine case”), 33.19 Dimension of fibres (“surjective f”)
(14.4.2): 37.74 Universally open morphisms (“quasi-finite, b)=>a)”)
(14.4.4): 37.74 Universally open morphisms (“loc. quasi-finite case”)
(14.4.5): 37.74 Universally open morphisms (“quasi-finite case”)
(14.4.9): 29.24 Open morphisms
(14.4.10): 41.10 Topological properties of flat morphisms (“special case”), 59.26 Étale morphisms (“special case”), 29.35 Smooth morphisms (“special case”), 29.37 Étale morphisms (“special case”)
(14.5.5): 10.5 Finite modules and finitely presented modules, 10.36 Finite and integral ring extensions, 10.37 Normal rings (“partial”), 10.161 Japanese rings (“finiteness of A₁ over A”), 11.2 Noncommutative algebras, 9.7 Finite extensions
(14.5.6): 37.74 Universally open morphisms (“(d) ⇒ (b)”)
(14.5.7): 10.119 Around Krull-Akizuki (“more general”)
(14.5.8): 90.6 Prorepresentable functors and predeformation categories, 26.18 Base change in algebraic geometry
(14.5.9): 37.74 Universally open morphisms (“(d) ⇒ (a), special case”)
(14.5.11): 65.4 Lists of useful properties of morphisms of schemes (“Necessity direction”)

§15

(15.1.1): 10.112 Homomorphisms and dimension (“affine, special case”)
(15.1.1.6): 10.119 Around Krull-Akizuki (“DVR dom. A, frac field K”)
(15.2.1): 41.10 Topological properties of flat morphisms, 59.26 Étale morphisms, 29.26 Flat morphisms, 29.35 Smooth morphisms, 29.37 Étale morphisms, 67.30 Flat morphisms (“flat+lfp ⇒ univ open”), 67.31 Flat modules (“more general”)
(15.2.4): 10.78 Finite projective modules (“Freeness clause”), 41.10 Topological properties of flat morphisms (“(ii), special case”), 59.26 Étale morphisms (“(ii), special case”), 29.35 Smooth morphisms (“(ii), special case”), 29.37 Étale morphisms (“(ii), special case”), 29.46 Universal homeomorphisms (“more general”)
(15.4.1): 10.103 Cohen-Macaulay modules (“local form”; “special case”), 10.104 Cohen-Macaulay rings (“special case”), 10.112 Homomorphisms and dimension (“special case”), 37.15 Openness of the flat locus (“partial”), 26.18 Base change in algebraic geometry
(15.4.2): 10.128 More flatness criteria (“e′)⇒b), partial”), 10.163 Ascending properties (“partial”)
(15.4.3): 37.22 Cohen-Macaulay morphisms (“F = O_X, f flat”)
(15.5.2.2): 28.5 Noetherian schemes (“1°”)
(15.5.3): 37.28 Connected components of fibres (“more general”)
(15.5.4): 37.53 Stein factorization (“Flat case”)
(15.5.6): 37.28 Connected components of fibres (“more general”)
(15.5.7): 37.53 Stein factorization (“Flat”)
(15.5.9): 37.53 Stein factorization (“(ii), special case”), 76.36 Stein factorization (“more general”)
(15.6.4): 37.29 Connected components meeting a section (“special case”)
(15.6.5): 37.29 Connected components meeting a section (“DVR base”; “In particular: X⁰ open”; “Noeth. source”; “nbhd of X⁰_y”)
(15.6.8): 37.29 Connected components meeting a section (“special case”)
(15.6.10): 39.7 Properties of group schemes over a field (“partial”), 39.8 Properties of algebraic group schemes (“dim(G_y^0) = dim(G_y)”; “special case”), 39.9 Abelian varieties (“special case”), 40.10 Properties of groupoids on fields (“partial”)
(15.7.5): 32.15 Noetherian valuative criterion (“a⇔c”; “c⇒a”), 32.16 Refined Noetherian valuative criteria (“c⇒a”)
(15.7.8): 41.10 Topological properties of flat morphisms (“partial”), 29.25 Submersive morphisms, 67.7 Submersive morphisms (“more general”), 67.30 Flat morphisms (“fppf ⇒ submersive”), 5.6 Submersive maps
(15.7.11.1): 35.23 Properties of morphisms local in the fpqc topology on the target (“partial”), 65.4 Lists of useful properties of morphisms of schemes (“fpqc-locality conjunct”)

§16

(16.1.2): 31.20 The normal cone of an immersion (“Gr_•(f), immersions”), 37.5 Infinitesimal neighbourhoods (“Y^{(1)}, immersions”; “Y^{(n)} inclusion”; “special case”), 76.6 The normal cone of an immersion (“I^n/I^{n+1} pieces”)
(16.2.2): 10.134 The naive cotangent complex (“(iii)”), 29.32 Conormal sheaf of an immersion (“(iii)”)
(16.2.3): 41.6 Topological properties of unramified morphisms (“special case”), 29.3 Immersions (“more general”), 26.21 Separation axioms (“more general”)
(16.2.4): 10.134 The naive cotangent complex (“affine case”)
(16.2.7): 29.32 Conormal sheaf of an immersion (“surjectivity at right”)
(16.3.1): 10.131 Differentials (“affine Ω¹_{X/S}”), 29.33 Sheaf of differentials of a morphism (“Ω¹_{X/S}=Gr₁(Δ_f)”)
(16.3.6): 10.131 Differentials (“special case”), 17.28 Modules of differentials (“dt”), 18.33 Modules of differentials (“dt”)
(16.3.7): 10.131 Differentials
(16.4.3): 17.28 Modules of differentials (“(16.4.3.6),(16.4.3.7)”), 18.33 Modules of differentials (“(16.4.3.6),(16.4.3.7)”)
(16.4.5): 10.131 Differentials (“special case”)
(16.4.7): 27.6 Vector bundles
(16.4.19): 10.131 Differentials (“exact at Ω, affine”; “second assertion”), 17.28 Modules of differentials (“exactness at middle”; “surjectivity on right”)
(16.4.21): 10.131 Differentials (“special case”), 90.3 The base category (“special case”), 29.33 Sheaf of differentials of a morphism (“special case”; “→ 0 (surjectivity)”)
(16.4.22): 10.131 Differentials (“Ω¹ only, affine”), 29.33 Sheaf of differentials of a morphism
(16.4.23): 29.33 Sheaf of differentials of a morphism
(16.4.24): 17.29 Finite order differential operators (“(ii)”), 18.34 Finite order differential operators (“(ii)”)
(16.5.1): 17.28 Modules of differentials (“more general”), 18.33 Modules of differentials (“more general”)
(16.5.2): 29.33 Sheaf of differentials of a morphism
(16.5.3): 10.131 Differentials (“(i), affine case”; “(ii), affine case”; “(ii), affine/ring case”), 17.28 Modules of differentials (“(i)”; “(ii)”), 37.9 Infinitesimal deformations of maps (“(ii)”), 29.33 Sheaf of differentials of a morphism (“(ii)”), 18.33 Modules of differentials (“(i)”)
(16.5.7): 10.131 Differentials (“special case”)
(16.5.8): 10.131 Differentials (“affine space over a ring”)
(16.5.10): 10.131 Differentials (“(16.5.10.1) affine case”)
(16.5.12): 27.6 Vector bundles (“more general”)
(16.5.13): 29.33 Sheaf of differentials of a morphism (“more general”), 33.16 Tangent spaces (“(16.5.13.1)”; “(16.5.13.5)”; “partial”; “partial, special case”)
(16.5.14): 10.131 Differentials (“partial”), 91.7 Deformations of ringed spaces and the naive cotangent complex (“more general”), 37.9 Infinitesimal deformations of maps (“more general”)
(16.5.15): 20.4 First cohomology and torsors (“abelian sheaf of groups”), 20.11 Čech cohomology and cohomology (“partial”), 95.14 Classifying torsors (“more general”), 21.4 First cohomology and torsors (“abelian sheaf of groups”), 21.10 Čech cohomology and cohomology (“Abelian”; “partial”; “special case”)
(16.5.17): 91.2 Deformations of rings and the naive cotangent complex (“special case”)
(16.6.2): 17.30 The de Rham complex (“existence”; “uniqueness”)
(16.6.4): 10.132 The de Rham complex (“affine case”)
(16.6.5): 10.131 Differentials (“special case”)
(16.8.1): 10.133 Finite order differential operators (“affine (module) case”; “special case”)
(16.8.2): 10.133 Finite order differential operators (“factorization of D”)
(16.8.3): 10.133 Finite order differential operators (“affine case”), 18.34 Finite order differential operators (“(16.8.3.1)”)
(16.8.9): 10.133 Finite order differential operators (“affine case”), 18.34 Finite order differential operators (“more general”)
(16.8.10): 18.34 Finite order differential operators (“more general”)
(16.8.11): 18.30 Towards constructible modules (“(i), special case”)
(16.9.1): 31.21 Regular ideal sheaves (“quasi-regular”)
(16.9.2): 31.22 Regular immersions (“quasi-regular”)
(16.9.4): 31.21 Regular ideal sheaves, 15.33 Regular ideals (“(i)”; “(ii)”)
(16.9.5): 31.21 Regular ideal sheaves (“(b') => (a)”)
(16.9.7): 15.33 Regular ideals
(16.9.8): 31.22 Regular immersions (“(i)”; “(i), one direction”)
(16.9.9): 31.22 Regular immersions (“l.f.p. is necessary”)
(16.9.10): 31.21 Regular ideal sheaves, 31.22 Regular immersions
(16.9.11): 10.69 Quasi-regular sequences (“local Noetherian ring”), 31.21 Regular ideal sheaves, 31.22 Regular immersions (“more general”)
(16.9.12): 10.69 Quasi-regular sequences (“parenthetical”), 10.106 Regular local rings (“special case”; “sufficiency”), 15.31 Koszul regular sequences (“parenthetical”)
(16.9.13): 31.22 Regular immersions (“more general”)
(16.12.2): 33.25 Schemes smooth over fields (“base field, finite type”)

§17

(17.1.1): 10.148 Formally unramified maps (“special case”), 37.6 Formally unramified morphisms (“partial”), 37.8 Formally étale morphisms, 37.11 Formally smooth morphisms (“étale ⟺ smooth+unram”), 76.13 Formally smooth, étale, unramified transformations (“étale ⟺ smooth+unram”)
(17.1.2): 37.6 Formally unramified morphisms (“special case”)
(17.1.3): 10.138 Formally smooth maps (“(ii), special case”), 10.148 Formally unramified maps (“(iii), special case”), 10.151 Unramified ring maps (“(ii) unram., affine”; “(iii) unram., affine”), 41.14 Topological properties of étale morphisms (“more general”), 37.6 Formally unramified morphisms (“clause (ii)”; “clause (iii)”), 29.37 Étale morphisms (“more general”), 76.13 Formally smooth, étale, unramified transformations (“(i) open immersion”; “(ii) smooth”; “(ii) unramified”; “(ii) étale”; “(iii) smooth”; “(iii) unramified”; “(iii) étale”; “(v), b fmly unramified”), 76.14 Formally unramified morphisms (“more general”), 76.15 Universal first order thickenings (“more general”)
(17.1.4): 76.13 Formally smooth, étale, unramified transformations (“formally smooth case”; “formally étale case”)
(17.1.5): 59.29 Neighborhoods, stalks and points (“special case”), 37.6 Formally unramified morphisms (“partial”), 76.13 Formally smooth, étale, unramified transformations (“necessity, smooth”; “necessity, unramified”; “necessity, étale”; “sufficiency, smooth”; “sufficiency, unramified”; “sufficiency, étale”)
(17.2.1): 10.148 Formally unramified maps (“affine case”; “affine/ring case”), 37.6 Formally unramified morphisms, 76.14 Formally unramified morphisms (“more general”)
(17.2.2): 37.6 Formally unramified morphisms (“case Z = Y”)
(17.2.3): 10.138 Formally smooth maps (“(i), affine case”; “(ii), affine case”), 37.11 Formally smooth morphisms (“(i) locally projective”), 29.35 Smooth morphisms, 76.19 Formally smooth morphisms (“(i) locally projective”)
(17.2.4): 10.150 Formally étale maps (“affine case”)
(17.2.5): 10.131 Differentials (“affine middle exactness”), 10.138 Formally smooth maps (“injectivity, affine”), 10.139 Smoothness and differentials (“affine case”), 90.3 The base category (“affine middle exactness”), 37.11 Formally smooth morphisms (“more general”)
(17.2.6): 41.7 Universally injective, unramified morphisms, 41.14 Topological properties of étale morphisms (“a) ⇒ radicial part of b)”), 76.14 Formally unramified morphisms (“(b) => (a)”)
(17.3.1): 10.138 Formally smooth maps (“smooth case, ring maps”), 10.150 Formally étale maps (“étale case, ring maps”), 41.11 Étale morphisms, 41.16 The functorial characterization, 39.4 Group schemes, 37.6 Formally unramified morphisms (“unramified (net)”), 37.8 Formally étale morphisms, 37.11 Formally smooth morphisms, 29.35 Smooth morphisms, 29.37 Étale morphisms
(17.3.2): 10.138 Formally smooth maps (“(ii), smooth case”), 10.150 Formally étale maps (“(ii), étale case”), 10.151 Unramified ring maps (“(ii) unramified”), 35.32 Properties of morphisms étale local on source-and-target (“Remark 17.3.2(iii)”; “partial”), 41.16 The functorial characterization (“(ii) etale”), 32.6 Limits and morphisms of finite presentation (“more general”), 37.6 Formally unramified morphisms (“(ii) unramified”), 37.8 Formally étale morphisms (“(ii) etale”), 37.11 Formally smooth morphisms (“(ii) smooth”)
(17.3.3): 10.137 Smooth ring maps (“(ii), special case”; “(iii), special case”), 10.151 Unramified ring maps (“(ii) unramified, affine”; “(iii) unramified, affine”), 41.14 Topological properties of étale morphisms, 59.26 Étale morphisms (“(ii), étale case”; “(iii), étale case”), 29.35 Smooth morphisms (“smooth branch of (ii)”; “smooth branch of (iii)”), 29.36 Unramified morphisms (“(i), special case”; “(ii)”; “(ii), special case”), 29.37 Étale morphisms (“(ii), étale case”; “(iii), étale case”), 65.4 Lists of useful properties of morphisms of schemes (“(ii), étale case”; “(iii)”; “(iii), étale case”; “smooth branch of (ii)”; “smooth branch of (iii)”), 67.37 Smooth morphisms (“more general”)
(17.3.5): 59.26 Étale morphisms (“étale clause”), 59.29 Neighborhoods, stalks and points (“étale clause”), 29.37 Étale morphisms (“étale clause”), 65.4 Lists of useful properties of morphisms of schemes (“étale clause”)
(17.3.6): 29.33 Sheaf of differentials of a morphism (“more general”)
(17.3.7): 29.36 Unramified morphisms (“unramified case”)
(17.3.9): 33.34 Projective space (“free E, P^n_S”)
(17.4.1): 10.148 Formally unramified maps (“(c) <=> (e)”), 10.158 Formal smoothness of fields (“field extension case”), 41.4 Three other characterizations of unramified morphisms (“Y loc. Noeth.; a),b),c)”), 37.6 Formally unramified morphisms (“a) <=> e)”), 29.36 Unramified morphisms (“a) => b)”; “a) => d'' finiteness”; “a) => d'' separability”; “b) => a)”; “more general”)
(17.4.1.2): 10.131 Differentials (“special case”), 10.151 Unramified ring maps (“quasi-finite (ring case)”), 37.6 Formally unramified morphisms (“a) formally unramified”), 29.22 Morphisms of finite presentation (“Delta loc. of fin. pres.”; “special case”), 29.33 Sheaf of differentials of a morphism (“special case”), 29.36 Unramified morphisms (“Omega = 0 criterion”; “diagonal => unramified”), 67.28 Morphisms of finite presentation (“more general”)
(17.4.2): 10.148 Formally unramified maps (“Affine case of a ⇔ e”), 10.151 Unramified ring maps (“affine case: a) ⇔ c)”), 10.158 Formal smoothness of fields (“field extension K/k”), 41.4 Three other characterizations of unramified morphisms (“Y loc. Noetherian, at x”), 37.6 Formally unramified morphisms (“a) ⇔ c)”; “a) ⇔ e)”), 29.36 Unramified morphisms (“Implication d′ ⇒ a”; “a) ⇒ b)”; “b) ⇒ a)”), 67.38 Unramified morphisms (“a) ⇒ b)”; “b) ⇒ a)”)
(17.4.3): 10.151 Unramified ring maps (“affine case”), 29.36 Unramified morphisms, 67.38 Unramified morphisms
(17.4.4): 41.3 Unramified morphisms (“f'') k sep closed”; “f'') when k(x)=k(y)”; “f')”; “f) finiteness of B̂/Â”)
(17.4.5): 26.17 Fibre products of schemes
(17.4.7): 29.36 Unramified morphisms (“open nbhd U, f|U=g|U”)
(17.4.9): 41.6 Topological properties of unramified morphisms (“"In particular" clause”; “bijection g↦g(Y)”; “section is comp. iso”)
(17.5.1): 29.35 Smooth morphisms (“a)⇒flatness in b)”; “b)⇒a), global form”)
(17.5.2): 37.21 Regular morphisms (“Necessity”), 29.35 Smooth morphisms (“necessity”), 33.12 Geometrically regular schemes (“base a field, pointwise”), 33.25 Schemes smooth over fields (“field base, smooth⇒reg”)
(17.5.3): 10.139 Smoothness and differentials (“partial”), 15.38 Formally smooth maps of topological rings (“d) ⇔ d')”), 15.39 Formally smooth maps of local rings (“a) ⇔ d)”)
(17.5.4): 37.12 Smoothness over a Noetherian base
(17.5.5): 33.12 Geometrically regular schemes (“base = Spec k”)
(17.5.8): 10.163 Ascending properties (“(iii)”), 35.18 Properties of schemes local in the smooth topology (“(iii)”), 66.7 Properties of Spaces defined by properties of schemes (“(iii), special case”)
(17.6.1): 41.16 The functorial characterization (“a) <=> d)”), 37.8 Formally étale morphisms (“a) <=> d)”), 76.16 Formally étale morphisms (“a) <=> d)”)
(17.6.2): 10.143 Étale ring maps (“a ⇔ c', special case”), 68.4 Finiteness conditions and points (“base = Spec k”), 59.26 Étale morphisms (“base = Spec k”), 29.37 Étale morphisms (“a) => c')”; “c') => a)”; “more general”)
(17.6.3): 41.3 Unramified morphisms (“partial”), 41.11 Étale morphisms (“a) <=> e)”; “e') freeness”)
(17.6.4): 35.21 Local properties of germs, 15.45 Permanence of properties under étale maps (“affine/ring case”)
(17.7.1): 29.35 Smooth morphisms (“(ii) smooth”), 29.37 Étale morphisms (“(ii) étale”)
(17.7.2): 10.137 Smooth ring maps (“(ii), affine case”), 29.35 Smooth morphisms (“(ii)”), 29.36 Unramified morphisms (“more general”), 100.9 Immersions of algebraic stacks (“more general”)
(17.7.3): 10.137 Smooth ring maps (“(ii)”), 10.148 Formally unramified maps (“(i), (ii)”), 35.23 Properties of morphisms local in the fpqc topology on the target (“special case”), 59.26 Étale morphisms (“(ii)”), 37.6 Formally unramified morphisms (“partial”), 29.35 Smooth morphisms (“(ii)”), 29.37 Étale morphisms (“(ii)”), 65.4 Lists of useful properties of morphisms of schemes (“(ii)”)
(17.7.4): 29.35 Smooth morphisms (“(v)”), 29.37 Étale morphisms (“(vi)”), 67.37 Smooth morphisms (“(v), special case”; “more general”)
(17.7.5): 35.14 Descent of finiteness and smoothness properties of morphisms (“(i)”; “(ii)”; “affine schemes”; “special case”)
(17.7.8): 32.8 Descending properties of morphisms (“(ii) smooth”; “(ii) étale”), 70.6 Descending properties of morphisms (“(ii) smooth”; “(ii) étale”)
(17.7.11): 26.17 Fibre products of schemes (“partial”), 26.18 Base change in algebraic geometry (“partial”)
(17.8.1): 29.36 Unramified morphisms (“case Y = S”)
(17.8.2): 29.35 Smooth morphisms (“special case”), 26.18 Base change in algebraic geometry (“special case”)
(17.9.1): 35.25 Application of fpqc descent of properties of morphisms (“c) => a)”), 41.14 Topological properties of étale morphisms (“(a)”; “(a) ⇒ flatness in (b)”; “(a) ⇒ étaleness in (c)”; “(b) ⇒ (a)”), 29.22 Morphisms of finite presentation (“(a)”), 29.27 Flat closed immersions (“(b) ⇒ (a)”), 29.37 Étale morphisms (“(a) ⇒ étaleness in (c)”)
(17.9.2): 41.14 Topological properties of étale morphisms (“partial”), 29.27 Flat closed immersions (“partial”)
(17.9.3): 41.6 Topological properties of unramified morphisms (“more general”), 29.3 Immersions (“partial”), 26.21 Separation axioms (“partial”)
(17.9.4): 41.6 Topological properties of unramified morphisms (“bijection g ↦ g(Y)”; “injectivity of g↦g(Y)”; “iso onto conn. component”)
(17.10.1): 10.125 Dimension of fibres (“affine case”), 107.5 Dimension theory of algebraic stacks
(17.10.2): 29.35 Smooth morphisms (“locally free”; “rank_x = dim_x f”), 107.5 Dimension theory of algebraic stacks (“more general”; “special case”)
(17.10.4): 10.139 Smoothness and differentials (“special case”)
(17.11.3): 26.15 A representability criterion (“base Spec ℤ”)
(17.11.4): 41.13 Étale and smooth morphisms (“smooth ⇒ étale/A^d”), 29.37 Étale morphisms (“smooth ⇒ étale/A^d”)
(17.11.5): 41.14 Topological properties of étale morphisms (“necessity”)
(17.12.1): 31.20 The normal cone of an immersion, 31.21 Regular ideal sheaves (“more general”), 31.22 Regular immersions
(17.12.3): 31.23 Relative regular immersions (“necessity”)
(17.12.6): 32.8 Descending properties of morphisms (“f smooth ⇒ ∃λ”), 70.6 Descending properties of morphisms (“f smooth ⇒ ∃λ”)
(17.12.8): 110.49 A non-flat group scheme with flat identity component (“non-flat example”)
(17.13.1): 10.131 Differentials (“special case”), 90.3 The base category (“special case”), 37.7 Universal first order thickenings (“more general”), 29.2 Closed immersions (“closed immersions”), 29.32 Conormal sheaf of an immersion (“special case”), 26.17 Fibre products of schemes (“closed immersions”), 26.18 Base change in algebraic geometry (“closed immersions”), 65.4 Lists of useful properties of morphisms of schemes (“closed immersions”)
(17.13.5): 10.131 Differentials (“Affine case”), 29.33 Sheaf of differentials of a morphism
(17.13.9): 83.5 Quotients as orbit spaces (“binary diagonal”)
(17.14.2): 10.138 Formally smooth maps (“partial”)
(17.15.1): 10.140 Smooth algebras over fields (“last assertion”), 33.43 Curves (“the smoothness item”)
(17.15.2): 33.12 Geometrically regular schemes (“more general”), 33.25 Schemes smooth over fields (“k perfect case”; “necessity”; “smooth ⇒ reduced”; “smooth ⇒ regular”), 33.43 Curves (“the smoothness item”)
(17.15.5): 10.140 Smooth algebras over fields (“affine case”; “affine, char 0”; “affine, k alg closed”), 29.35 Smooth morphisms (“a) => c) (free)”; “a) => c) rank”), 33.25 Schemes smooth over fields (“special case”)
(17.15.6): 33.25 Schemes smooth over fields (“characteristic zero”; “perfect k, reduced X”)
(17.15.7): 10.140 Smooth algebras over fields (“affine case”), 33.25 Schemes smooth over fields (“free ⇒ smooth”)
(17.15.8): 10.18 Local rings, 10.60 Dimension, 10.114 Dimension of finite type algebras over fields, 10.116 Dimension of finite type algebras over fields, reprise (“special case”), 10.131 Differentials, 29.29 Morphisms and dimensions of fibres (“more general”), 33.20 Algebraic schemes
(17.15.9): 10.134 The naive cotangent complex, 10.140 Smooth algebras over fields (“a)”; “k alg closed, closed pt”), 91.2 Deformations of rings and the naive cotangent complex, 90.3 The base category
(17.15.11): 10.143 Étale ring maps (“a ⇔ c, affine case”), 10.158 Formal smoothness of fields (“X = Spec of a field”), 41.8 Examples of unramified morphisms (“a) <=> b)”; “affineness in c)”; “connected X, b) => c)”), 29.36 Unramified morphisms (“b) <=> c)”)
(17.15.12): 33.25 Schemes smooth over fields (“X reduced, 1st equiv”; “special case”)
(17.15.13): 33.25 Schemes smooth over fields (“special case”)
(17.15.14): 33.43 Curves (“Proper case of 17.15.14”; “special case”)
(17.15.14.1): 29.55 Normalization (“(i), special case”)
(17.15.14.2): 33.6 Geometrically reduced schemes (“geom. reduced reduction”)
(17.15.15): 29.35 Smooth morphisms (“necessity: flatness”; “necessity: Ω loc free”; “smooth ⇒ rank=dim_x f”)
(17.16.1): 37.23 Slicing Cohen-Macaulay morphisms
(17.16.2): 10.168 Colimits and maps of finite presentation, II (“affine case”)
(17.16.3): 37.38 Slicing smooth morphisms (“Clause (i)”)

§18

(18.1.1): 10.143 Étale ring maps (“special case”), 59.48 Big sites and pushforward (“smooth case”; “étale case”)
(18.1.2): 41.15 Topological invariance of the étale topology (“fully faithful”), 59.45 Topological invariance of the small étale site (“more general”), 15.11 Henselian pairs (“affine case”), 58.8 Topological invariance of the fundamental group (“special case”)
(18.1.3): 41.15 Topological invariance of the étale topology, 59.45 Topological invariance of the small étale site (“more general”)
(18.2.1): 9.20 Trace and norm
(18.2.3): 10.143 Étale ring maps (“a') ⇒ a)”), 29.37 Étale morphisms (“a') ⇒ a)”)
(18.2.7): 10.78 Finite projective modules (“partial”), 41.10 Topological properties of flat morphisms (“partial”), 59.26 Étale morphisms (“partial”), 29.35 Smooth morphisms (“partial”), 29.37 Étale morphisms (“partial”)
(18.2.8): 37.76 More on weightings (“semicontinuity of n(y)”)
(18.3.2): 10.153 Henselian local rings (“local, I = maximal ideal”), 15.13 Lifting and henselian pairs
(18.3.2.1): 10.5 Finite modules and finitely presented modules, 10.14 Base change, 11.2 Noncommutative algebras, 9.7 Finite extensions, 15.13 Lifting and henselian pairs (“(ii)”)
(18.3.3): 10.143 Étale ring maps, 15.107 Weakly étale algebras over fields (“partial”), 58.5 Finite étale morphisms (“special case”)
(18.4.3): 10.144 Local structure of étale ring maps (“(F)”)
(18.5.2): 27.6 Vector bundles
(18.5.8): 59.32 Henselian rings (“special case”), 15.11 Henselian pairs
(18.5.9): 10.153 Henselian local rings (“(ii)”; “(ii) necessity”; “more general”), 10.160 The Cohen structure theorem (“(ii), special case”), 59.32 Henselian rings (“(ii)”; “(ii) necessity”)
(18.5.10): 10.153 Henselian local rings (“R henselian local”), 59.32 Henselian rings (“R henselian local”)
(18.5.11): 10.153 Henselian local rings (“a), a'), b)”), 10.155 Henselization and strict henselization (“special case”), 59.32 Henselian rings (“(F), partial”; “a), a'), b)”)
(18.5.12): 10.155 Henselization and strict henselization (“special case”)
(18.5.13): 10.153 Henselian local rings (“a'') existence half”), 59.32 Henselian rings (“a'') existence half”)
(18.5.14): 10.153 Henselian local rings (“local case”), 59.32 Henselian rings (“more general”), 15.11 Henselian pairs (“more general”)
(18.5.15): 10.153 Henselian local rings (“henselian local ring”), 15.13 Lifting and henselian pairs, 58.8 Topological invariance of the fundamental group (“more general”)
(18.5.16): 10.153 Henselian local rings (“special case”), 10.160 The Cohen structure theorem (“special case”), 59.32 Henselian rings (“more general”), 15.10 Zariski pairs (“partial”), 15.11 Henselian pairs (“more general”)
(18.5.17): 10.155 Henselization and strict henselization (“affine étale morphisms”)
(18.6.1): 10.18 Local rings, 10.151 Unramified ring maps (“more general”)
(18.6.2): 10.155 Henselization and strict henselization
(18.6.3): 59.33 Stalks of the structure sheaf (“more general”), 37.35 Étale neighbourhoods (“more general”)
(18.6.6): 10.155 Henselization and strict henselization (“(i), part of (iii)”), 59.32 Henselian rings (“(ii)”), 15.46 Permanence of properties under henselization (“(iv), (v)”), 16.13 The approximation property for G-rings (“(v)”)
(18.6.7): 59.32 Henselian rings (“special case”), 93.15 Application to isolated singularities (“more general”)
(18.6.9): 15.46 Permanence of properties under henselization (“(i), special case”; “more general”), 33.39 The delta invariant (“(i), special case”)
(18.6.10): 15.46 Permanence of properties under henselization (“a)”)
(18.6.12): 15.108 Local irreducibility (“a) ⇔ c), irreducible”)
(18.6.14): 10.154 Filtered colimits of étale ring maps, 15.51 G-rings (“(iii)”)
(18.7.2): 15.52 Properties of formal fibres (“Forward implication”; “partial”)
(18.7.3): 33.39 The delta invariant (“Forward implication”)
(18.8.1): 10.155 Henselization and strict henselization (“(a)⇒(c), special case”), 59.29 Neighborhoods, stalks and points (“(a)⇒(c), special case”)
(18.8.4): 10.155 Henselization and strict henselization (“partial”; “special case”)
(18.8.6): 10.159 Constructing flat ring maps (“(18.8.6.2)”), 37.35 Étale neighbourhoods (“more general”)
(18.8.7): 10.155 Henselization and strict henselization
(18.8.8): 10.155 Henselization and strict henselization (“(i), part of (iii)”; “(ii)”; “more general”), 15.46 Permanence of properties under henselization (“(iv)”)
(18.8.8.1): 10.155 Henselization and strict henselization (“existence of A^hs->B”; “induces \psi on residues”; “locality of A^hs->B”; “uniqueness of A^hs->B”)
(18.8.9): 10.155 Henselization and strict henselization (“special case”)
(18.8.12): 15.46 Permanence of properties under henselization (“(i), special case”; “(ii), special case”), 33.39 The delta invariant (“(i)”)
(18.8.13): 15.46 Permanence of properties under henselization (“a)”)
(18.8.15): 15.108 Local irreducibility (“a) ⇔ c), irreducible”)
(18.8.18): 10.154 Filtered colimits of étale ring maps (“i)”)
(18.9.7): 37.28 Connected components of fibres (“special case”)
(18.9.7.8): 10.119 Around Krull-Akizuki (“more general”), 10.120 Factorization (“more general”), 33.41 Normalization of one dimensional schemes (“more general”)
(18.10.1): 15.109 Miscellaneous on branches (“(i)+(ii) ⇒ étale”), 37.37 Unramified and étale morphisms (“suff.; generic pt in”)
(18.10.2): 37.74 Universally open morphisms (“special case”)
(18.10.3): 37.37 Unramified and étale morphisms (“X irreducible”; “f étale”)
(18.10.5): 41.14 Topological properties of étale morphisms (“necessity”), 37.20 Normal morphisms (“more general”)
(18.10.8): 10.36 Finite and integral ring extensions (“special case”), 29.54 Relative normalization (“partial”)
(18.10.10): 10.36 Finite and integral ring extensions (“affine case”), 29.54 Relative normalization (“more general”; “partial”)
(18.10.15): 10.161 Japanese rings (“partial”)
(18.10.17.1): 37.74 Universally open morphisms (“globally univ. open”)
(18.10.19): 29.50 Rational maps
(18.10.20): 32.15 Noetherian valuative criterion (“partial”), 32.16 Refined Noetherian valuative criteria (“partial”)
(18.11.6): 33.21 Complete local rings (“counterexample”)
(18.11.12): 10.9 Localization, 9.3 Examples of fields
(18.12.1): 59.32 Henselian rings (“affine case”), 37.41 Étale localization of quasi-finite morphisms (“more general”; “separated case”; “separated, finite fibre”)
(18.12.3): 37.41 Étale localization of quasi-finite morphisms (“more general”; “partial”)
(18.12.4): 37.44 Applications of Zariski's Main Theorem, I, 76.35 Applications of Zariski's Main Theorem, I (“finite⇒proper+discrete”; “proper q-finite⇒finite”)
(18.12.6): 41.7 Universally injective, unramified morphisms (“b) <=> c)”), 29.45 Integral and finite morphisms (“(a)⇔(b), special case”)
(18.12.8): 29.45 Integral and finite morphisms, 67.45 Integral and finite morphisms (“more general”)
(18.12.9): 29.45 Integral and finite morphisms (“partial”)
(18.12.11): 29.46 Universal homeomorphisms (“integral case”)
(18.12.12): 37.43 Zariski's Main Theorem, 76.34 Zariski's Main Theorem (“more general”)
(18.12.13): 10.123 Zariski's Main Theorem (“affine case”), 63.10 Properties of derived lower shriek (“special case”), 63.11 Derived upper shriek (“special case”), 37.43 Zariski's Main Theorem (“special case”), 29.3 Immersions (“more general”), 29.57 Zariski's Main Theorem (algebraic version) (“affine case”), 26.21 Separation axioms (“more general”)
(18.12.14): 10.36 Finite and integral ring extensions (“partial”), 29.54 Relative normalization (“partial”)
(18.12.15): 10.147 Integral closure and smooth base change (“more general”), 37.19 Normalization revisited
(18.12.17): 27.8 Proj of a graded ring, 26.21 Separation axioms (“u is separated”), 26.23 Monomorphisms (“special case”)

§19

(19.1.2): 10.106 Regular local rings (“principal J, sufficiency”; “sufficiency direction”)
(19.1.3): 31.22 Regular immersions, 15.33 Regular ideals (“special case”)
(19.1.5): 35.23 Properties of morphisms local in the fpqc topology on the target (“(ii), special case”; “more general”), 31.22 Regular immersions (“(ii)”; “(iv)”; “more general”; “partial”)
(19.1.6): 23.7 Application to complete intersections (“more general”)
(19.1.7): 17.5 Supports of modules and sections (“closed support of O_X/J”)
(19.2.4): 31.23 Relative regular immersions (“b) => c)”; “c') => b')”; “c') => c)”)
(19.2.9): 37.22 Cohen-Macaulay morphisms (“more general”), 37.23 Slicing Cohen-Macaulay morphisms (“x closed in fibre”)
(19.3.1): 23.8 Local complete intersection rings (“local ring definition”), 15.33 Regular ideals
(19.3.2): 10.135 Local complete intersections (“ess. f.t. over a field”; “special case”), 23.8 Local complete intersection rings (“reg seq ⇒ CI”; “special case”)
(19.3.2.1): 15.40 Some results on power series rings (“(iii)”)
(19.3.4): 10.135 Local complete intersections (“affine case”), 29.31 Syntomic morphisms
(19.3.6): 10.135 Local complete intersections, 10.136 Syntomic morphisms (“affine case”)
(19.4.1): 10.70 Blow up algebras (“partial”), 42.29 Gysin homomorphisms (“more general”), 31.33 Blowing up (“(1)”; “X'=Proj(⊕J^n)”; “Y'=f^{-1}(Y)”; “structural morphism f”), 26.17 Fibre products of schemes (“more general”)
(19.4.6): 37.5 Infinitesimal neighbourhoods
(19.4.11): 54.3 Quadratic transformations (“(iii), special case”)
(19.5.1): 10.68 Regular sequences, 10.69 Quasi-regular sequences (“a) ⇒ c)”; “a) ⇒ c) for M = A”; “special case”), 31.21 Regular ideal sheaves (“a) => b)”; “b) => b')”), 15.31 Koszul regular sequences (“a)–c), local case”)
(19.5.4): 10.69 Quasi-regular sequences (“more general”)
(19.5.5.2): 10.69 Quasi-regular sequences (“more general”)
(19.6.1): 12.19 Filtrations (“more general”; “rows of (19.6.1.1)”)
(19.6.2): 10.69 Quasi-regular sequences (“special case”)
(19.7.1): 10.69 Quasi-regular sequences (“special case”)
(19.7.4): 17.14 Locally free sheaves (“partial”), 28.21 Locally free modules (“more general”)
(19.7.5): 31.22 Regular immersions (“more general”)
(19.9.6): 37.23 Slicing Cohen-Macaulay morphisms (“n = 1, F = O_X”)

§20

(20.1.3): 31.24 Meromorphic functions and sections (“M(X)”; “M(X,F)”; “M_X(F)”), 111.49 Divisors
(20.1.3.1): 10.25 Zerodivisors and total rings of fractions (“field-stalk clause”; “special case”)
(20.1.5): 31.11 Torsion free modules (“quasi-coherent case”), 15.22 Torsion free modules (“special case”)
(20.1.8): 31.24 Meromorphic functions and sections (“regular section defn”), 31.28 The Weil divisor class associated to an invertible module (“partial”), 71.10 Meromorphic functions and sections (“regular section defn”)
(20.1.10): 31.28 The Weil divisor class associated to an invertible module (“integral; tensor clause”)
(20.1.11): 31.24 Meromorphic functions and sections (“M_f = M_X condition”; “mero section pullback”)
(20.1.13): 71.10 Meromorphic functions and sections (“Algebraic clause only”)
(20.2.1): 29.50 Rational maps
(20.2.3): 29.50 Rational maps (“special case”), 67.47 Rational maps (“defined at a point”; “dom(ω)”)
(20.2.4): 29.50 Rational maps (“special case”)
(20.2.5): 29.50 Rational maps (“special case”), 26.21 Separation axioms (“special case”)
(20.2.7): 29.50 Rational maps
(20.2.8): 39.5 Examples of group schemes (“Morphisms V → Spec Z[T”), 29.50 Rational maps (“special case”)
(20.2.9): 10.17 The spectrum of a ring (“special case”)
(20.2.11): 29.50 Rational maps (“(i)”; “(ii) reduced case”)
(20.2.13): 10.63 Associated primes (“S”), 31.26 Meromorphic functions and sections; reduced case (“(ii) global sections”)
(20.2.14): 31.24 Meromorphic functions and sections (“definition”; “special case”)
(20.2.16): 26.14 Glueing schemes (“more general”)
(20.3.2): 29.50 Rational maps (“special case”)
(20.3.6): 28.5 Noetherian schemes (“partial”)
(20.3.7): 29.50 Rational maps (“bijectivity (fin pres)”; “injectivity”)
(20.3.13): 71.10 Meromorphic functions and sections (“flat: pullback defined”)
(20.4.3): 29.50 Rational maps (“Identification”)
(20.4.12): 28.7 Normal schemes (“partial”)
(20.5.7): 37.16 Critère de platitude par fibres (“more general”; “partial”), 67.17 Scheme theoretic closure and density (“special case”)
(20.6.9): 10.39 Flat modules and flat ring maps (“partial”), 42.15 Push and pull (“flatness of X' over S'”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“flatness of X' over S'”; “partial”), 29.22 Morphisms of finite presentation (“partial”), 29.26 Flat morphisms (“flatness of X' over S'”), 65.4 Lists of useful properties of morphisms of schemes (“flatness of X' over S'”; “partial”), 34.7 The fppf topology (“special case”)

§21

(21.1.3): 31.27 Weil divisors (“(21.1.3.1)”)
(21.2.7): 38.41 Blowing up perfect modules (“partial”)
(21.2.8): 31.15 Effective Cartier divisors and invertible sheaves (“(21.2.8.3), special case”)
(21.2.12): 31.14 Effective Cartier divisors (“partial”), 31.16 Effective Cartier divisors on Noetherian schemes (“special case”), 26.10 Immersions of schemes (“more general”)
(21.3.2): 17.25 Invertible modules (“Pic(X) comm. group”; “Pic(X) definition”; “cl(L)cl(L')=cl(L⊗L')”; “partial”), 18.32 Invertible modules (“Pic(X) definition”; “cl(L)”; “cl(L)=cl(L') iff iso”)
(21.3.3): 31.15 Effective Cartier divisors and invertible sheaves (“(ii)”)
(21.3.6): 20.6 First cohomology and invertible sheaves (“special case”)
(21.4.2): 31.14 Effective Cartier divisors (“effective Cartier div.”)
(21.4.5): 31.14 Effective Cartier divisors (“(i), eff. Cartier div”; “special case”), 31.24 Meromorphic functions and sections (“(i)”; “(ii), to generic points”; “(iii), X,X' integral”)
(21.5.3): 10.39 Flat modules and flat ring maps (“partial”), 10.78 Finite projective modules (“special case”), 17.14 Locally free sheaves (“partial”), 15.22 Torsion free modules (“partial”), 29.45 Integral and finite morphisms (“partial”)
(21.5.6): 31.14 Effective Cartier divisors (“(D)”)
(21.6.1): 42.8 Cycles, 82.3 Cycles (“fixed δ-dimension k”)
(21.6.2): 42.8 Cycles (“(X)”), 28.10 Dimension (“partial”)
(21.6.5): 31.14 Effective Cartier divisors (“partial”), 26.10 Immersions of schemes
(21.6.7): 42.17 Principal divisors (“cyc(div(f))”), 31.25 Meromorphic functions and sections; Noetherian case (“partial”), 31.27 Weil divisors (“partial”), 31.28 The Weil divisor class associated to an invertible module (“o_x(f) = mult_x(div f)”)
(21.6.9): 31.16 Effective Cartier divisors on Noetherian schemes (“(ii) c) ⇒ b)”)
(21.6.10): 31.28 The Weil divisor class associated to an invertible module (“(i), injectivity, (ii)”; “(i), integral case”)
(21.6.11): 55.9 The geometry of a regular model (“special case”)
(21.7.3): 31.16 Effective Cartier divisors on Noetherian schemes (“normal ⇒ (S_1)”)
(21.7.4): 31.16 Effective Cartier divisors on Noetherian schemes (“c) => a'')”)
(21.7.7): 10.120 Factorization
(21.9.1): 6.27 Skyscraper sheaves and stalks (“stalk off closure”; “stalk on closure”), 6.32 Closed immersions and (pre)sheaves (“special case”), 7.32 Points (“more general”)
(21.9.10): 10.119 Around Krull-Akizuki (“Domain case”)
(21.10.1): 10.114 Dimension of finite type algebras over fields (“partial”), 20.24 Alternative view of the Čech complex (“partial”), 31.14 Effective Cartier divisors (“partial”; “special case”), 5.28 Partitions and stratifications (“partial”)
(21.10.6): 31.14 Effective Cartier divisors (“(D), special case”)
(21.10.11): 10.52 Length (“partial”)
(21.10.13): 15.127 Principal radical ideals (“(i) for M=A”)
(21.10.15): 29.45 Integral and finite morphisms (“Condition a)”), 65.4 Lists of useful properties of morphisms of schemes (“Condition a)”)
(21.10.16): 29.45 Integral and finite morphisms (“f₁ is finite”), 65.4 Lists of useful properties of morphisms of schemes (“f₁ is finite”)
(21.10.19): 82.19 Intersecting with an invertible sheaf and push and pull (“special case”)
(21.12.1): 27.3 Relative spectrum via glueing (“more general”), 27.4 Relative spectrum as a functor, 67.20 Affine morphisms (“more general”)
(21.12.7): 31.17 Complements of affine opens (“codimension = 1 if dense”; “codimension ≤ 1”; “dim = 1, U affine open”; “dim ≤ 1, U affine open”)
(21.12.9): 31.16 Effective Cartier divisors on Noetherian schemes (“special case”), 31.17 Complements of affine opens (“special case”), 33.38 One dimensional Noetherian schemes (“more general”)
(21.12.14): 58.21 Purity of branch locus (“(v)”; “(v), (v), special case”), 58.28 Purity of ramification locus (“more general”), 58.29 Affineness of complement of ramification locus (“special case”)
(21.13.3): 6.24 Continuous maps and sheaves of modules, 18.21 Localization of ringed topoi
(21.13.9): 10.37 Normal rings (“(iv)”), 76.11 Picard groups of thickenings (“The exact segment H¹(U”)
(21.13.12.1): 10.59 Noetherian local rings
(21.13.13): 20.6 First cohomology and invertible sheaves (“special case”)
(21.13.15): 5.19 Specialization
(21.14.4): 10.165 Geometrically normal algebras (“special case”), 33.10 Geometrically normal schemes (“more general”)
(21.15.3.3): 31.19 Relative effective Cartier divisors (“converse”; “forward direction”), 26.10 Immersions of schemes (“more general”)
(21.15.9): 10.39 Flat modules and flat ring maps (“partial”), 42.15 Push and pull (“partial”), 31.19 Relative effective Cartier divisors (“partial”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“partial”), 29.22 Morphisms of finite presentation (“partial”), 29.26 Flat morphisms (“partial”), 65.4 Lists of useful properties of morphisms of schemes (“partial”)

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EGA IV

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