SGA 1

The items of SGA 1 that a declaration of an article carries, numbered as printed, each with the articles it is in and the attribute's comments where there are any.

Exposé I

• (2.1): 10.5 Finite modules and finitely presented modules, 10.18 Local rings, 10.59 Noetherian local rings, 10.97 Completion for Noetherian rings (“(i) => (iii)”), 11.2 Noncommutative algebras, 9.7 Finite extensions
• (3.1): 10.143 Étale ring maps (“(i)”), 10.158 Formal smoothness of fields (“(i)<=>(ii), field ext”), 29.36 Unramified morphisms (“(ii) => (iii)”; “(ii)=>(i) finiteness”; “(ii)=>(i) separability”; “(iii) => (ii)”; “(iii)=>(ii), fin pres”)
• (3.3): 29.36 Unramified morphisms (“loc. finite presentation”)
• (3.4): 29.36 Unramified morphisms (“diagonal case (g = id)”), 67.38 Unramified morphisms (“diagonal case (g = id)”)
• (3.5): 10.151 Unramified ring maps (“(ii), affine case”; “(iii), affine case”), 29.36 Unramified morphisms (“(iii)”; “more general”), 67.38 Unramified morphisms (“(iii)”)
• (3.6): 29.36 Unramified morphisms (“(v)”)
• (3.7): 41.3 Unramified morphisms (“unram ⇒ surj, κ sep cl”; “unram ⇒ surj, κ(A)=κ(B)”)
• (4.1): 41.11 Étale morphisms (“loc. Noetherian base”; “special case”), 29.37 Étale morphisms
• (4.2): 41.11 Étale morphisms (“special case”)
• (4.6): 41.14 Topological properties of étale morphisms, 59.26 Étale morphisms, 29.37 Étale morphisms, 65.4 Lists of useful properties of morphisms of schemes
• (4.10): 9.20 Trace and norm (“special case”), 29.49 Finite locally free morphisms (“more general”)
• (5.1): 35.25 Application of fpqc descent of properties of morphisms (“sufficiency”), 41.14 Topological properties of étale morphisms (“more general”; “necessity”; “special case”), 29.27 Flat closed immersions (“special case”), 29.37 Étale morphisms (“necessity”)
• (5.3): 41.6 Topological properties of unramified morphisms (“bijection with comps”; “determined by one point”; “section iso onto comp”)
• (5.4): 41.6 Topological properties of unramified morphisms
• (5.5): 41.15 Topological invariance of the étale topology (“special case”)
• (5.6): 37.6 Formally unramified morphisms (“special case”)
• (5.8): 29.36 Unramified morphisms (“Net clause, special case”)
• (5.9): 10.99 Criteria for flatness (“partial”), 10.128 More flatness criteria (“partial”), 37.16 Critère de platitude par fibres (“partial”)
• (6.2): 10.155 Henselization and strict henselization (“special case”)
• (7.2): 49.7 The Kähler different (“more general”), 49.12 A formula for the different (“equality case”)
• (7.3): 10.144 Local structure of étale ring maps (“partial”)
• (7.4): 10.144 Local structure of étale ring maps (“partial”)
• (7.5): 10.18 Local rings, 11.2 Noncommutative algebras (“more general”), 9.7 Finite extensions (“more general”), 26.2 Locally ringed spaces (“more general”)
• (7.7): 41.12 The structure theorem (“unramified ⇒ quotient”)
• (9.1): 10.110 Regular rings and global dimension (“B regular ⇒ A regular”), 10.112 Homomorphisms and dimension (“A regular ⇒ B regular”), 41.19 Permanence properties
• (9.2): 35.19 Variants on descending properties (“converse (X reg ⇒ Y reg)”), 74.9 Descending properties of spaces (“converse (X reg ⇒ Y reg)”)
• (9.3): 41.19 Permanence properties (“Noetherian case”)
• (9.4): 10.162 Nagata rings
• (9.5): 41.19 Permanence properties (“(i), Noetherian”), 15.109 Miscellaneous on branches (“(ii)”)
• (9.11): 37.37 Unramified and étale morphisms (“X irreducible”; “f étale”), 58.11 Fundamental groups of normal schemes (“special case”)
• (10.1): 10.143 Étale ring maps (“special case”), 58.11 Fundamental groups of normal schemes (“finite étale case”; “integrality, finite case”)
• (10.2): 58.11 Fundamental groups of normal schemes (“finite => normalization”)
• (10.3): 29.50 Rational maps (“more general”; “special case”)
• (10.4): 15.113 Extensions of discrete valuation rings (“(ii) transitivity”)
• (10.7): 37.74 Universally open morphisms (“more general”)
• (10.11): 26.12 Reduced schemes

Exposé II

• (1.1): 10.137 Smooth ring maps, 39.4 Group schemes, 29.35 Smooth morphisms
• (1.3): 10.137 Smooth ring maps (“(ii), affine case”; “(iii), special case”), 10.143 Étale ring maps (“clause (i), special case”), 35.32 Properties of morphisms étale local on source-and-target (“special case”), 59.26 Étale morphisms (“clause (i), special case”; “more general”), 29.35 Smooth morphisms (“more general”), 29.54 Relative normalization (“more general”), 65.4 Lists of useful properties of morphisms of schemes (“more general”), 67.37 Smooth morphisms (“(ii)”)
• (1.5): 29.35 Smooth morphisms (“special case”)
• (2.1): 10.137 Smooth ring maps (“(a) + (b) ⇒ smooth”), 29.35 Smooth morphisms (“necessity of (a)”; “sufficiency, global form”)
• (2.2): 29.35 Smooth morphisms (“sufficiency with Y=S”)
• (2.4): 26.12 Reduced schemes, 26.18 Base change in algebraic geometry
• (3.1): 10.163 Ascending properties (“normal, O_y ⇒ O_x”; “reduced, O_y ⇒ O_x”; “regular, O_y ⇒ O_x”), 35.18 Properties of schemes local in the smooth topology (“special case”), 74.9 Descending properties of spaces (“reduced, Y ⇒ X”), 33.25 Schemes smooth over fields (“base a field, normal”; “base a field, reduced”; “base a field, regular”)
• (4.1): 29.36 Unramified morphisms (“S=Y, unramified clause”)
• (4.2): 10.131 Differentials (“exact at Ω, affine”; “exact at Ω¹_{C/A}”; “second assertion”)
• (4.3): 10.131 Differentials (“(i)”), 10.138 Formally smooth maps (“(i) inj., affine case”), 29.35 Smooth morphisms (“(ii) local freeness”; “(ii) rank formula”)
• (4.4): 10.138 Formally smooth maps (“special case”)
• (4.5): 10.82 Universally injective module maps
• (4.8): 10.112 Homomorphisms and dimension (“more general”), 26.15 A representability criterion (“base Spec Z”)
• (4.10): 10.139 Smoothness and differentials (“(iv)”)
• (4.12): 37.38 Slicing smooth morphisms (“dF(x)≠0 ⇒ Y smooth”)
• (4.13): 10.137 Smooth ring maps (“affine ascent”; “affine case”), 29.35 Smooth morphisms (“global ascent”; “special case”), 26.17 Fibre products of schemes, 65.4 Lists of useful properties of morphisms of schemes (“global ascent”)
• (4.14): 10.68 Regular sequences, 15.33 Regular ideals, 15.40 Some results on power series rings (“special case”), 33.25 Schemes smooth over fields (“special case”)
• (4.16): 31.23 Relative regular immersions (“X smooth ⇒ i regular”)
• (4.17): 10.139 Smoothness and differentials (“affine case, (i)⇒(iii)”), 29.3 Immersions (“more general”), 26.21 Separation axioms (“more general”)
• (4.18): 43.13 Proper intersections (“variety over perfect fld”), 37.62 Local complete intersection morphisms (“(1) ⇒ (2), special case”)
• (4.19): 37.22 Cohen-Macaulay morphisms
• (5.1): 10.151 Unramified ring maps (“(i) ⟺ (iii)”)
• (5.3): 10.140 Smooth algebras over fields (“alg. closed k, max ideal”; “first assertion”; “regular ⇒ smooth”)
• (5.4): 10.140 Smooth algebras over fields (“affine local, κ(q)/k sep”; “char 0, affine local”), 33.12 Geometrically regular schemes (“more general”), 33.25 Schemes smooth over fields (“more general”; “perfect k”), 33.43 Curves (“k perfect”; “κ(x)/k separable”)
• (5.5): 10.140 Smooth algebras over fields (“affine (i)<=>(ii bis)”; “alg closed k, closed pt”; “char 0, affine, k'=k”), 33.25 Schemes smooth over fields (“(ii)=>(i), X reduced”)
• (5.6): 15.39 Formally smooth maps of local rings (“(iii)”)
• (5.7): 10.131 Differentials, 10.140 Smooth algebras over fields (“char 0, local at prime”), 10.158 Formal smoothness of fields (“partial”), 33.25 Schemes smooth over fields (“k perfect, X reduced”)
• (5.8): 29.29 Morphisms and dimensions of fibres (“n = p + trdeg_k κ(x)”), 33.20 Algebraic schemes (“partial”)
• (5.9): 10.114 Dimension of finite type algebras over fields, 10.131 Differentials (“partial”), 10.134 The naive cotangent complex (“more general”), 10.140 Smooth algebras over fields (“(i) <=> smooth in (ii)”; “characteristic zero”; “k alg closed, closed pt”), 91.2 Deformations of rings and the naive cotangent complex (“more general”), 90.3 The base category, 5.11 Codimension and catenary spaces, 33.20 Algebraic schemes (“n = trdeg_k κ(x) + p”)
• (5.10): 10.140 Smooth algebras over fields (“affine case”)

Exposé III

• (1.2): 41.11 Étale morphisms (“partial”), 9.6 Field extensions
• (1.3): 15.41 Geometric regularity and formal smoothness (“more general”), 15.50 Formal smoothness and regularity (“special case”)
• (1.4): 10.9 Localization, 10.31 Noetherian rings (“partial”; “special case”), 10.36 Finite and integral ring extensions (“partial”), 10.160 The Cohen structure theorem (“special case”)
• (1.5): 10.139 Smoothness and differentials (“necessity, special case”), 90.8 Smooth morphisms (“complete local case”)
• (1.7): 10.18 Local rings, 10.112 Homomorphisms and dimension, 15.41 Geometric regularity and formal smoothness (“Noetherian local rings”; “if direction”; “only if, flatness”), 15.50 Formal smoothness and regularity (“Noetherian complete case”)
• (1.9): 15.39 Formally smooth maps of local rings (“Noetherian A”)
• (2.1): 15.39 Formally smooth maps of local rings (“(i) ⇔ (iv bis)”)
• (2.2): 90.3 The base category, 15.39 Formally smooth maps of local rings (“(iv ter)”)
• (2.3): 15.40 Some results on power series rings (“special case”)
• (3.1): 37.12 Smoothness over a Noetherian base (“(i)<=>(iii), Noetherian”), 76.20 Smoothness over a Noetherian base (“(iii), (iii)⇒(i)”)
• (4.2): 10.126 Algebras and modules of finite presentation (“affine case + finiteness”)
• (5.2): 10.131 Differentials (“special case”), 10.138 Formally smooth maps (“special case”), 10.140 Smooth algebras over fields (“special case”)
• (5.3): 91.2 Deformations of rings and the naive cotangent complex (“special case”)
• (6.1): 93.8 Rings (“special case”), 93.9 Schemes (“special case”)
• (7.1): 31.14 Effective Cartier divisors (“partial”), 26.10 Immersions of schemes (“partial”)

Exposé IV

• (1.1): 10.39 Flat modules and flat ring maps (“(i)”; “(ii)”), 17.17 Flat modules (“(ii) M flat => M' flat”; “(ii) M' flat => M flat”), 18.28 Flat modules (“(i)”; “more general”)
• (1.2): 10.39 Flat modules and flat ring maps (“(i) 2nd assertion”; “(ii)”; “(ii) with all primes”; “B = A”; “B = A, primes”; “M = B = A”)
• (1.3): 15.22 Torsion free modules (“more general”)
• (2.1): 10.39 Flat modules and flat ring maps (“(i bis) <=> (i ter)”), 35.4 Descent for universally injective morphisms (“special case”), 41.9 Flat morphisms
• (2.2): 10.39 Flat modules and flat ring maps (“case M = B”), 67.30 Flat morphisms (“special case”)
• (2.3): 10.39 Flat modules and flat ring maps (“M = B”)
• (2.4): 10.41 Going up and going down (“more general”)
• (2.6): 10.14 Base change, 10.39 Flat modules and flat ring maps (“(i) ⇔ (ii) ⇔ (ii bis)”; “(i), special case”; “(i)⇒(iv)”), 10.82 Universally injective module maps (“(i)”), 67.30 Flat morphisms (“(i), special case”)
• (3.1): 10.97 Completion for Noetherian rings (“flatness, I ⊆ rad”), 38.8 Completion and Mittag-Leffler modules, 15.28 Completion and flatness (“more general”)
• (3.2): 10.97 Completion for Noetherian rings (“if direction”)
• (4.1): 10.106 Regular local rings (“(b), special case”)
• (4.3): 10.12 Tensor products, 10.39 Flat modules and flat ring maps, 10.83 Descent for finite projective modules (“(ii) ⇒ (iii)”; “case (b): (iii) ⇒ (ii)”), 10.99 Criteria for flatness (“case (b): (iv) ⇒ (iii)”), 10.101 Flatness criteria over Artinian rings (“(i),(ii),(iii)”), 41.9 Flat morphisms
• (4.4): 11.2 Noncommutative algebras, 9.7 Finite extensions
• (5.2): 10.99 Criteria for flatness (“(ii) ⇒ (i), special case”)
• (5.3): 10.99 Criteria for flatness (“=> (iii)”)
• (5.5): 10.99 Criteria for flatness (“(ii) => (i)”; “more general”), 10.101 Flatness criteria over Artinian rings (“(i)⇔(ii), special case”)
• (5.6): 10.99 Criteria for flatness (“(ii)⇒(i), special case”; “(ii)⇒(iv)”), 10.128 More flatness criteria (“(ii)⇒(i), special case”)
• (5.7): 10.128 More flatness criteria (“(ii) ⟹ (i)”)
• (5.9): 10.39 Flat modules and flat ring maps (“(i)”), 10.99 Criteria for flatness (“(ii) ⇒ (i)”), 10.101 Flatness criteria over Artinian rings (“(ii) ⇒ (i), special case”), 10.128 More flatness criteria (“(ii) ⇒ (i)”; “(ii) ⇒ (i), special case”)
• (6.1): 5.15 Constructible sets (“necessity”), 5.16 Constructible sets and Noetherian spaces
• (6.2): 10.29 Images of ring maps of finite presentation (“affine case”), 10.35 Jacobson rings (“affine source and target”), 67.29 Constructible sets
• (6.3): 5.15 Constructible sets (“special case”), 5.19 Specialization (“special case”)
• (6.4): 5.15 Constructible sets (“(b)”), 5.19 Specialization (“(a)”; “special case”)
• (6.5): 5.19 Specialization (“partial”)
• (6.6): 41.10 Topological properties of flat morphisms (“F = O_X”), 59.26 Étale morphisms (“F = O_X”), 29.24 Open morphisms, 29.35 Smooth morphisms (“F = O_X”), 29.37 Étale morphisms (“F = O_X”)
• (6.7): 10.118 Generic flatness (“more general”), 37.54 Generic flatness stratification (“flatness clause”)
• (6.10): 10.129 Openness of the flat locus (“affine case”), 41.10 Topological properties of flat morphisms (“F = O_X case”)

Exposé IX

• (1.1): 29.37 Étale morphisms
• (1.2): 10.143 Étale ring maps (“converse direction”; “special case”), 29.35 Smooth morphisms (“more general”), 29.37 Étale morphisms (“converse direction”)
• (1.3): 10.144 Local structure of étale ring maps (“standard étale case”), 59.26 Étale morphisms, 29.37 Étale morphisms, 65.4 Lists of useful properties of morphisms of schemes, 67.39 Étale morphisms, 66.16 Étale morphisms of algebraic spaces
• (1.4): 35.25 Application of fpqc descent of properties of morphisms (“étale+radicial ⇒ open”), 41.14 Topological properties of étale morphisms (“Necessity”), 29.37 Étale morphisms (“Necessity”)
• (1.7): 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 (“partial”; “special case”)
• (1.8): 10.143 Étale ring maps (“more general”), 10.153 Henselian local rings (“more general”), 15.46 Permanence of properties under henselization (“decomposition direction”), 15.107 Weakly étale algebras over fields (“decomposition direction”), 58.5 Finite étale morphisms (“special case”)
• (1.9): 10.143 Étale ring maps, 15.46 Permanence of properties under henselization (“partial, special case”), 58.5 Finite étale morphisms (“special case”)
• (2.1): 29.25 Submersive morphisms (“submersive”; “univ. submersive”), 67.7 Submersive morphisms (“submersive”; “univ. submersive”), 101.12 Submersive morphisms (“submersive”; “univ. submersive”), 5.6 Submersive maps (“partial”)
• (2.2): 41.10 Topological properties of flat morphisms (“partial”), 67.30 Flat morphisms (“flat qc case, closed”; “flat qc case, open”; “flat qc submersive”), 5.6 Submersive maps (“a) surjective open case”)
• (2.3): 26.10 Immersions of schemes
• (2.4): 42.15 Push and pull (“partial”), 53.8 The genus of a curve (“partial”), 29.42 Proper morphisms (“partial”), 26.21 Separation axioms (“partial”), 65.4 Lists of useful properties of morphisms of schemes (“partial”; “special case”)
• (3.1): 59.45 Topological invariance of the small étale site (“special case”)
• (3.4): 41.20 Descending étale morphisms (“fully faithful”), 33.7 Geometrically connected schemes
• (4.2): 59.26 Étale morphisms (“Étaleness of X₁' → S₁'”), 29.37 Étale morphisms (“Étaleness of X₁' → S₁'”), 26.21 Separation axioms (“partial”), 65.4 Lists of useful properties of morphisms of schemes (“partial”; “Étaleness”; “Étaleness of X₁' → S₁'”)
• (4.3): 41.20 Descending étale morphisms (“sufficiency”)
• (4.7): 10.31 Noetherian rings (“special case”), 41.10 Topological properties of flat morphisms (“partial”), 41.20 Descending étale morphisms, 29.22 Morphisms of finite presentation (“partial”)
• (4.10): 10.31 Noetherian rings (“affine Noetherian base”), 41.10 Topological properties of flat morphisms (“more general”), 59.45 Topological invariance of the small étale site (“full faithfulness”), 15.11 Henselian pairs (“affine, nilpotent ideal”), 29.22 Morphisms of finite presentation (“more general”), 58.8 Topological invariance of the fundamental group (“special case”)
• (4.11): 41.20 Descending étale morphisms (“more general”)
• (4.12): 41.10 Topological properties of flat morphisms (“partial”), 29.22 Morphisms of finite presentation (“partial”)
• (6.1): 58.6 Fundamental groups (“π₁(Spec k, special case”), 58.14 Geometric and arithmetic fundamental groups (“special case”), 26.18 Base change in algebraic geometry
• (6.3): 59.83 Cohomology of torsion sheaves on curves (“more general”), 58.9 Finite étale covers of proper schemes (“more general”)
• (6.6): 37.53 Stein factorization (“partial”)
• (6.8): 58.8 Topological invariance of the fundamental group (“special case”)
• (6.9): 41.20 Descending étale morphisms (“fully faithful half”)

Exposé V

• (1.1): 10.47 Geometrically irreducible algebras (“(ii), special case”), 39.23 Finite flat groupoids, affine case (“more general”), 15.112 Group actions and integral closure (“more general”; “special case”), 15.114 Galois extensions and ramification (“special case”), 58.13 Ramification theory (“closed points”), 106.13 The Keel-Mori theorem (“(iv), special case”)
• (1.3): 15.112 Group actions and integral closure (“special case”), 15.114 Galois extensions and ramification (“special case”), 58.13 Ramification theory (“special case”)
• (1.6): 66.14 Obtaining a scheme (“partial”)
• (1.8): 39.24 Finite flat groupoids (“orbit⊂affine⇒inv affine”)
• (1.9): 106.13 The Keel-Mori theorem (“affine case with Z = Y”)
• (2.1): 58.12 Group actions and integral closure (“special case”)
• (2.2): 58.13 Ramification theory (“(ii)”)
• (2.3): 58.13 Ramification theory (“special case”)
• (3.3): 59.26 Étale morphisms (“special case”), 59.29 Neighborhoods, stalks and points (“special case”), 29.37 Étale morphisms (“special case”), 65.4 Lists of useful properties of morphisms of schemes (“special case”)
• (3.6): 41.14 Topological properties of étale morphisms (“monomorphism”), 87.26 Monomorphisms, 39.3 Equivalence relations, 29.45 Integral and finite morphisms (“monomorphism”), 26.23 Monomorphisms (“monomorphism”), 78.4 Equivalence relations, 67.10 Monomorphisms
• (3.7): 58.5 Finite étale morphisms (“more general”)
• (4.1): 58.3 Galois categories (“partial”)
• (5.1): 58.3 Galois categories (“partial”), 5.30 Topological groups, rings, modules
• (5.3): 58.3 Galois categories (“connected => transitive”)
• (5.7): 58.3 Galois categories (“Finite-G-Sets case”; “more general”)
• (5.11): 39.11 Principal homogeneous spaces (“formally princ. homog.”)
• (6.2): 58.3 Galois categories (“converse direction”; “existence of t”; “h indep. of t”)
• (6.3): 58.3 Galois categories (“U(F')=F'∘H lies in Γ”; “essential surjectivity”)
• (6.5): 58.4 Functors and homomorphisms (“more general”)
• (6.8): 58.4 Functors and homomorphisms
• (6.9): 4.2 Definitions, 58.4 Functors and homomorphisms (“more general”)
• (6.10): 58.4 Functors and homomorphisms (“a) <=> fully faithful”), 58.8 Topological invariance of the fundamental group (“special case”)
• (6.13): 90.6 Prorepresentable functors and predeformation categories, 15.114 Galois extensions and ramification
• (8.1): 58.6 Fundamental groups (“Canonicity clause”; “special case”)
• (8.2): 58.10 Local connectedness (“special case”), 58.11 Fundamental groups of normal schemes (“special case”; “surjectivity”)

Exposé VI

• (4.1): 4.31 2-fibre products
• (4.2): 4.35 Categories fibred in groupoids (“(iii)”)
• (4.5): 4.35 Categories fibred in groupoids (“both displayed maps”)
• (6.1): 4.33 Fibred categories
• (6.8): 4.35 Categories fibred in groupoids (“Well-definedness of F̃”)
• (6.10): 4.35 Categories fibred in groupoids (“fibred in groupoids”)
• (6.11): 4.33 Fibred categories (“Sufficiency”)
• (6.12): 4.33 Fibred categories (“iso ⇒ Cartesian”)
• (6.13): 4.35 Categories fibred in groupoids
• (7.2): 4.33 Fibred categories (“partial”)

Exposé VIII

• (1.1): 35.3 Descent for modules (“affine case”; “affine case, effectivity”), 35.5 Fpqc descent of quasi-coherent sheaves (“effectivity”; “full faithfulness”; “standard affine case”), 59.16 Faithfully flat descent (“affine case”; “effectivity”), 95.4 Quasi-coherent sheaves (“special case”)
• (1.3): 35.5 Fpqc descent of quasi-coherent sheaves (“Cor. 1.3, special case”), 95.4 Quasi-coherent sheaves (“special case”)
• (1.6): 35.3 Descent for modules (“effectivity”; “more general”), 59.16 Faithfully flat descent (“effectivity”)
• (1.7): 35.3 Descent for modules (“affine case”), 59.16 Faithfully flat descent (“affine case”)
• (1.8): 99.7 The functor of quotients (“more general”)
• (1.10): 10.83 Descent for finite projective modules (“fin. pres., affine”; “loc. free, affine”), 35.7 Descent of finiteness properties of modules (“finite locally free”; “finite presentation”)
• (1.11): 10.83 Descent for finite projective modules (“fin. pres., descent”), 35.4 Descent for universally injective morphisms (“finite presentation”; “finite type”)
• (2.1): 29.11 Affine morphisms, 65.4 Lists of useful properties of morphisms of schemes (“partial”)
• (3.1): 29.9 Surjective morphisms (“Surjectivity clause”), 65.4 Lists of useful properties of morphisms of schemes (“Surjectivity clause”), 100.5 Surjective morphisms (“f' surj => f surj”)
• (3.3): 10.126 Algebras and modules of finite presentation (“finite type, affine”), 56.5 Functors between categories of quasi-coherent modules (“quasi-compact clause”), 29.15 Morphisms of finite type (“quasi-compact clause”), 29.21 Quasi-finite morphisms (“quasi-compact clause”), 26.19 Quasi-compact morphisms (“quasi-compact clause”), 65.4 Lists of useful properties of morphisms of schemes (“quasi-compact clause”)
• (3.4): 10.14 Base change (“forward implication”), 10.126 Algebras and modules of finite presentation
• (3.5): 35.23 Properties of morphisms local in the fpqc topology on the target (“more general”; “special case”)
• (3.6): 10.6 Ring maps of finite type and of finite presentation (“special case”), 10.14 Base change (“partial”), 10.126 Algebras and modules of finite presentation (“affine case”; “partial”), 67.28 Morphisms of finite presentation (“more general”)
• (4.2): 41.10 Topological properties of flat morphisms (“special case”)
• (4.3): 41.10 Topological properties of flat morphisms, 67.30 Flat morphisms (“closed subsets”; “open subsets”)
• (4.4): 35.13 Fpqc coverings are universal effective epimorphisms (“more general”), 26.10 Immersions of schemes
• (4.7): 35.23 Properties of morphisms local in the fpqc topology on the target (“more general”), 29.42 Proper morphisms (“necessity”), 65.4 Lists of useful properties of morphisms of schemes (“more general”; “necessity”)
• (4.8): 42.15 Push and pull (“partial”), 53.8 The genus of a curve (“partial”), 35.23 Properties of morphisms local in the fpqc topology on the target (“more general”), 29.42 Proper morphisms (“partial”), 26.21 Separation axioms (“partial”), 65.4 Lists of useful properties of morphisms of schemes (“partial”), 74.11 Descending properties of morphisms in the fpqc topology (“more general”; “partial”)
• (4.9): 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”)
• (5.2): 35.35 Fully faithfulness of the pullback functors (“more general”), 59.15 The fpqc topology (“more general”), 90.6 Prorepresentable functors and predeformation categories (“fiber over S”)
• (5.3): 35.13 Fpqc coverings are universal effective epimorphisms (“more general”)
• (5.5): 35.23 Properties of morphisms local in the fpqc topology on the target (“closed-immersion clause”; “open-immersion clause”), 29.2 Closed immersions (“closed-immersion clause”), 29.14 Types of morphisms defined by properties of ring maps (“special case”), 26.17 Fibre products of schemes (“closed-immersion clause”; “open-immersion clause”), 26.18 Base change in algebraic geometry (“closed-immersion clause”; “open-immersion clause”), 65.4 Lists of useful properties of morphisms of schemes (“closed-immersion clause”; “more general”; “open-immersion clause”)
• (5.6): 65.4 Lists of useful properties of morphisms of schemes (“necessity”)
• (5.7): 35.23 Properties of morphisms local in the fpqc topology on the target (“detection (integral)”; “more general”; “partial”; “special case”), 29.45 Integral and finite morphisms (“preservation (finite)”; “preservation (integral)”), 65.4 Lists of useful properties of morphisms of schemes (“more general”; “preservation (finite)”; “preservation (integral)”; “special case”)
• (5.8): 29.39 Very ample sheaves (“Very ample clause”)
• (6.2): 37.43 Zariski's Main Theorem (“more general”)
• (6.4): 10.123 Zariski's Main Theorem (“affine case”)
• (6.6): 29.45 Integral and finite morphisms, 65.4 Lists of useful properties of morphisms of schemes
• (7.2): 35.35 Fully faithfulness of the pullback functors (“local eff ⇒ effective”)
• (7.3): 41.20 Descending étale morphisms (“partial”)
• (7.6): 10.78 Finite projective modules (“affine module condition”), 39.25 Descending quasi-projective schemes (“more general”; “special case”; “sufficiency, fibres ≤ d”)
• (7.7): 39.25 Descending quasi-projective schemes (“Effectivity clause”; “more general”; “special case”)
• (7.8): 35.25 Application of fpqc descent of properties of morphisms (“more general”; “special case”), 39.25 Descending quasi-projective schemes (“partial”), 40.15 Descending ind-quasi-affine morphisms (“partial”), 8.3 Descent data in fibred categories (“more general”)
• (7.9): 35.38 Descending quasi-affine morphisms (“partial”; “surjective étale cover”), 40.15 Descending ind-quasi-affine morphisms (“partial”; “special case”)

Exposé X

• (1.1): 33.6 Geometrically reduced schemes (“special case”)
• (1.2): 37.53 Stein factorization (“Under Y′ ≅ Spec f_O_X”; “Y′ is finite over Y”; “Y′ ≅ Spec f_O_X over Y”), 58.15 Homotopy exact sequence (“Y′ → Y is étale”)
• (1.4): 58.15 Homotopy exact sequence (“Surjectivity of π₁(X”)
• (1.8): 58.9 Finite étale covers of proper schemes (“more general”)
• (1.10): 59.83 Cohomology of torsion sheaves on curves (“partial”)
• (2.1): 58.9 Finite étale covers of proper schemes (“more general”)
• (2.4): 58.30 Specialization maps in the smooth proper case (“more general”)
• (2.5): 58.30 Specialization maps in the smooth proper case (“more general”)
• (2.13): 58.27 Lefschetz for the fundamental group (“more general”; “partial”)
• (3.1): 58.21 Purity of branch locus (“more general”; “partial”; “partial, special case”), 58.28 Purity of ramification locus (“more general”)
• (3.2): 58.21 Purity of branch locus (“Case B finite over A”; “more general”; “special case”)
• (3.3): 58.31 Tame ramification (“partial”)
• (3.8): 58.30 Specialization maps in the smooth proper case (“more general”; “special case”)
• (3.9): 58.30 Specialization maps in the smooth proper case

Exposé XI

• (4.1): 39.11 Principal homogeneous spaces, 78.9 Principal homogeneous spaces
• (4.6): 39.25 Descending quasi-projective schemes (“partial”)
• (4.7): 39.11 Principal homogeneous spaces, 78.9 Principal homogeneous spaces (“more general”)
• (4.8): 20.2 Cohomology of sheaves (“more general”), 21.2 Cohomology of sheaves (“more general”)
• (5.3): 59.23 Examples of sheaves (“G_a clause”), 59.24 Picard groups (“G_m clause, special case”)
• (6.1): 59.28 Kummer theory (“exact sequence (big)”; “exact sequence (small)”), 39.5 Examples of group schemes (“special case”), 78.7 Examples of group algebraic spaces
• (6.4): 20.6 First cohomology and invertible sheaves (“more general”), 59.24 Picard groups (“picardSpace field”)
• (6.5): 31.29 More on invertible modules (“factorial-ring example”)
• (6.7): 59.63 The Artin-Schreier sequence

Exposé XIII

• (1.3): 95.14 Classifying torsors (“partial”)
• (1.5): 29.15 Morphisms of finite type, 29.21 Quasi-finite morphisms, 26.19 Quasi-compact morphisms, 26.21 Separation axioms, 33.20 Algebraic schemes
• (1.11): 59.55 Vanishing of finite higher direct images (“partial”)
• (2.0): 10.143 Étale ring maps (“more general”), 15.113 Extensions of discrete valuation rings (“L a field extension”), 15.114 Galois extensions and ramification (“inertia structure”; “partial”), 58.13 Ramification theory (“inertia structure”)
• (2.0.2): 15.116 Abhyankar's lemma and tame ramification (“⇒: tameness ascends”), 58.31 Tame ramification (“partial”)
• (2.0.3): 15.116 Abhyankar's lemma and tame ramification (“subalgebra of tame”; “tame after DVR extension”), 58.31 Tame ramification (“more general”)
• (2.1.0): 41.21 Normal crossings divisors (“absolute case”), 111.33 Schemes, 26.10 Immersions of schemes
• (2.1.3): 58.5 Finite étale morphisms (“(C_t, special case”)
• (2.9): 63.15 Applications (“special case”)
• (2.13): 59.83 Cohomology of torsion sheaves on curves (“partial”)
• (4.0): 58.14 Geometric and arithmetic fundamental groups (“vu = 1”)
• (4.1): 58.15 Homotopy exact sequence (“Surjectivity of v”)
• (4.4): 58.14 Geometric and arithmetic fundamental groups (“special case”), 58.15 Homotopy exact sequence (“(X)”; “partial”)
• (4.7): 58.26 Purity in local case, III (“special case”)
• (5.1): 58.31 Tame ramification (“r = 1, codim 1 tame”; “r = 1, finite part”; “r = 1, tame part”; “r = 1, étale part”)
• (6.1): 26.17 Fibre products of schemes

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