SGA 3

The items of SGA 3 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

(1.1): 4.3 Opposite Categories and the Yoneda Lemma, 59.9 Presheaves, 59.18 Čech cohomology, 26.15 A representability criterion, 7.2 Presheaves, 7.12 Representable sheaves (“special case”), 18.4 Free abelian presheaves, 18.5 Free abelian sheaves, 65.6 Algebraic spaces
(1.1.1): 59.9 Presheaves (“Hom_D(X”), 59.18 Čech cohomology (“more general”), 7.2 Presheaves (“more general”), 7.12 Representable sheaves (“more general”), 18.4 Free abelian presheaves (“more general”), 18.5 Free abelian sheaves (“more general”), 103.14 The lisse-étale and the flat-fppf sites (“Hom_C(X,Y) = Hom_D(X,Y)”), 34.3 The Zariski topology (“Hom_C(X,Y) = Hom_D(X,Y)”)
(1.2): 26.15 A representability criterion, 26.23 Monomorphisms (“more general”), 6.16 Exactness and points (“special case”), 7.10 Sheafification (“special case”), 7.11 Injective and surjective maps of sheaves (“more general”)
(1.3): 90.6 Prorepresentable functors and predeformation categories, 61.12 The pro-étale site, 34.3 The Zariski topology, 34.4 The étale topology, 34.8 The ph topology
(1.6): 92.3 The cotangent complex of a ring map (“more general”), 61.12 The pro-étale site, 26.18 Base change in algebraic geometry, 7.25 Localization (“partial”), 7.27 More localization (“i_S ⊣ p_S hom bijection”), 7.30 Localization of topoi (“p_S clause: i_S ⊣ p_S”), 73.7 Fppf topology, 73.8 The ph topology, 34.3 The Zariski topology, 34.4 The étale topology, 34.7 The fppf topology, 34.8 The ph topology, 33.36 Frobenii
(1.6.1): 59.18 Čech cohomology (“Hom(h_S, F) ≃ F(S)”), 7.2 Presheaves (“Hom(h_S, F) ≃ F(S)”), 7.12 Representable sheaves (“Hom(h_S, F) ≃ F(S)”), 18.4 Free abelian presheaves (“Hom(h_S, F) ≃ F(S)”), 18.5 Free abelian sheaves (“Hom(h_S, F) ≃ F(S)”)
(1.7.0): 59.18 Čech cohomology (“partial”), 7.2 Presheaves (“partial”), 7.12 Representable sheaves (“partial”), 18.4 Free abelian presheaves (“partial”), 18.5 Free abelian sheaves (“partial”)
(1.8): 39.5 Examples of group schemes (“special case”), 14.13 Products with simplicial sets (“(1.8.2)”)
(2.1.1): 4.3 Opposite Categories and the Yoneda Lemma, 59.9 Presheaves, 6.30 Bases and sheaves (“morphism of Ĉ-groups”), 6.31 Open immersions and (pre)sheaves (“restriction G ↦ G_S”), 7.2 Presheaves, 7.10 Sheafification (“partial”), 65.6 Algebraic spaces (“more general”)
(2.3.3): 15.114 Galois extensions and ramification (“special case”)
(2.3.6): 39.8 Properties of algebraic group schemes (“center”)
(3.1): 6.6 Presheaves of modules (“F(S) is an O(S)-module”; “Hom_O(F,F')”; “O-module definition”; “category (O-Mod.)”; “morphism compatibility”), 6.30 Bases and sheaves (“special case”), 18.9 Presheaves of modules (“Hom_O(F,F')”; “O-module definition”; “category (O-Mod.)”)
(3.1.1): 17.3 The abelian category of sheaves of modules (“more general”), 18.14 The abelian category of sheaves of modules (“more general”)
(3.1.2): 19.7 Abelian Sheaves on a site (“special case”), 19.8 Modules on a ringed site (“more general”)
(3.1.4): 6.31 Open immersions and (pre)sheaves (“F ↦ F_U for C = Opens(X)”), 18.19 Localization of ringed sites (“more general”)
(3.2.1): 19.13 Additional remarks on Grothendieck abelian categories (“special case”)
(4.1): 78.7 Examples of group algebraic spaces (“Hom = loc const maps”)
(4.2): 27.3 Relative spectrum via glueing, 27.4 Relative spectrum as a functor (“A ≅ π_*O_Spec(A)”; “Spec(A) affine over S”), 39.5 Examples of group schemes (“partial”), 29.11 Affine morphisms (“more general”; “partial”)
(4.3.1): 59.23 Examples of sheaves (“more general”), 39.5 Examples of group schemes (“G_{a,S} group over S”; “more general”; “partial”; “the scheme G_a”; “Δ(T)=T⊗1+1⊗T”)
(4.3.2): 39.5 Examples of group schemes (“G_{m,S}(T)=Γ(T,O_T)^*”; “special case”)
(4.4.4): 39.5 Examples of group schemes (“μ_n = Spec Z[x]/(x^n-1)”; “μ_n(T)={f : f^n=1}”; “μ_{n,S}”)
(4.5): 39.5 Examples of group schemes (“GL_n representability”)
(4.6.1): 18.19 Localization of ringed sites (“more general”), 18.21 Localization of ringed topoi (“more general”)
(4.6.3.1): 27.6 Vector bundles (“partial”)
(4.6.4): 6.26 Morphisms of ringed spaces and modules (“more general”)
(4.6.4.1): 65.4 Lists of useful properties of morphisms of schemes (“partial”)
(4.6.5.1): 51.22 A bit of uniformity, III (“more general”)
(4.7.1.1): 17.3 The abelian category of sheaves of modules (“more general”), 6.6 Presheaves of modules, 6.10 Sheaves of modules, 18.9 Presheaves of modules, 18.10 Sheaves of modules, 18.14 The abelian category of sheaves of modules (“more general”)
(4.7.2.1): 39.14 Quasi-coherent sheaves on groupoids (“Mod.q.c. case”)
(4.7.3): 10.12 Tensor products (“special case”)
(5.2): 20.2 Cohomology of sheaves (“more general”), 12.31 Inverse systems (“partial”), 21.2 Cohomology of sheaves (“more general”)
(5.2.1): 12.12 Cohomological delta-functors (“special case”), 64.29 Cohomology of curves, revisited (“special case”)
(5.3.1.1): 10.39 Flat modules and flat ring maps (“partial”), 59.14 Cohomology (“more general”), 12.27 Injectives, 19.8 Modules on a ringed site (“more general”), 14.34 Standard resolutions
(6.3.1): 39.12 Equivariant quasi-coherent sheaves, 78.10 Equivariant quasi-coherent sheaves (“special case”)
(6.4): 6.24 Continuous maps and sheaves of modules (“more general”)
(6.5): 39.12 Equivariant quasi-coherent sheaves
(6.5.1): 39.12 Equivariant quasi-coherent sheaves (“equivariant module”)
(6.5.2): 39.12 Equivariant quasi-coherent sheaves
(6.6.2): 39.14 Quasi-coherent sheaves on groupoids (“more general”)
(6.8.5): 10.134 The naive cotangent complex (“special case”), 29.32 Conormal sheaf of an immersion, 76.5 Conormal sheaf of an immersion (“étale base change”)

Exposé II

(2.1): 33.16 Tangent spaces (“special case”)
(3.14): 10.131 Differentials (“affine case”), 93.11 Algebraic spaces (“special case”), 33.16 Tangent spaces (“partial”)
(4.5.1): 33.16 Tangent spaces
(4.7.1): 33.16 Tangent spaces (“partial”; “special case”)
(4.11.4): 41.6 Topological properties of unramified morphisms (“more general”), 39.6 Properties of group schemes (“(a)”; “(a) over a field”), 39.22 Separation conditions (“more general”)

Exposé VIIA

(1.1): 6.26 Morphisms of ringed spaces and modules (“more general”)
(1.1.1): 10.133 Finite order differential operators (“partial”)
(1.2): 10.133 Finite order differential operators (“affine, deviations of id”), 18.34 Finite order differential operators (“sheaf, deviations of id”)
(1.2.0): 10.131 Differentials (“special case”), 17.28 Modules of differentials (“more general”), 33.16 Tangent spaces (“partial”)
(1.3): 29.6 Scheme theoretic image (“special case”), 18.13 Morphisms of ringed topoi and modules (“more general”), 18.19 Localization of ringed sites (“more general”)
(1.3.1): 10.131 Differentials (“special case”), 10.133 Finite order differential operators (“Dev≤1 ≃ O_S ⊕ Der_S”), 90.11 Tangent spaces of functors (“special case”)
(1.4): 10.133 Finite order differential operators (“affine case”; “partial”), 18.34 Finite order differential operators (“partial”)
(1.4.1): 10.131 Differentials (“special case”), 10.133 Finite order differential operators (“affine case”)
(1.5): 93.8 Rings (“partial”)
(2.5): 93.9 Schemes (“first component”)
(3.1.1): 51.22 A bit of uniformity, III (“affine dual object”)
(3.3): 39.5 Examples of group schemes (“partial”; “special case”)
(3.3.0): 39.5 Examples of group schemes (“partial”)
(4.0): 90.6 Prorepresentable functors and predeformation categories (“more general”)
(4.1): 33.36 Frobenii (“Fr(X/S) for S-schemes”; “X^{(p/S)} for S-schemes”; “absolute Frobenius fr(X)”; “special case”)
(4.1.3): 33.36 Frobenii (“Case n=1”; “special case”)
(4.1.4): 39.5 Examples of group schemes (“partial”)
(4.2.1): 10.81 Characterizing flatness (“more general”)
(4.3.1): 33.36 Frobenii
(4.3.2): 33.36 Frobenii (“more general”)
(4.3.3): 51.22 A bit of uniformity, III (“more general”; “special case”)
(5.2): 10.131 Differentials (“special case”)
(6.4.3): 93.9 Schemes (“special case”)
(7.4.2): 33.36 Frobenii (“more general”)
(7.4.3): 10.101 Flatness criteria over Artinian rings (“special case”)
(8.1.4): 59.15 The fpqc topology (“Converse clause”), 7.11 Injective and surjective maps of sheaves (“partial”), 34.7 The fppf topology (“Converse clause”)
(8.3): 39.8 Properties of algebraic group schemes (“Smoothness clause”)
(8.3.1): 35.23 Properties of morphisms local in the fpqc topology on the target (“Final descent step”), 65.4 Lists of useful properties of morphisms of schemes (“Smoothness entry (17)”; “partial”)
(8.4): 10.101 Flatness criteria over Artinian rings (“partial”), 39.8 Properties of algebraic group schemes (“more general”)
(8.5.1): 39.8 Properties of algebraic group schemes (“more general”)
(8.5.2): 29.11 Affine morphisms (“affine implies separated”)

Exposé VIIB

(0.1): 10.60 Dimension (“partial”)
(0.1.1): 10.21 Open and closed subsets of spectra (“partial”), 10.53 Artinian rings (“partial”; “partial, special case”), 10.60 Dimension (“partial”), 10.112 Homomorphisms and dimension (“partial”), 10.160 The Cohen structure theorem (“special case”), 33.39 The delta invariant (“special case”), 33.41 Normalization of one dimensional schemes (“special case”)
(0.1.2): 87.4 Topological rings and modules
(0.1.3): 87.4 Topological rings and modules (“more general”; “r(A) ↦ r(B)”), 5.30 Topological groups, rings, modules
(0.2): 15.37 Topological rings and modules, 5.30 Topological groups, rings, modules
(0.2.1): 10.18 Local rings
(0.2.2): 10.83 Descent for finite projective modules (“partial”; “projective implies flat”), 10.101 Flatness criteria over Artinian rings (“special case”), 15.37 Topological rings and modules, 5.30 Topological groups, rings, modules
(0.3): 87.4 Topological rings and modules (“P = lim quotients”; “def of L⊗̂M”; “lim (L/L')⊗(M/M')”)
(0.3.2): 10.12 Tensor products (“Corollary, special case”; “special case”)
(0.3.4): 15.3 Stably free modules (“Corollary (ii) of 0.3.4”)
(0.3.5): 10.11 Characterizing finite and finitely presented modules (“(3), special case”), 10.12 Tensor products
(0.3.6): 15.37 Topological rings and modules, 5.30 Topological groups, rings, modules
(0.3.7): 10.101 Flatness criteria over Artinian rings (“local Artinian rings”)
(0.4.2): 10.52 Length (“more general”), 10.60 Dimension (“more general”), 59.18 Čech cohomology (“more general”)
(0.5.A): 87.4 Topological rings and modules (“def of M⊗̂_k N”)
(0.5.B): 15.37 Topological rings and modules, 5.30 Topological groups, rings, modules
(1.1): 87.2 Formal schemes à la EGA (“partial”)
(1.2.1): 10.39 Flat modules and flat ring maps (“partial”), 10.83 Descent for finite projective modules (“special case”), 10.101 Flatness criteria over Artinian rings (“special case”)
(1.2.6): 10.52 Length (“partial”)
(1.3): 26.23 Monomorphisms (“geometric clause”)
(1.3.3): 10.39 Flat modules and flat ring maps (“(i), special case”; “(iii), special case”), 42.14 Flat pullback (“(i), special case”), 42.15 Push and pull (“(iii), special case”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“(iii), special case”), 29.26 Flat morphisms (“(i), special case”; “(iii), special case”), 65.4 Lists of useful properties of morphisms of schemes (“(i), special case”; “(iii), special case”)
(1.3.4): 15.44 Permanence of properties under completion (“partial”), 88.13 Flat morphisms (“Flatness clause”)
(1.3.5.A): 15.37 Topological rings and modules, 5.30 Topological groups, rings, modules
(1.3.5.C): 51.22 A bit of uniformity, III (“more general”)
(1.4.3): 12.28 Projectives (“more general”)
(1.6): 10.143 Étale ring maps (“special case”), 10.150 Formally étale maps (“discrete base change”), 10.153 Henselian local rings (“partial”), 9.7 Finite extensions (“(F)”)
(1.6.B): 10.150 Formally étale maps (“(a)⇒(c), discrete case”)
(1.6.C): 10.150 Formally étale maps (“special case”), 59.26 Étale morphisms (“special case”), 29.37 Étale morphisms (“special case”), 65.4 Lists of useful properties of morphisms of schemes (“special case”)
(1.6.E): 10.143 Étale ring maps (“special case”), 10.153 Henselian local rings (“(b), special case”)
(2.1): 39.5 Examples of group schemes (“(i), special case”)
(2.2.1): 10.12 Tensor products (“special case”), 10.39 Flat modules and flat ring maps (“special case”), 41.9 Flat morphisms (“special case”), 39.5 Examples of group schemes (“partial”)
(2.3.2): 10.14 Base change (“partial”), 4.3 Opposite Categories and the Yoneda Lemma (“more general”), 59.9 Presheaves (“more general”), 7.2 Presheaves (“more general”), 65.6 Algebraic spaces (“definition of Hom_k-Gr(G”)
(2.4.1): 87.26 Monomorphisms, 39.3 Equivalence relations, 29.9 Surjective morphisms (“special case”), 26.17 Fibre products of schemes, 26.23 Monomorphisms, 65.4 Lists of useful properties of morphisms of schemes (“special case”), 78.4 Equivalence relations, 67.10 Monomorphisms
(2.5): 10.143 Étale ring maps (“special case”), 58.2 Schemes étale over a point (“k a field”)
(2.5.2.A): 39.5 Examples of group schemes (“more general”; “partial”)
(2.5.A): 15.114 Galois extensions and ramification
(2.6): 33.16 Tangent spaces
(3.2): 10.60 Dimension (“U = C”)
(3.3.3): 59.56 Galois action on stalks (“more general”)
(4.1.1): 33.36 Frobenii (“more general”)
(5.5): 23.8 Local complete intersection rings
(5.5.2): 10.68 Regular sequences (“special case”), 10.97 Completion for Noetherian rings (“Noetherianity of Â”; “flatness of Â over A”; “partial”), 10.139 Smoothness and differentials (“partial”), 15.40 Some results on power series rings (“partial”)

Exposé VIII

(1.0.3): 90.6 Prorepresentable functors and predeformation categories, 39.5 Examples of group schemes
(1.3): 39.5 Examples of group schemes (“partial”; “special case”)
(1.4): 39.5 Examples of group schemes (“partial”; “special case”), 18.43 Locally constant sheaves (“partial”)
(1.7.1): 35.35 Fully faithfulness of the pullback functors (“more general”), 59.15 The fpqc topology (“special case”)
(1.7.2): 35.37 Descending affine morphisms (“(ii), special case”; “more general”), 35.38 Descending quasi-affine morphisms (“more general”), 40.15 Descending ind-quasi-affine morphisms (“(ii), special case”; “more general”)
(1.7.3): 35.23 Properties of morphisms local in the fpqc topology on the target (“Closed immersion clause”)
(2.1): 29.35 Smooth morphisms (“partial”), 78.5 Group algebraic spaces (“(e), special case”)
(3.4): 10.5 Finite modules and finitely presented modules, 11.2 Noncommutative algebras, 9.7 Finite extensions, 87.26 Monomorphisms, 39.3 Equivalence relations, 12.3 Preadditive and additive categories, 26.23 Monomorphisms, 78.4 Equivalence relations, 67.10 Monomorphisms
(4.4): 20.6 First cohomology and invertible sheaves (“Pic(S) ≅ H¹(S, O_S^×)”), 59.24 Picard groups (“partial”), 21.6 First cohomology and invertible sheaves (“Pic ≅ H¹(O*)”)
(6.2): 10.101 Flatness criteria over Artinian rings (“flat clause”; “more general”)
(6.4): 35.37 Descending affine morphisms (“Object effectivity”), 38.23 Flattening a map (“special case”)
(6.5): 39.7 Properties of group schemes over a field (“partial”), 39.8 Properties of algebraic group schemes (“special case”)
(6.7): 39.8 Properties of algebraic group schemes (“first item, special case”)
(6.8): 51.22 A bit of uniformity, III
(7.1): 39.8 Properties of algebraic group schemes (“affine group schemes”; “more general”)
(7.11): 39.9 Abelian varieties (“special case”)

Exposé X

(2.0): 41.15 Topological invariance of the étale topology, 59.45 Topological invariance of the small étale site (“full faithfulness”)
(3.5): 10.99 Criteria for flatness (“Flat case”), 10.128 More flatness criteria (“Flat case”)
(3.7): 41.14 Topological properties of étale morphisms (“partial”), 29.27 Flat closed immersions (“partial”)
(4.0): 10.97 Completion for Noetherian rings (“partial”), 10.153 Henselian local rings (“partial”), 59.32 Henselian rings (“Recall 4.0”), 15.11 Henselian pairs (“Recall 4.0”), 15.13 Lifting and henselian pairs (“ÉtCov(S)≃ÉtCov(S₀)”), 58.8 Topological invariance of the fundamental group (“ÉtCov(S)≃ÉtCov(S₀)”), 58.9 Finite étale covers of proper schemes (“more general”), 58.13 Ramification theory (“Recall 4.0, special case”), 33.39 The delta invariant (“partial”), 33.41 Normalization of one dimensional schemes (“partial”)
(4.4): 68.11 Residue fields and henselian local rings (“Case 2°”), 37.16 Critère de platitude par fibres (“partial”), 76.23 Critère de platitude par fibres (“partial”)
(4.5): 59.32 Henselian rings (“special case”)
(5.4): 35.37 Descending affine morphisms (“special case”), 35.38 Descending quasi-affine morphisms (“special case”), 41.10 Topological properties of flat morphisms (“partial”), 59.26 Étale morphisms (“partial”), 40.15 Descending ind-quasi-affine morphisms (“more general”; “special case”), 37.43 Zariski's Main Theorem (“more general”), 37.57 Descending separated locally quasi-finite morphisms (“more general”; “special case”), 29.35 Smooth morphisms (“partial”), 29.37 Étale morphisms (“partial”)
(5.11.0): 5.7 Connected components (“more general”)
(5.14.0): 37.36 Étale neighbourhoods and branches (“partial”), 29.55 Normalization (“univ. homeomorphism”), 28.16 Local irreducibility
(6.3): 14.3 Simplicial objects, 85.2 Simplicial topological spaces
(7.0): 28.5 Noetherian schemes, 5.9 Noetherian topological spaces (“local connectedness”)
(7.0.3): 58.5 Finite étale morphisms (“special case”)
(8.9): 35.18 Properties of schemes local in the smooth topology (“more general”), 37.19 Normalization revisited (“more general”), 29.35 Smooth morphisms (“partial”), 29.54 Relative normalization (“partial”), 76.25 Normalization revisited (“partial”), 67.49 Normalization (“partial”)
(9.2.1): 4.3 Opposite Categories and the Yoneda Lemma, 59.9 Presheaves, 59.64 Locally constant sheaves (“(2), (4)”), 7.2 Presheaves, 18.43 Locally constant sheaves (“(2), (4)”; “(4)”)
(9.7.1): 10.48 Geometrically connected algebras (“affine case, i = 1”), 14.3 Simplicial objects, 85.2 Simplicial topological spaces
(9.7.2): 10.48 Geometrically connected algebras (“affine case”)
(9.7.3): 5.7 Connected components (“more general”)

Exposé XI

(1.1): 10.148 Formally unramified maps (“partial”), 41.11 Étale morphisms, 41.16 The functorial characterization, 39.4 Group schemes, 37.6 Formally unramified morphisms (“partial”), 37.8 Formally étale morphisms, 37.11 Formally smooth morphisms, 29.35 Smooth morphisms, 29.37 Étale morphisms
(1.2): 10.139 Smoothness and differentials (“(ii), special case”), 29.35 Smooth morphisms (“(i)”), 33.25 Schemes smooth over fields (“Proposition 1.2(i)”)
(1.3): 10.158 Formal smoothness of fields (“field extension case”), 41.3 Unramified morphisms (“(ii) completion surj.”), 29.36 Unramified morphisms (“(a) => (b)”; “(a) => (e)”; “(a)<=>(e) over field”; “(b) => (a)”; “(i)(e)⇒(a)”; “(ii) k(B)/k(A) finite”; “(ii) k(B)/k(A) separable”; “more general”), 67.38 Unramified morphisms (“more general”)
(1.4): 29.37 Étale morphisms (“more general”)
(1.5): 10.138 Formally smooth maps (“partial”), 37.6 Formally unramified morphisms (“partial”), 29.36 Unramified morphisms, 76.20 Smoothness over a Noetherian base (“partial”)
(1.6): 10.148 Formally unramified maps (“affine case, unramified”; “partial”), 41.11 Étale morphisms, 41.16 The functorial characterization (“étale case”), 39.4 Group schemes, 37.6 Formally unramified morphisms (“partial”; “represented functors”; “special case”), 37.8 Formally étale morphisms (“étale case”), 29.35 Smooth morphisms, 29.37 Étale morphisms, 76.13 Formally smooth, étale, unramified transformations (“formally smooth”; “formally unramified”; “formally étale”; “necessity direction”), 76.14 Formally unramified morphisms (“formally unramified”), 76.16 Formally étale morphisms (“formally étale”), 76.19 Formally smooth morphisms (“formally smooth”)
(1.9): 90.11 Tangent spaces of functors (“partial”), 29.35 Smooth morphisms (“finite locally free”; “rank = fiber dimension”), 33.16 Tangent spaces (“partial”)
(1.10): 37.38 Slicing smooth morphisms (“more general”)
(1.11): 10.155 Henselization and strict henselization (“special case”)
(2.1): 39.4 Group schemes (“The value of M_H”)
(3.2): 98.25 Limit preserving functors on Noetherian schemes (“partial”), 32.6 Limits and morphisms of finite presentation (“definition of ()”; “partial”; “representable case”), 70.3 Morphisms of finite presentation (“definition of ()”)
(3.3): 90.7 Formal objects and completion categories (“representable, affine”), 87.33 Maps out of affine formal schemes (“injectivity”)
(3.4): 98.25 Limit preserving functors on Noetherian schemes (“special case”), 59.15 The fpqc topology (“fpqc example”), 34.9 The fpqc topology (“special case”)
(3.6): 29.9 Surjective morphisms (“special case”)
(3.9): 10.107 Epimorphisms of rings (“Affine case”), 35.13 Fpqc coverings are universal effective epimorphisms (“partial”)
(3.11): 70.5 Descending properties (“more general”)
(4.2): 40.15 Descending ind-quasi-affine morphisms (“more general”; “partial”)
(4.3): 29.11 Affine morphisms (“T is separated over S”), 26.21 Separation axioms (“special case”), 26.23 Monomorphisms (“F is separated over T”), 70.4 Limits of algebraic spaces (“partial”)
(6.4): 67.28 Morphisms of finite presentation (“more general”)
(6.8): 59.15 The fpqc topology (“Fpqc covering step”), 38.23 Flattening a map (“special case”), 65.4 Lists of useful properties of morphisms of schemes (“partial”)

Exposé XII

(1.0): 39.7 Properties of group schemes over a field (“partial”), 39.8 Properties of algebraic group schemes (“partial”)
(4.4.2): 39.9 Abelian varieties (“partial”)
(6.1): 39.8 Properties of algebraic group schemes
(9.4): 37.9 Infinitesimal deformations of maps (“partial”)

Exposé XIV

(2.13): 10.16 Cayley-Hamilton (“special case”)
(2.15): 4.8 Fibre products and representability (“more general”)
(2.17): 9.26 Transcendence
(6.1): 9.26 Transcendence

Exposé XIX

(2.1): 10.14 Base change (“special case”), 10.39 Flat modules and flat ring maps (“special case”), 10.49 Geometrically integral algebras (“partial”), 10.126 Algebras and modules of finite presentation (“special case”), 10.138 Formally smooth maps (“special case”), 42.15 Push and pull, 35.14 Descent of finiteness and smoothness properties of morphisms (“special case”), 63.5 Weightings and trace maps for locally quasi-finite morphisms, 29.26 Flat morphisms, 29.35 Smooth morphisms (“(i)”), 65.4 Lists of useful properties of morphisms of schemes (“(i)”; “more general”), 33.9 Geometrically integral schemes (“partial”)
(5.10): 37.29 Connected components meeting a section

Exposé XV

(1.1): 37.11 Formally smooth morphisms (“more general”), 29.35 Smooth morphisms (“partial”)
(1.4): 87.33 Maps out of affine formal schemes (“injectivity of (*)”), 26.13 Points of schemes (“partial”)
(1.5): 10.160 The Cohen structure theorem (“partial”)
(2.7): 91.5 Infinitesimal deformations of modules on ringed spaces (“more general”), 37.10 Infinitesimal deformations of schemes (“affine case”)
(2.10): 10.39 Flat modules and flat ring maps (“partial”), 10.71 Ext groups (“more general”), 10.78 Finite projective modules, 12.6 Extensions (“more general”)
(3.3): 29.23 Constructible sets (“partial”), 28.5 Noetherian schemes (“partial”), 5.19 Specialization (“special case”)
(3.8): 10.39 Flat modules and flat ring maps (“partial”), 99.7 The functor of quotients (“more general”)
(3.9): 10.160 The Cohen structure theorem
(4.6): 39.8 Properties of algebraic group schemes (“(2), special case”), 39.9 Abelian varieties (“special case”)
(4.7): 15.22 Torsion free modules (“partial”)
(5.5): 26.14 Glueing schemes (“more general”)
(5.7): 17.26 Rank and determinant (“partial”)
(6.5): 39.8 Properties of algebraic group schemes (“partial”; “special case”)
(6.6): 5.19 Specialization (“special case”)
(7.2): 37.16 Critère de platitude par fibres (“partial”)
(8.10): 59.15 The fpqc topology (“(i)”)

Exposé XVI

(1.1): 66.14 Obtaining a scheme (“more general”)
(2.1): 41.10 Topological properties of flat morphisms (“more general”), 59.26 Étale morphisms (“more general”), 29.26 Flat morphisms (“more general”), 29.35 Smooth morphisms (“more general”), 29.37 Étale morphisms (“more general”), 67.30 Flat morphisms (“more general”)

Exposé XVII

(4.3.2): 33.36 Frobenii (“Notation X^(p^n)”; “partial”)
(5.2.1): 10.143 Étale ring maps (“partial”)
(5.5): 39.8 Properties of algebraic group schemes (“more general”; “partial”), 40.10 Properties of groupoids on fields (“partial”)
(5.7.1): 37.11 Formally smooth morphisms

Exposé XVIII

(1.4): 4.6 Fibre products, 4.8 Fibre products and representability
(1.5): 29.50 Rational maps (“representatives f”; “special case”)
(2.2): 39.6 Properties of group schemes, 79.9 Properties of groups over fields and groupoids on fields (“special case”)

Exposé XX

(5.1): 39.5 Examples of group schemes (“special case”)

Exposé XXI

(1.1.1): 10.5 Finite modules and finitely presented modules (“partial”), 11.2 Noncommutative algebras (“partial”), 9.7 Finite extensions (“partial”)
(1.1.6): 11.2 Noncommutative algebras, 9.7 Finite extensions
(6.7.2): 15.114 Galois extensions and ramification (“more general”)

Exposé XXII

(4.1.11): 59.11 Sheaves (“partial”), 7.10 Sheafification, 7.49 Sheafification in a topology
(5.1.1): 15.25 Content ideals (“more general”)
(5.3.0): 17.21 Symmetric and exterior powers (“partial”)
(5.3.1): 11.2 Noncommutative algebras, 9.7 Finite extensions
(5.4.10): 41.14 Topological properties of étale morphisms (“Footnote (36)”; “more general”), 29.27 Flat closed immersions (“more general”)
(5.9.10): 10.15 Miscellany (“partial”)

Exposé XXIV

(3.9.1): 10.15 Miscellany (“partial”)
(3.11.2): 35.37 Descending affine morphisms (“partial”; “special case”), 35.38 Descending quasi-affine morphisms (“partial”), 40.15 Descending ind-quasi-affine morphisms (“partial”)
(7.1.9): 4.6 Fibre products, 4.8 Fibre products and representability, 26.18 Base change in algebraic geometry
(8.1.2): 58.9 Finite étale covers of proper schemes (“more general”)

Exposé XXV

(4.2): 66.14 Obtaining a scheme (“special case”), 89.6 Base change to the completion (“more general”)
(4.3): 30.2 Čech cohomology of quasi-coherent sheaves (“partial”), 29.2 Closed immersions (“partial”), 29.11 Affine morphisms (“partial”), 33.19 Dimension of fibres (“special case”)

Exposé XXVI

(7.15): 26.18 Base change in algebraic geometry

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