SGA 4½

The items of SGA 4½ 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.

Arcata

• (1.1): 35.15 Local properties of schemes (“more general”), 7.6 Sites, 7.47 Topologies (“generated sieve”)
• (1.2.3): 59.11 Sheaves (“more general”), 26.15 A representability criterion (“more general”), 6.7 Sheaves (“more general”; “partial”), 6.8 Abelian sheaves (“more general”), 6.9 Sheaves of algebraic structures (“special case”), 7.6 Sites (“more general”), 7.47 Topologies (“more general”), 85.2 Simplicial topological spaces (“more general”), 34.9 The fpqc topology
• (1.3.4): 7.47 Topologies (“more general”)
• (1.4): 35.3 Descent for modules (“module case”; “module equivalence”), 35.5 Fpqc descent of quasi-coherent sheaves (“fpqc qcoh case”), 59.16 Faithfully flat descent (“fpqc qcoh case”; “module case”)
• (1.4.1): 10.39 Flat modules and flat ring maps (“special case”)
• (1.4.2): 10.23 Glueing properties (“(ii) fin pres, Zariski”; “(ii) fin type, Zariski”), 10.39 Flat modules and flat ring maps (“Clause (i), special case”; “more general”), 10.78 Finite projective modules, 10.83 Descent for finite projective modules (“(ii), fin. presentation”), 35.4 Descent for universally injective morphisms (“(ii) finite presentation”; “(ii) finite type”; “Clause (i), special case”)
• (1.4.3): 7.47 Topologies (“generated sieve”)
• (1.4.4): 8.3 Descent data in fibred categories
• (1.4.5): 35.3 Descent for modules (“affine case”; “ess. surj. (affine)”), 59.16 Faithfully flat descent (“ess. surj. (affine)”)
• (1.5.1): 35.6 Galois descent for quasi-coherent sheaves (“a), b), c) equivalence”)
• (1.5.3): 59.67 Galois cohomology (“more general”)
• (1.6.1): 7.47 Topologies (“generated sieve”; “restriction U_V”; “sieve generated by U_i”)
• (1.6.3): 4.3 Opposite Categories and the Yoneda Lemma, 59.9 Presheaves, 59.11 Sheaves (“topological site”), 59.14 Cohomology (“more general”), 19.7 Abelian Sheaves on a site (“enough injectives”), 19.9 Embedding abelian categories (“partial”), 6.3 Presheaves (“special case”), 6.4 Abelian presheaves (“special case”), 6.5 Presheaves of algebraic structures (“special case”), 6.7 Sheaves (“topological site”), 6.8 Abelian sheaves (“topological site”), 7.2 Presheaves, 7.6 Sites (“special case”; “topological site”), 7.7 Sheaves (“topological site”), 18.3 Abelian sheaves (“more general”), 85.2 Simplicial topological spaces (“topological site”), 34.9 The fpqc topology (“partial”)
• (1.6.4): 59.10 Sites (“partial”), 59.15 The fpqc topology (“special case”), 90.6 Prorepresentable functors and predeformation categories (“partial”), 7.6 Sites (“partial”)
• (1.6.5): 20.2 Cohomology of sheaves (“more general”), 21.2 Cohomology of sheaves (“more general”)
• (1.6.6): 20.4 First cohomology and torsors (“topological space”), 21.4 First cohomology and torsors (“H¹ = torsor classes”)
• (2.1.1): 41.16 The functorial characterization (“more general”), 59.26 Étale morphisms (“special case”), 37.8 Formally étale morphisms (“more general”), 29.37 Étale morphisms (“b) <=> c)”)
• (2.1.2): 10.143 Étale ring maps, 68.4 Finiteness conditions and points (“a)”), 59.26 Étale morphisms (“a)”), 15.107 Weakly étale algebras over fields (“partial”)
• (2.1.3): 59.26 Étale morphisms, 59.29 Neighborhoods, stalks and points, 29.37 Étale morphisms, 65.4 Lists of useful properties of morphisms of schemes
• (2.1.4): 59.29 Neighborhoods, stalks and points (“more general”), 66.18 The étale site of an algebraic space (“étale X-schemes”; “étale topology”)
• (2.2): 35.8 Quasi-coherent sheaves and topologies, I (“structure sheaf O_S”), 59.23 Examples of sheaves (“structure sheaf O_S”)
• (2.2.1): 59.23 Examples of sheaves, 7.10 Sheafification, 21.4 First cohomology and torsors (“H^1 = torsor classes”)
• (2.2.3): 20.6 First cohomology and invertible sheaves (“ringed space case”), 59.24 Picard groups, 17.25 Invertible modules, 21.6 First cohomology and invertible sheaves
• (2.2.5): 59.28 Kummer theory (“more general”)
• (2.3): 59.35 Direct images (“R^p f_ definition”; “f_ definition”)
• (2.3.1): 59.29 Neighborhoods, stalks and points (“special case”), 66.19 Points of the small étale site (“étale neighbourhood”), 66.22 Stalks of the structure sheaf (“partial, special case”; “special case”)
• (2.3.3): 20.2 Cohomology of sheaves (“more general”), 13.22 Composition of right derived functors (“more general”), 59.33 Stalks of the structure sheaf (“Ỹ = Spec(O_{Y”), 59.35 Direct images (“higher direct images”), 59.36 Inverse image (“(f^F)_x̄ = F_f(x̄)”), 59.51 Colimits, 7.21 Cocontinuous functors and morphisms of topoi (“left exactness of f_”), 21.2 Cohomology of sheaves (“more general”), 21.14 The Leray spectral sequence (“more general”), 18.21 Localization of ringed topoi (“more general”), 103.11 Pushforward of quasi-coherent modules (“more general”), 64.3 Frobenii (“special case”)
• (2.3.4): 13.22 Composition of right derived functors (“more general”), 59.54 The Leray spectral sequence (“first spectral sequence”), 21.14 The Leray spectral sequence (“more general”)
• (2.3.5): 20.13 The Leray spectral sequence (“H^p(X,F)=H^p(Y,f_F)”), 21.7 Locality of cohomology (“H^p(Y, special case”), 21.14 The Leray spectral sequence (“H^p(X,F)=H^p(Y,f_F)”)
• (2.3.6): 59.55 Vanishing of finite higher direct images (“more general”), 59.92 Applications of proper base change (“more general”), 69.4 Finite morphisms (“more general”)
• (2.4.1): 58.6 Fundamental groups
• (2.4.2): 10.143 Étale ring maps, 59.56 Galois action on stalks (“more general”), 15.46 Permanence of properties under henselization (“Only-if direction”), 15.107 Weakly étale algebras over fields (“more general”), 58.6 Fundamental groups (“partial”)
• (2.4.3): 58.5 Finite étale morphisms (“special case”), 58.6 Fundamental groups
• (2.4.4): 66.19 Points of the small étale site (“sep. closed field, G=1”)
• (3.1): 11.5 The Brauer group of a field (“abelian group structure”)
• (3.1.1): 11.2 Noncommutative algebras, 11.8 Splitting fields
• (3.1.2): 11.6 Skolem-Noether (“more general”), 59.61 Brauer groups (“definition of δ”; “δ is a homomorphism”; “δ is an isomorphism”; “δ is injective”; “δ is surjective”; “θ_n bijection”)
• (3.1.6): 59.67 Galois cohomology (“clause i), degree q=2”; “clause i), special case”; “more general”)
• (3.2.1): 59.67 Galois cohomology (“more general”)
• (3.2.2): 11.5 The Brauer group of a field (“special case”), 59.67 Galois cohomology (“special case”)
• (3.2.4): 59.67 Galois cohomology (“q = 2 (Brauer group)”)
• (3.3.1): 59.24 Picard groups (“more general”)
• (3.3.2): 59.68 Higher vanishing for the multiplicative group (“more general”)
• (3.3.5): 59.69 Picard groups of curves (“partial”)
• (3.3.6): 59.83 Cohomology of torsion sheaves on curves (“(ii), degree q=2”; “more general”)
• (3.4.3): 30.18 Chow's Lemma (“Chow's lemma step”)
• (4.1.1): 84.4 Proper base change (“more general”)
• (4.2.1): 58.9 Finite étale covers of proper schemes (“more general”)
• (4.2.2): 58.9 Finite étale covers of proper schemes (“special case”)
• (4.3): 28.5 Noetherian schemes
• (4.3.2): 59.23 Examples of sheaves, 59.71 Constructible sheaves (“(i)”), 59.74 Constructible sheaves on Noetherian schemes (“(i) ⇒ (ii)”), 7.10 Sheafification
• (4.3.3): 59.73 More on constructible sheaves (“more general”), 59.74 Constructible sheaves on Noetherian schemes (“more general”)
• (4.4): 59.63 The Artin-Schreier sequence (“H^q(X_0, special case”; “special case”)
• (4.5.1): 63.16 More on derived upper shriek (“more general”)
• (4.5.3): 38.33 Nagata compactification (“special case”), 63.9 Derived lower shriek via compactifications (“more general”)
• (4.5.4): 63.12 Compactly supported cohomology (“more general”; “special case”)
• (4.6.4): 59.95 Cohomological dimension (“more general”; “special case”)
• (5.1.1): 59.33 Stalks of the structure sheaf, 66.22 Stalks of the structure sheaf
• (5.1.4): 59.93 Local acyclicity (“more general”; “special case”)
• (5.1.6): 59.93 Local acyclicity (“more general”)
• (5.2.1): 59.93 Local acyclicity (“more general”)
• (5.2.4): 58.21 Purity of branch locus (“special case”)
• (5.3.1): 59.94 The cospecialization map (“more general”)
• (5.3.3): 59.90 Applications of smooth base change (“special case”)
• (6.2.1): 59.69 Picard groups of curves (“partial”)

Cycle

• (1.1): 21.4 First cohomology and torsors (“H^1 ≅ torsor classes”)
• (1.1.1): 20.4 First cohomology and torsors (“topological space”), 21.4 First cohomology and torsors (“as a group map”; “normalization by ∂h”; “the bijection”)
• (1.1.2): 20.11 Čech cohomology and cohomology (“cocycle ↔ torsor class”)
• (1.2.2): 20.34 Cohomology with support in a closed subset, II (“(1.2.2.2)”; “(1.2.2.3), special case”)
• (1.3): 10.14 Base change (“Clause a)”)
• (2.2.1): 10.106 Regular local rings (“special case”), 50.17 Blowing up and de Rham cohomology (“more general”)
• (2.3.1): 59.92 Applications of proper base change (“special case”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“special case”)
• (2.3.9): 43.14 Intersection multiplicities using Tor formula
• (3.5): 43.14 Intersection multiplicities using Tor formula

Dualite

• (2.3): 63.5 Weightings and trace maps for locally quasi-finite morphisms (“special case”)

Etat 0

• (1.1.0): 13.3 The definition of a triangulated category
• (1.1.2): 10.71 Ext groups (“special case”), 12.6 Extensions (“special case”), 15 More on Algebra: the chapter's own tag (“special case”)
• (1.2.1): 42.2 Periodic complexes and Herbrand quotients, 13.9 Cones and termwise split sequences, 12.14 Homotopy and the shift functor, 64.7 Filtered derived category
• (1.2.2): 13.10 Distinguished triangles in the homotopy category (“partial”), 22.10 The homotopy category is triangulated (“special case”), 22.27 Obtaining triangulated categories (“TR1, TR2, TR3”; “more general”), 64.7 Filtered derived category
• (1.2.3): 42.2 Periodic complexes and Herbrand quotients (“more general”), 13.6 Quotients of triangulated categories (“more general”), 13.8 The homotopy category (“bounded-above complexes”), 13.10 Distinguished triangles in the homotopy category (“(A)”), 13.11 Derived categories (“(A)”), 12.13 Complexes (“more general”), 64.7 Filtered derived category
• (1.2.4): 13.9 Cones and termwise split sequences, 13.10 Distinguished triangles in the homotopy category (“triangle from semi-split”; “ρ ess. surjective”; “ρ essentially surjective”), 22.8 Distinguished triangles (“splitting independence”), 22.32 Variant of derived Hom
• (1.3): 13.3 The definition of a triangulated category
• (1.3.2): 10.71 Ext groups (“partial”), 12.13 Complexes (“partial”)
• (1.3.3): 13.10 Distinguished triangles in the homotopy category (“partial”), 22.24 Linear categories (“partial”), 22.26 Differential graded categories (“H⁰(Hom•(A”; “The final clause of 3.3”), 90.11 Tangent spaces of functors (“partial”), 12.17 Additive monoidal categories (“special case”), 15.59 Tensor products of complexes (“tensor over ModuleCat R”), 21.17 Flat resolutions (“special case”)
• (2.1.1): 13.6 Quotients of triangulated categories (“partial”)
• (2.2): 13.5 Localization of triangulated categories (“partial”), 13.6 Quotients of triangulated categories (“S=φ(B) sat. mult. system”; “partial”; “thickness”), 12.10 Serre subcategories
• (2.3.1): 4.27 Localization in categories (“partial”), 13.6 Quotients of triangulated categories (“special case”)
• (2.3.2): 4.27 Localization in categories (“partial”), 13.5 Localization of triangulated categories (“partial”), 13.6 Quotients of triangulated categories (“(A_S”), 12.8 Localization (“partial”), 24.26 The derived category (“partial”)
• (2.3.3): 13.6 Quotients of triangulated categories (“more general”), 13.36 Generators of triangulated categories (“more general”)
• (2.4.1): 13.6 Quotients of triangulated categories (“more general”; “thickness”), 13.11 Derived categories (“case D = K(A), H = H^0”), 13.13 Filtered derived categories (“filtered complexes”)
• (2.4.2): 13.5 Localization of triangulated categories (“(a)”)
• (2.5): 13.6 Quotients of triangulated categories
• (2.6.2): 13.40 Admissible subcategories (“thickness”)
• (2.6.6): 13.40 Admissible subcategories
• (2.6.7): 13.7 Adjoints for exact functors (“more general”)
• (3.1.1): 13.11 Derived categories
• (3.1.2): 13.11 Derived categories
• (3.1.3): 13.11 Derived categories
• (3.1.4): 13.23 Resolution functors (“the equivalence”), 64.6 Derived categories
• (3.1.5): 13.12 The canonical delta-functor (“partial”)
• (3.1.6): 13.6 Quotients of triangulated categories (“factors through quotient”), 13.11 Derived categories (“H^i factors via D(A)”)
• (3.1.7): 12.13 Complexes
• (3.1.8): 10.71 Ext groups (“special case”), 13.18 Injective resolutions (“injective case”), 13.19 Projective resolutions (“projective case”)
• (3.1.9): 13.19 Projective resolutions (“Part 1, special case”), 13.32 Bounded cohomological dimension (“(2) left, σ=F-acyclics”; “(2) right, σ=F-acyclics”)
• (3.2.1): 13.27 Ext groups (“degree-0 objects”), 12.6 Extensions (“special case”)
• (3.2.2): 10.71 Ext groups (“Exactness at Ext^n(X”; “Exactness at Ext^{n+1}(V”; “special case”), 12.6 Extensions (“special case”), 15 More on Algebra: the chapter's own tag (“special case”)
• (3.2.3): 13.18 Injective resolutions (“(4) existence”), 13.19 Projective resolutions (“partial”), 13.23 Resolution functors (“(4) existence”; “(4) uniqueness”), 64.6 Derived categories (“(3)”; “(4) proj. analogue”)
• (4.1): 13.16 Higher derived functors (“right derived R^iF”)
• (4.1.2): 64.6 Derived categories (“left derived functor”; “right derived functor”)
• (4.1.3): 64.6 Derived categories (“L⁻ f”)
• (4.1.4): 64.6 Derived categories (“LF”; “RF”)
• (4.1.5): 20.2 Cohomology of sheaves (“special case”), 13.16 Higher derived functors (“R^i f”), 21.2 Cohomology of sheaves (“special case”), 64.6 Derived categories (“L^- f”)
• (4.2.3): 13.6 Quotients of triangulated categories (“(b)”), 13.11 Derived categories (“partial”)
• (4.2.3.1): 13.18 Injective resolutions (“Corollary 1”), 13.19 Projective resolutions (“Corollary 1”)
• (4.2.3.2): 13.15 Derived functors on derived categories (“(A)”; “SES-stable σ, + case”; “σ left f-acyclic”; “σ right f-acyclic”), 13.32 Bounded cohomological dimension (“left case: Lf on D(A)”; “right case: Rf on D(A)”)
• (4.3.1): 13.22 Composition of right derived functors (“enough injectives”)
• (6.2): 10.12 Tensor products (“special case”), 10.39 Flat modules and flat ring maps (“special case”), 20.26 Flat resolutions (“Tor_i, ringed space”), 41.9 Flat morphisms (“special case”), 21.17 Flat resolutions (“X ⊗^L Y on a site”; “existence of ⊗^L”; “⊗^L via K-flat”)
• (Problem1): 13.6 Quotients of triangulated categories
• (Problem2): 4.27 Localization in categories (“partial”), 13.5 Localization of triangulated categories (“A_S exists, Q exact”; “G is exact”), 13.6 Quotients of triangulated categories (“partial”), 24.26 The derived category (“exactness of Q, partial”)

Fonctions L

• (1.4): 14.25 Dold-Kan for cosimplicial objects (“N is an equivalence”), 14.27 Homotopies in abelian categories (“N preserves homotopy”)
• (2.2): 64.20 Cohomological interpretation (“Theorem 2.2 case a)”)
• (3.4): 41.14 Topological properties of étale morphisms, 37.64 Weakly étale morphisms (“more general”), 33.36 Frobenii (“F_abs definition”)
• (3.7): 48.27 Duality for proper schemes over fields (“special case”)
• (3.8): 10.14 Base change
• (4.5): 33.16 Tangent spaces

Rapport

• (1.1): 64.3 Frobenii, 33.36 Frobenii (“q = p”)
• (1.2): 37.64 Weakly étale morphisms (“special case”), 64.3 Frobenii
• (1.3): 33.36 Frobenii (“special case”)
• (1.4): 64.3 Frobenii (“special case”)
• (1.6): 64.19 L-functions (“Euler product”; “L-function of a sheaf”; “adic coefficients”; “partial”)
• (2.1): 64.18 On l-adic sheaves (“Z_ℓ-sheaf definition”)
• (2.2): 64.18 On l-adic sheaves (“fiber definition”)
• (2.5): 61.28 Constructible adic sheaves (“more general”)
• (2.6): 64.18 On l-adic sheaves (“abelian”; “ess.const ⇒ Zℓ-sheaf”; “hom(F,G)”)
• (2.9): 64.18 On l-adic sheaves (“more general”)
• (2.10): 64.18 On l-adic sheaves, 64.20 Cohomological interpretation (“special case”)
• (3.2): 64.14 Lefschetz numbers (“partial”), 64.16 Proof of the trace formula (“special case”)
• (3.6): 63.12 Compactly supported cohomology
• (4.1): 11.2 Noncommutative algebras (“partial”), 13.11 Derived categories, 15.58 Computing Tor (“special case”)
• (4.2): 64.4 Traces (“free module tr”), 64.11 Filtrations and perfect complexes (“(4.2.2) ungraded”)
• (4.3): 15.75 Derived hom (“more general”), 64.7 Filtered derived category (“arrows of K_parf(Λ)”), 64.10 Perfectness (“D_parf(Λ)”; “K_parf(Λ)”)
• (4.4): 13.13 Filtered derived categories (“filtered quasi-iso”), 64.11 Filtrations and perfect complexes (“(4.4.1)”)
• (4.5): 20.48 Tor dimension (“sheaf case, finite”; “sheaf module case”; “sheaf tor-dimension ≤ r”), 15.68 Tor dimension (“finite tor dimension”; “module case”; “tor-dimension ≤ r”), 21.46 Tor dimension (“sheaf case, finite”; “sheaf module case”; “sheaf tor-dimension ≤ r”), 64.12 Characterizing perfect objects (“finite tor dimension”)
• (4.5.1): 64.12 Characterizing perfect objects
• (4.6): 59.77 Tor finite with constructible cohomology (“ii)”; “partial”; “special case”), 64.14 Lefschetz numbers (“D_ctf(X, Λ)”)
• (4.7): 59.76 Complexes with constructible cohomology (“(1) ⇒ (3)”; “partial”)
• (4.10): 64.20 Cohomological interpretation (“special case”)
• (4.11): 10.5 Finite modules and finitely presented modules, 10.86 Mittag-Leffler systems, 10.102 What makes a complex exact?, 11.2 Noncommutative algebras, 9.7 Finite extensions, 12.31 Inverse systems
• (5.5): 64.15 Preliminaries and sorites (“tr_Λ^H definition”; “ε factorization”)
• (5.6): 64.15 Preliminaries and sorites
• (5.7): 64.15 Preliminaries and sorites (“more general”)
• (5.8): 64.15 Preliminaries and sorites
• (5.9): 64.15 Preliminaries and sorites (“(5.9.1)”; “(5.9.2)”; “P⊗M projective”)
• (5.10): 64.15 Preliminaries and sorites (“P_H projective over Λ”; “action of G^+ on P_H”)
• (5.11): 64.15 Preliminaries and sorites
• (5.13): 64.20 Cohomological interpretation (“more general”)

Sommes trig

• (1.3): 21.4 First cohomology and torsors (“partial”)
• (1.18): 64.29 Cohomology of curves, revisited (“Clause b)”)
• (4.5): 10.143 Étale ring maps (“partial”)
• (4.6): 27.6 Vector bundles (“partial”)
• (4.7): 59.28 Kummer theory (“special case”)
• (6.1): 10.143 Étale ring maps (“partial”)

Th finitude

• (1.3): 59.54 The Leray spectral sequence (“partial”), 59.92 Applications of proper base change (“special case”), 21.14 The Leray spectral sequence (“more general”)
• (1.7): 59.77 Tor finite with constructible cohomology
• (1.11): 59.97 Künneth in étale cohomology (“Künneth arrow RΓ(X”; “special case”)
• (2.7): 59.45 Topological invariance of the small étale site (“more general”; “partial”), 58.8 Topological invariance of the fundamental group (“partial”), 33.25 Schemes smooth over fields (“Perfect-field clause”)
• (2.8): 38.33 Nagata compactification (“more general”)
• (2.11): 59.33 Stalks of the structure sheaf
• (3.9): 59.90 Applications of smooth base change (“more general”)
• (5.1.3): 59.91 The proper base change theorem (“1.3.1, special case”)
• (5.2.4): 59.94 The cospecialization map (“for proper f”; “partial”)

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