SGA 7

The items of SGA 7 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

(0.0.2): 59.32 Henselian rings (“ring case”)
(0.0.3): 10.153 Henselian local rings (“partial”)
(0.0.4): 59.33 Stalks of the structure sheaf (“strict henselization”)
(0.0.5): 9.6 Field extensions
(0.1): 64.18 On l-adic sheaves (“b)”; “more general”)
(0.1.1): 64.20 Cohomological interpretation (“Q_ℓ clause of case 0.1.1”)
(0.1.5): 63.15 Applications (“special case”)
(0.3): 59.56 Galois action on stalks, 15.114 Galois extensions and ramification (“finite Galois case”; “partial”), 58.13 Ramification theory (“I/P ≅ Ẑ(1), P pro-p”)
(0.5): 16.4 Intermezzo: Néron desingularization (“Lemma 0.5.1”)
(0.5.1): 16.4 Intermezzo: Néron desingularization (“ind-smooth colimit”)
(0.5.3): 10.160 The Cohen structure theorem
(1.3): 15.22 Torsion free modules (“partial”)
(2.1): 59.94 The cospecialization map (“more general”)
(2.4): 59.93 Local acyclicity (“partial”)
(3.5): 55.8 Models (“partial”)
(5.1): 109.22 Stable curves (“(2), special case”; “special case”)
(5.2): 58.14 Geometric and arithmetic fundamental groups (“more general”; “surjectivity of π₁(X_η”)
(6.3): 64.3 Frobenii

Exposé VII

(1.3.8): 10.39 Flat modules and flat ring maps (“flatness clause”), 42.15 Push and pull (“partial”), 41.9 Flat morphisms (“flatness of p”), 59.15 The fpqc topology (“special case”), 56.5 Functors between categories of quasi-coherent modules (“partial”), 39.4 Group schemes (“flatness of p”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“partial”), 29.15 Morphisms of finite type (“partial”), 29.21 Quasi-finite morphisms (“partial”), 29.26 Flat morphisms (“flatness of p”; “partial”), 26.19 Quasi-compact morphisms (“partial”), 26.21 Separation axioms (“partial”), 7.12 Representable sheaves, 65.4 Lists of useful properties of morphisms of schemes (“partial”), 33.20 Algebraic schemes (“partial”)
(1.4): 17.4 Sections of sheaves of modules (“special case”)
(2.0): 7.30 Localization of topoi (“partial”)
(2.9.6): 39.4 Group schemes (“partial”)
(3.1): 12.5 Abelian categories (“more general”)
(3.1.7): 10.71 Ext groups (“special case”)
(3.2): 13.27 Ext groups (“c(X, special case”), 12.6 Extensions (“special case”)
(3.2.5): 13.27 Ext groups (“partial”)
(3.3.6): 10.71 Ext groups (“Exactness at Extⁿ(L•,A)”; “Exactness at Extⁿ(L•,A″)”; “Exactness at Extⁿ⁺¹(L•”; “partial”), 12.6 Extensions (“partial”)
(3.4): 15.56 Injective modules (“special case”)
(3.4.9): 10.71 Ext groups (“partial”), 15 More on Algebra: the chapter's own tag (“partial”)
(3.5): 6.4 Abelian presheaves (“special case”)
(3.7.5): 10.71 Ext groups (“Exactness at Ext^n(P”; “Exactness at Ext^{n+1}(P”; “special case”), 12.6 Extensions (“special case”), 12.12 Cohomological delta-functors, 15 More on Algebra: the chapter's own tag (“special case”)
(3.7.6): 10.39 Flat modules and flat ring maps (“partial”), 41.9 Flat morphisms (“partial”)

Exposé VIII

(0.2): 34.7 The fppf topology (“partial”)
(1.1): 20.43 Ext sheaves (“partial”), 15.75 Derived hom (“(1.1.2) in D(R)”)
(1.6): 4.43 Monoidal categories (“more general”)
(2.1): 15.63 Spectral sequences for Tor (“special case”)
(4.1): 39.9 Abelian varieties (“Proper group schemes”)
(5.1): 59.73 More on constructible sheaves (“special case”)
(5.2): 42.13 Preparation for flat pullback (“partial”), 62.6 Relative cycles (“special case”)
(6.5.1): 15.123 A regular local ring is a UFD (“special case”)
(7.0): 26.18 Base change in algebraic geometry

Exposé X

(1.8): 55.9 The geometry of a regular model (“partial”; “special case”)
(1.10): 54.16 Contracting exceptional curves (“more general”)
(1.13): 31.15 Effective Cartier divisors and invertible sheaves (“more general”)
(1.14): 54.16 Contracting exceptional curves (“partial”)
(1.15): 55.9 The geometry of a regular model (“partial”)
(2.4): 31.34 Strict transform (“more general”)
(2.6): 31.33 Blowing up (“partial”), 43.15 Algebraic multiplicities
(2.10): 33.39 The delta invariant
(2.24): 53.19 Nodal curves (“more general”)
(2.26): 10.36 Finite and integral ring extensions, 10.131 Differentials, 15.22 Torsion free modules, 33.39 The delta invariant (“partial”)
(2.27): 93.9 Schemes (“more general”)

Exposé XI

(1.3): 37.7 Universal first order thickenings (“partial”), 37.11 Formally smooth morphisms (“partial”), 29.33 Sheaf of differentials of a morphism (“partial”), 29.35 Smooth morphisms (“partial”)

Exposé XII

(1.9): 22.3 Differential graded algebras (“more general”)

Exposé XIII

(0.2): 10.36 Finite and integral ring extensions (“partial”), 59.11 Sheaves (“more general”), 59.32 Henselian rings (“(0.2.3) ring case”), 59.56 Galois action on stalks (“(0.2.5)”), 15.114 Galois extensions and ramification
(1.1.3): 59.56 Galois action on stalks (“Y = Spec k”)
(1.2.1): 10.153 Henselian local rings (“the equivalence”), 58.8 Topological invariance of the fundamental group (“FÉt equivalence”), 58.9 Finite étale covers of proper schemes (“more general”)
(1.2.2): 59.56 Galois action on stalks (“(a)”)
(1.4.2): 13.9 Cones and termwise split sequences (“partial”), 13.10 Distinguished triangles in the homotopy category (“partial”)
(2.1.3): 59.33 Stalks of the structure sheaf (“The scheme X_(x̄)”)
(2.1.6): 38.33 Nagata compactification (“2.1.6 c), special case”; “more general”), 63.9 Derived lower shriek via compactifications (“more general”), 63.11 Derived upper shriek (“more general”), 63.16 More on derived upper shriek (“more general”)

Exposé XIV

(2.1): 39.4 Group schemes (“partial”), 29.35 Smooth morphisms (“partial”)
(4.5): 10.16 Cayley-Hamilton (“forward implication”), 26.17 Fibre products of schemes (“partial”), 18.19 Localization of ringed sites (“more general”), 18.21 Localization of ringed topoi (“more general”)

Exposé XIX

(1.2): 30.8 Cohomology of projective space (“special case”)
(1.4): 27.22 Grassmannians (“partial”), 59.29 Neighborhoods, stalks and points, 58.10 Local connectedness (“partial”), 58.11 Fundamental groups of normal schemes (“π1(ℓ−S”)

Exposé XV

(1.2): 5.10 Krull dimension
(1.2.1): 53.16 Multicross and nodal singularities (“dim 1, alg. closed k”)
(1.2.5): 53.19 Nodal curves (“ordinary case”)
(2.1.1): 27.7 Cones (“more general”)
(2.2.5): 58.13 Ramification theory (“partial”)

Exposé XVI

(1.6): 59.33 Stalks of the structure sheaf (“S_(s̄) only”), 59.56 Galois action on stalks
(1.13): 55.14 Semistable reduction (“Case 1, special case”)
(2.1): 45.10 Further properties (“partial”)

Exposé XVII

(1.1): 53.16 Multicross and nodal singularities (“n=2, k alg. closed”)
(5.1): 50.11 de Rham cohomology of projective space (“more general”), 37.11 Formally smooth morphisms (“more general”)
(5.2): 43.27 Pullback for a general morphism (“partial”)
(5.3): 42.36 Projective space bundle formula (“more general”)

Exposé XVIII

(2.1): 31.33 Blowing up (“more general”), 37.17 Closed immersions between smooth schemes (“Smoothness of tilde X”)
(3.1): 37.17 Closed immersions between smooth schemes (“more general”)
(5.1): 26.18 Base change in algebraic geometry (“k is a preimmersion”; “partial”)
(5.6.8): 29.35 Smooth morphisms
(6.1.6.1): 37.32 Bertini theorems (“Case s = 1 of 6.1.6.1”; “special case”)

Exposé XX

(4.4): 45.3 Correspondences (“partial”)

Exposé XXI

(1.0): 33.36 Frobenii
(2.0): 64.18 On l-adic sheaves (“more general”)
(5.0): 10.9 Localization (“more general”), 64.3 Frobenii (“F_x acts on G_x̄”)

Exposé XXII

(1.2): 59.63 The Artin-Schreier sequence (“alg. closed, q=p”)
(2.2): 26.17 Fibre products of schemes, 64.3 Frobenii (“more general”)
(2.2.4): 64.3 Frobenii (“The element φ_q”)
(3.3): 50.6 Two spectral sequences (“First page of (3.3.2)”), 50.7 The Hodge filtration (“more general”), 50.12 The spectral sequence for a smooth morphism (“more general”)
(3.4): 63.12 Compactly supported cohomology (“more general”)
(6.2): 53.9 Plane curves (“(6.2.5), special case”), 33.36 Frobenii (“(6.2.1) F_q for q = p”)

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