SGA 4

The items of SGA 4 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 IX

(1.2): 59.78 Torsion sheaves (“(ii), (v)”), 7.17 Quasi-compact objects and colimits
(2.0): 59.23 Examples of sheaves, 59.64 Locally constant sheaves (“locally constant”), 6.3 Presheaves (“partial”), 6.7 Sheaves (“special case”), 7.10 Sheafification, 18.43 Locally constant sheaves (“locally constant”)
(2.1): 59.64 Locally constant sheaves (“(i), étale site”; “(ii) map, étale abelian”; “(ii) map, étale site”), 18.43 Locally constant sheaves (“(i)”; “(ii) ker/coker, (iii)”; “(ii) map, abelian”)
(2.2): 59.64 Locally constant sheaves
(2.3): 10.71 Ext groups (“partial”), 59.71 Constructible sheaves (“A-modules, A noetherian”; “sheaves of groups”; “sheaves of sets”; “special case”)
(2.4): 59.71 Constructible sheaves (“(i)”; “(i), (ii)”; “(i), modules”; “special case”)
(2.6): 59.71 Constructible sheaves (“(i bis) finite colimits”; “(i bis) finite limits”; “(ii)+ker/coker abelian”)
(2.7): 59.73 More on constructible sheaves (“Module sheaves”; “Sheaves of sets”; “special case”)
(2.7.1): 59.73 More on constructible sheaves (“Sufficiency”)
(2.7.4): 59.73 More on constructible sheaves (“partial”; “partial, special case”)
(2.9): 59.73 More on constructible sheaves (“(ii)”), 59.74 Constructible sheaves on Noetherian schemes (“(ii)”; “(ii), special case”)
(2.10): 59.74 Constructible sheaves on Noetherian schemes
(2.13): 59.75 Specializations and étale sheaves (“(i)”; “(i), special case”)
(2.14): 59.73 More on constructible sheaves (“(i), A-module variant”; “(i), sets variant”; “(i), special case”), 59.74 Constructible sheaves on Noetherian schemes (“(ii), A-modules”; “(ii), sheaves of sets”; “(ii), special case”)
(2.14.1): 59.73 More on constructible sheaves (“special case”), 28.16 Local irreducibility
(2.14.2): 28.13 Japanese and Nagata schemes
(3.2): 59.28 Kummer theory (“big étale site”)
(3.3): 59.24 Picard groups, 17.25 Invertible modules
(3.5): 59.63 The Artin-Schreier sequence
(4.0): 29.50 Rational maps (“partial”; “special case”)
(4.1.1): 59.53 Stalks of higher direct images (“more general”)
(4.2): 59.54 The Leray spectral sequence (“more general”)
(5.1): 63.5 Weightings and trace maps for locally quasi-finite morphisms (“more general”)
(5.7): 59.83 Cohomology of torsion sheaves on curves (“special case”)

Exposé V

(0.1.1): 59.14 Cohomology, 12.27 Injectives, 19.4 Abelian sheaves on a space (“sheaves on a space”), 19.7 Abelian Sheaves on a site (“sheaves on a site”), 19.8 Modules on a ringed site, 19.11 Injectives in Grothendieck categories (“enough injectives”), 19.13 Additional remarks on Grothendieck abelian categories
(0.2): 12.29 Injectives and adjoint functors (“1) (i) ⇒ (ii)”; “1) (i)⇔(ii)”; “2)”)
(0.2.1): 59.14 Cohomology, 19.7 Abelian Sheaves on a site (“special case”), 19.8 Modules on a ringed site, 66.32 Quasi-coherent sheaves and presentations (“special case”)
(0.3): 13.22 Composition of right derived functors (“(ii) ⇒ (i)”)
(0.4): 13.15 Derived functors on derived categories (“M acyclic for u”), 13.16 Higher derived functors
(1.1): 10.12 Tensor products (“special case”), 10.39 Flat modules and flat ring maps (“special case”), 41.9 Flat morphisms (“special case”), 17.17 Flat modules (“commutative case”), 18.28 Flat modules (“commutative case”), 18.31 Flat morphisms (“commutative case”)
(1.2): 10.39 Flat modules and flat ring maps (“special case”), 10.81 Characterizing flatness (“Reverse implication only”), 35.4 Descent for universally injective morphisms (“(ii) ⇒ (i)”)
(1.6): 18.39 Pullbacks of flat modules (“2)”; “assertion 1”)
(1.6.1): 18.28 Flat modules
(1.7): 17.20 Flat morphisms of ringed spaces (“(ii) ⇒ (i)”)
(1.7.1): 10.39 Flat modules and flat ring maps (“special case”), 35.8 Quasi-coherent sheaves and topologies, I (“special case”), 17.17 Flat modules (“ringed spaces”), 18.39 Pullbacks of flat modules (“representably flat u”)
(1.7.2): 10.39 Flat modules and flat ring maps (“(i) ⇔ (ii), special case”), 41.9 Flat morphisms (“(i) ⇔ (ii), special case”)
(1.8): 18.31 Flat morphisms (“left flat”)
(1.10): 14.2 The category of finite ordered sets, 14.11 Simplicial sets, 85.2 Simplicial topological spaces (“special case”)
(1.11): 59.18 Čech cohomology (“partial”), 21.9 Čech cohomology as a functor on presheaves (“partial”)
(2.1.1): 10.71 Ext groups (“Exactness Ext^n(X2”; “Exactness Ext^n(X3”; “more general”; “partial”), 12.6 Extensions, 15 More on Algebra: the chapter's own tag (“partial”)
(2.2): 21.7 Locality of cohomology (“special case”)
(2.3.2): 21.7 Locality of cohomology (“more general”)
(2.3.6): 21.13 Totally acyclic sheaves (“e)”)
(2.4.1): 7.12 Representable sheaves
(2.4.2): 20.2 Cohomology of sheaves, 21.2 Cohomology of sheaves, 21.10 Čech cohomology and cohomology (“special case”), 21.12 Cohomology of modules (“(X), special case”; “special case”)
(2.4.3): 20.9 The Čech complex and Čech cohomology (“more general”), 59.18 Čech cohomology (“Čech cohomology of 𝒰”; “Čech complex”), 21.8 The Čech complex and Čech cohomology (“Čech cohomology of 𝒰”; “Čech complex”)
(2.4.5): 59.13 Sheafification (“special case”), 7.10 Sheafification (“special case”)
(2.5.2): 59.14 Cohomology (“more general”), 19.7 Abelian Sheaves on a site (“special case”), 19.8 Modules on a ringed site (“more general”)
(3.1): 20.7 Locality of cohomology (“special case”), 59.19 The Čech-to-cohomology spectral sequence (“special case”)
(3.2): 20.11 Čech cohomology and cohomology (“ringed space, open cover”), 59.19 The Čech-to-cohomology spectral sequence (“R = sieve of a family”; “special case”), 21.10 Čech cohomology and cohomology (“E₂-page (covering)”; “abutment (covering)”)
(3.3): 21.10 Čech cohomology and cohomology (“partial”)
(3.5): 6.10 Sheaves of modules, 21.5 First cohomology and extensions, 21.12 Cohomology of modules, 21.13 Totally acyclic sheaves (“H^q isomorphisms”), 18.10 Sheaves of modules, 18.14 The abelian category of sheaves of modules (“left exactness of M ↦ ℳ”)
(3.6): 25.6 Hypercoverings a la Verdier (“more general”), 21.10 Čech cohomology and cohomology (“more general”)
(4.1): 21.13 Totally acyclic sheaves (“flasque case”)
(4.4): 21.13 Totally acyclic sheaves
(4.6): 20.12 Flasque sheaves (“special case”), 21.14 The Leray spectral sequence (“flasque ⇒ Γ-acyclic”)
(4.9): 15.57 Derived categories of modules (“special case”), 21.14 The Leray spectral sequence (“clause 3”), 21.20 Some properties of K-injective complexes (“more general”)
(4.10): 20.12 Flasque sheaves (“2), F=A, ringed space”), 19.8 Modules on a ringed site (“1), A = Z”)
(4.11): 21.7 Locality of cohomology (“1) injective case”; “1) injective, etale site”)
(5.0): 6.26 Morphisms of ringed spaces and modules (“special case”), 18.14 The abelian category of sheaves of modules (“left exactness of u_*”)
(5.1): 20.7 Locality of cohomology (“Clause 3, special case”), 7.31 Localization and morphisms of topoi, 21.7 Locality of cohomology (“1)”), 96.21 Higher direct images and algebraic stacks (“clause 1”)
(5.2): 20.12 Flasque sheaves (“special case”), 21.14 The Leray spectral sequence (“totally acyclic modules”)
(5.3): 59.54 The Leray spectral sequence (“small étale sites”), 21.14 The Leray spectral sequence (“more general”), 103.11 Pushforward of quasi-coherent modules (“more general”)
(5.4): 20.13 The Leray spectral sequence (“convergence, ringed sp”; “ringed spaces”), 59.54 The Leray spectral sequence (“étale sites, Z = point”), 21.14 The Leray spectral sequence, 103.11 Pushforward of quasi-coherent modules (“more general”; “special case”)
(6.0): 10.71 Ext groups (“6.0.3, 6.0.4”), 20.2 Cohomology of sheaves (“more general”), 20.43 Ext sheaves (“6.0.1, special case”; “special case”), 17.22 Internal Hom (“6.0, special case”), 21.2 Cohomology of sheaves (“more general”)
(6.2): 10.71 Ext groups (“Exactness of (6.2.1)”; “Exactness of (6.2.2)”; “partial”), 12.6 Extensions (“Exactness of (6.2.1)”), 15 More on Algebra: the chapter's own tag (“Exactness of (6.2.2)”)
(6.3): 10.71 Ext groups (“(6.3.1)”), 17.6 Closed immersions and abelian sheaves (“special case”)
(6.5): 51.2 Generalities (“(6.5.3), special case”), 51.8 Finiteness of pushforwards, I (“(6.5.3), special case”; “special case”)
(6.13): 6.10 Sheaves of modules, 18.10 Sheaves of modules
(7.1.0): 14.3 Simplicial objects, 14.12 Truncated simplicial objects and skeleton functors, 14.19 Coskeleton functors (“adjunction i_n ⊣ i_{n}”; “partial”), 14.21 Left adjoints to the skeleton functors (“existence of i_{n!}”), 7.19 More functoriality of presheaves (“more general”), 85.2 Simplicial topological spaces
(7.1.2): 14.19 Coskeleton functors (“1) cosk composition”)
(7.2.1): 25.4 Acyclicity (“homology iso”)
(7.3.0): 59.11 Sheaves (“partial”), 25.2 Semi-representable objects (“equalizers in SR(C)”)
(7.3.1.2): 25.6 Hypercoverings a la Verdier
(7.3.1.3): 25.6 Hypercoverings a la Verdier (“a)”)
(7.3.1.6): 59.9 Presheaves (“more general”), 14.11 Simplicial sets
(7.3.2): 25.4 Acyclicity (“3) degree zero”; “3) positive degrees”; “more general”), 25.7 Covering hypercoverings (“2)”)
(7.3.3): 25.3 Hypercoverings (“object case, no type p”)
(7.3.6.0): 14.17 Hom from simplicial sets into simplicial objects (“U finite, C fin. limits”)
(7.4.0): 59.19 The Čech-to-cohomology spectral sequence (“Čech covering case”), 25.5 Čech cohomology and hypercoverings (“(7.4.0.3)”; “(7.4.0.3) E₂ page”; “(7.4.0.3) abutment”), 25.6 Hypercoverings a la Verdier (“(7.4.0.3)”)
(7.4.1): 25.10 Cohomology and hypercoverings (“Clause 2)”)
(8.1.0): 4.32 Categories over categories (“partial”)
(8.1.4): 4.17 Cofinal and initial categories (“special case”)
(8.1.5): 4.21 Limits and colimits over preordered sets (“special case”)
(8.1.12): 6.30 Bases and sheaves (“special case”)
(8.2.4): 10.12 Tensor products (“special case”)
(8.2.9): 10.39 Flat modules and flat ring maps (“special case”), 18.39 Pullbacks of flat modules (“representably flat u”)
(8.2.10): 15.67 Pseudo-coherent modules, II (“modules over a ring”)
(8.2.11): 18.28 Flat modules (“(a), special case”)
(8.2.12): 10.81 Characterizing flatness (“special case”)

Exposé Vbis

(1.2.6): 4.3 Opposite Categories and the Yoneda Lemma, 59.9 Presheaves, 14.3 Simplicial objects, 14.20 Augmentations, 7.2 Presheaves, 85.2 Simplicial topological spaces
(1.2.7): 14.3 Simplicial objects, 85.2 Simplicial topological spaces (“special case”)
(1.3.1): 6.25 Ringed spaces (“topological space case”), 18.7 Ringed topoi (“absolute case (D = pt)”)
(1.3.2): 18.7 Ringed topoi (“absolute case (D = pt)”)
(1.3.6): 6.24 Continuous maps and sheaves of modules (“special case”), 6.26 Morphisms of ringed spaces and modules (“ringed spaces”; “ringed spaces, φ^”; “ringed spaces, φ_”), 18.13 Morphisms of ringed topoi and modules (“special case”), 18.21 Localization of ringed topoi (“special case”)
(1.3.7): 6.31 Open immersions and (pre)sheaves (“special case”), 18.19 Localization of ringed sites (“special case”)
(1.3.8): 13.11 Derived categories
(2.1.1): 14.5 Cosimplicial objects (“special case”)
(2.2.2): 35.4 Descent for universally injective morphisms (“modules over rings”)
(2.2.7): 85.17 Cohomological descent for hypercoverings (“special case”), 85.18 Cohomological descent for hypercoverings: modules (“special case”)
(2.3.2): 12.18 Double complexes and associated total complexes (“differential formula”)
(2.3.3): 64.7 Filtered derived category (“(A)”)
(2.4.0): 17.3 The abelian category of sheaves of modules (“The fibers of Mod(E°”), 6.10 Sheaves of modules (“partial”), 18.10 Sheaves of modules (“partial”), 18.14 The abelian category of sheaves of modules (“The fibers of Mod(E°”)
(2.4.0.1): 17.20 Flat morphisms of ringed spaces (“ringed spaces”)
(2.4.1): 13.11 Derived categories
(2.4.3.1): 18.31 Flat morphisms (“non-fibered case”)
(2.4.10): 35.4 Descent for universally injective morphisms (“modules over a ring map”)
(2.4.12): 85.18 Cohomological descent for hypercoverings: modules (“special case”)
(2.5.1): 13.10 Distinguished triangles in the homotopy category (“more general”; “partial”), 13.13 Filtered derived categories (“Σ saturated”; “Σ: Gr_n quasi-iso”), 22.27 Obtaining triangulated categories (“more general”), 12.19 Filtrations (“more general”; “partial”), 64.7 Filtered derived category (“D^+F(A)”; “K⁺(F(A))”; “more general”; “partial”)
(2.5.2): 13.13 Filtered derived categories (“more general”), 12.24 Spectral sequences: filtered complexes (“partial”)
(2.5.6): 25.5 Čech cohomology and hypercoverings (“E_2 term”)
(3.0.1): 14.3 Simplicial objects (“special case”), 14.12 Truncated simplicial objects and skeleton functors, 14.19 Coskeleton functors (“more general”; “partial”), 85.2 Simplicial topological spaces
(3.0.2): 14.26 Homotopies
(3.0.2.2): 14.28 Homotopies and cosimplicial objects (“Dual form, special case”)
(3.0.2.4): 14.32 A homotopy equivalence (“simplicial sets”)
(3.2.3): 14.3 Simplicial objects
(3.3): 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”), 26.18 Base change in algebraic geometry, 65.4 Lists of useful properties of morphisms of schemes (“special case”)
(4.1.0): 6.7 Sheaves (“partial”), 6.10 Sheaves of modules (“fiber G_X = Mod(Z_X)”), 18.10 Sheaves of modules (“fiber G_X = Mod(Z_X)”), 5.4 Separated maps, 5.17 Characterizing proper maps (“proper morphism def”)
(4.1.1): 20.18 Proper base change in topology (“more general”)
(4.1.3): 20.16 Cohomology on Hausdorff quasi-compact spaces (“i = 0”)
(4.1.4): 5.12 Quasi-compact spaces and maps (“case A ∩ B = ∅”)
(4.2.0): 17.10 Quasi-coherent modules (“Fiber G_X”), 29.15 Morphisms of finite type (“partial”), 29.21 Quasi-finite morphisms (“partial”), 26.19 Quasi-compact morphisms (“partial”), 26.21 Separation axioms (“partial”), 6.7 Sheaves (“Fiber of E over a scheme”), 6.8 Abelian sheaves (“Fiber of E over a scheme”), 85.2 Simplicial topological spaces (“Fiber of E over a scheme”), 33.20 Algebraic schemes (“partial”)
(4.3.0): 6.10 Sheaves of modules (“fiber category F_{n_X}”), 18.10 Sheaves of modules (“fiber category F_{n_X}”)
(4.3.4): 59.55 Vanishing of finite higher direct images (“more general”), 59.92 Applications of proper base change (“more general”), 69.4 Finite morphisms (“Remarque 4.3.4”; “more general”)
(5.1.0): 14.2 The category of finite ordered sets, 14.3 Simplicial objects, 14.11 Simplicial sets, 85.2 Simplicial topological spaces
(5.2.0): 29.15 Morphisms of finite type (“partial”), 29.21 Quasi-finite morphisms (“partial”), 33.20 Algebraic schemes (“partial”)
(5.3.1): 48.16 Upper shriek functors

Exposé VI

(1.1): 7.17 Quasi-compact objects and colimits
(1.2): 7.17 Quasi-compact objects and colimits
(1.3): 7.17 Quasi-compact objects and colimits (“special case”)
(1.4): 7.17 Quasi-compact objects and colimits (“partial”)
(1.7): 67.4 Separation axioms (“quasi-separated”)
(1.8): 56.5 Functors between categories of quasi-coherent modules (“(ii), special case”; “(iv), special case”), 29.15 Morphisms of finite type (“(i), special case”; “(ii), special case”), 29.21 Quasi-finite morphisms (“(ii), special case”), 29.22 Morphisms of finite presentation (“(i), special case”), 26.19 Quasi-compact morphisms (“(i), special case”; “(ii), special case”), 26.21 Separation axioms (“(i), special case”; “(ii), special case”; “(iii), special case”), 65.4 Lists of useful properties of morphisms of schemes (“(i), special case”; “(ii), special case”)
(1.9.1): 56.5 Functors between categories of quasi-coherent modules (“(ii), special case”; “(iv), special case”), 29.3 Immersions (“(iv), special case”), 29.15 Morphisms of finite type (“(i), special case”; “(ii), special case”), 29.21 Quasi-finite morphisms (“(ii), special case”), 29.22 Morphisms of finite presentation (“(i), special case”), 26.19 Quasi-compact morphisms (“(i), special case”; “(ii), special case”), 26.21 Separation axioms (“(i), special case”; “(ii), special case”; “(iii), special case”; “(iv), special case”; “1.8 (iii) for schemes”; “special case”), 26.23 Monomorphisms (“special case”), 65.4 Lists of useful properties of morphisms of schemes (“(i), special case”; “(ii), special case”; “more general”), 65.5 Properties of representable morphisms of presheaves (“more general”), 67.4 Separation axioms (“(ii), special case”)
(1.14): 56.5 Functors between categories of quasi-coherent modules (“(i), special case”), 29.22 Morphisms of finite presentation (“(i), special case”; “(ii), special case”), 26.21 Separation axioms (“(ii), special case”), 67.8 Quasi-compact morphisms (“(i), algebraic spaces”), 101.7 Quasi-compact morphisms (“(i), algebraic stacks”; “(i), stacks, X also qs”), 33.7 Geometrically connected schemes (“(i), special case”)
(1.22.1): 6.7 Sheaves, 6.8 Abelian sheaves, 85.2 Simplicial topological spaces
(1.23): 6.29 Limits and colimits of sheaves (“topological space case”), 7.17 Quasi-compact objects and colimits (“(i) qc ⇒ inj”; “(i) qc⇒inj, X terminal”)
(1.24.2): 4.26 Categorically compact objects
(1.30.1): 12.9 Jordan-Hölder (“more general”)
(2.4.7): 26.11 Zariski topology of schemes (“partial”)
(2.15): 111.6 The Spectrum of a ring
(2.17): 59.9 Presheaves, 7.12 Representable sheaves (“more general”)
(2.18): 5.9 Noetherian topological spaces (“special case”)
(3.11): 7.12 Representable sheaves (“b)”)
(4.1): 90.6 Prorepresentable functors and predeformation categories (“partial”), 7.43 Subtopoi (“open inclusion j”; “partial”)
(4.10): 4.26 Categorically compact objects
(4.12): 7.39 Criterion for existence of points (“more general”)
(5.1): 69.5 Colimits and cohomology (“étale alg. spaces”)
(5.6): 10.5 Finite modules and finitely presented modules (“special case”), 11.2 Noncommutative algebras (“special case”), 9.7 Finite extensions (“special case”), 17.11 Modules of finite presentation (“special case”), 18.23 Local types of modules (“special case”)
(5.8): 18.27 Internal Hom (“special case”)
(6.0): 4.32 Categories over categories
(6.1): 4.33 Fibred categories
(6.1.1): 4.32 Categories over categories
(6.1.2): 4.33 Fibred categories (“(6.1.2.1)”)
(6.1.3): 4.36 Presheaves of categories (“Fiberedness clause”; “strict functor case”), 4.37 Presheaves of groupoids (“strict, groupoid fibers”)
(6.4): 4.35 Categories fibred in groupoids (“Fr1”)
(6.5): 4.27 Localization in categories (“6.5 c)”)
(7.1.3): 6.21 Continuous maps and sheaves (“special case”)
(7.1.4): 4.33 Fibred categories (“partial”), 4.36 Presheaves of categories (“partial”)
(7.4.1): 85.3 Simplicial sites and topoi (“simplicial case”; “simplicial, cocontinuous”)
(7.4.4): 85.3 Simplicial sites and topoi (“1)”)
(7.4.9): 85.5 Morphisms of simplicial sites (“simplicial case, m_*”; “simplicial, (7.4.9.2)”; “simplicial, continuous”; “simplicial, f^{-1}”)
(7.4.13.4): 7.13 Continuous functors, 18.16 Exactness of lower shriek, 85.3 Simplicial sites and topoi, 96.17 Injective sheaves
(7.4.13.5): 7.10 Sheafification (“special case”)
(7.4.13.6): 7.28 Localization and morphisms (“special case”)
(8.3.3): 7.18 Colimits of sites (“2) (PT0),(PT1)”)
(8.6.4): 10.39 Flat modules and flat ring maps (“special case”), 10.81 Characterizing flatness (“special case”), 18.39 Pullbacks of flat modules (“special case”)
(8.7.1): 6.10 Sheaves of modules, 18.10 Sheaves of modules, 18.19 Localization of ringed sites (“partial”), 18.21 Localization of ringed topoi (“partial”)
(9.0): 7.39 Criterion for existence of points
(9.2): 7.39 Criterion for existence of points
(9.3): 7.39 Criterion for existence of points
(9.4): 7.39 Criterion for existence of points
(9.5): 7.39 Criterion for existence of points (“existence of points”)

Exposé VII

(1.1): 41.11 Étale morphisms, 41.16 The functorial characterization (“functorial criterion”), 37.8 Formally étale morphisms (“functorial criterion”), 29.37 Étale morphisms (“flat+fibers => étale”), 26.9 Schemes
(1.2): 59.4 The étale topology, 59.20 Big and small sites of schemes (“more general”), 59.27 Étale coverings, 59.29 Neighborhoods, stalks and points (“more general”), 29.37 Étale morphisms, 34.4 The étale topology, 34.11 Change of topologies (“Cov X: étale coverings”; “étale topology on (Sch)”)
(1.3): 59.11 Sheaves, 34.4 The étale topology
(1.4): 59.26 Étale morphisms (“partial”), 59.35 Direct images, 29.37 Étale morphisms (“partial”), 65.4 Lists of useful properties of morphisms of schemes (“partial”)
(1.5): 13.22 Composition of right derived functors (“more general”), 59.54 The Leray spectral sequence (“H^p(Y,R^qf_*F) SS”), 21.14 The Leray spectral sequence (“more general”)
(2.1): 59.23 Examples of sheaves (“partial”), 7.10 Sheafification (“partial”), 7.12 Representable sheaves (“more general”)
(3.3): 69.5 Colimits and cohomology
(4.1): 7.21 Cocontinuous functors and morphisms of topoi (“more general”), 18.16 Exactness of lower shriek (“Clause (i)”), 96.16 Cohomology (“more general”)
(4.2): 59.15 The fpqc topology (“partial”; “special case”; “étale ≤ fpqc”), 34.3 The Zariski topology, 34.9 The fpqc topology
(4.2.1): 59.23 Examples of sheaves (“special case”), 59.24 Picard groups (“special case”)
(4.2.2): 26.10 Immersions of schemes (“partial”), 7.6 Sites
(4.3): 35.9 Cohomology of quasi-coherent modules and topologies, 59.22 Cohomology of quasi-coherent sheaves
(5.1): 32.2 Directed limits of schemes with affine transition maps (“all schemes affine”; “existence of lim”)
(5.2): 32.10 Descending relative objects (“(a) injectivity”; “(b)”; “more general”), 34.13 Extending functors (“a, special case”)
(5.3): 32.10 Descending relative objects (“essential surjectivity”; “faithfulness”; “more general”)
(5.4): 32.4 Descending properties (“partial”), 26.18 Base change in algebraic geometry
(5.11): 59.51 Colimits (“more general”; “special case”), 59.52 Colimits and complexes (“more general”)

Exposé VIII

(1.1): 59.45 Topological invariance of the small étale site (“partial”)
(1.3): 29.55 Normalization (“1.3 c), special case”)
(1.4): 29.46 Universal homeomorphisms
(2.1): 59.56 Galois action on stalks (“j is an equivalence”; “more general”), 58.2 Schemes étale over a point (“X_ét ≌ B_π”)
(2.2): 59.59 Cohomology of a point
(3.2): 59.29 Neighborhoods, stalks and points (“partial”), 58.5 Finite étale morphisms (“partial”)
(3.4): 59.36 Inverse image (“(F), special case”)
(3.5): 18.36 Stalks of modules (“Clause a)”; “a) colimits”; “a) finite limits”; “more general”), 66.19 Points of the small étale site (“3.5 a)”)
(3.6): 59.29 Neighborhoods, stalks and points (“epi ⇒ surj on stalks”; “epi ⟺ surj on stalks”; “mono ⟺ inj on stalks”), 66.19 Points of the small étale site (“injectivity clause”; “surjectivity clause”)
(3.7): 59.29 Neighborhoods, stalks and points (“special case”)
(3.9): 59.29 Neighborhoods, stalks and points (“objects of C_ξ”)
(3.13): 29.16 Points of finite type and Jacobson schemes (“more general”), 33.14 Change of fields and the Jacobson property (“partial”)
(4.1): 10.153 Henselian local rings (“(i)–(iv)”; “(ii)”; “partial”), 10.155 Henselization and strict henselization (“(iii)”; “henselization”; “more general”), 10.160 The Cohen structure theorem (“(i), special case”), 59.32 Henselian rings (“(i), (iii), (iv)”; “(i), forward direction”; “partial”), 38.2 Lemmas on étale localization (“more general”), 15.11 Henselian pairs, 16.13 The approximation property for G-rings
(4.2): 59.32 Henselian rings (“(i)”)
(4.3): 59.33 Stalks of the structure sheaf (“strictly local ring”)
(4.4): 10.155 Henselization and strict henselization (“more general”), 38.2 Lemmas on étale localization (“more general”)
(5.1): 59.51 Colimits (“partial”), 56.5 Functors between categories of quasi-coherent modules (“partial”), 29.15 Morphisms of finite type (“partial”), 29.21 Quasi-finite morphisms (“partial”), 29.22 Morphisms of finite presentation (“partial”), 26.19 Quasi-compact morphisms (“partial”), 26.21 Separation axioms (“partial”), 21.7 Locality of cohomology (“more general”), 65.4 Lists of useful properties of morphisms of schemes (“partial”), 33.7 Geometrically connected schemes (“partial”)
(5.4): 29.45 Integral and finite morphisms (“more general”), 65.4 Lists of useful properties of morphisms of schemes (“more general”)
(5.5): 59.55 Vanishing of finite higher direct images (“more general”; “partial”)
(5.6): 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”)
(5.7): 7.27 More localization (“open immersion case”), 67.13 Closed immersions (“partial”)
(6.2): 18.21 Localization of ringed topoi (“partial”)
(6.3): 67.13 Closed immersions (“fully faithful”)
(6.5): 59.70 Extension by zero (“partial”)
(6.6): 59.31 Supports of abelian sheaves (“support of a section”; “support of a sheaf”), 66.20 Supports of abelian sheaves (“support of a section”; “support of a sheaf”)
(7.1): 59.33 Stalks of the structure sheaf (“partial”)
(7.9): 59.29 Neighborhoods, stalks and points (“essential surjectivity”), 66.19 Points of the small étale site (“essential surjectivity”)
(7.9.3): 26.11 Zariski topology of schemes (“partial”), 7.32 Points, 5.8 Irreducible components
(8.2): 85.8 Cohomology on simplicial sites (“resolution of Z”)
(8.6): 59.55 Vanishing of finite higher direct images (“strictly henselian, n=0”; “strictly henselian, p≥1”)
(9.4): 59.104 Descending étale sheaves
(9.4.4): 67.6 Open morphisms (“more general”)

Exposé X

(5.2): 59.83 Cohomology of torsion sheaves on curves (“special case”), 59.95 Cohomological dimension (“special case”)

Exposé XII

(4.3): 29.2 Closed immersions (“Parenthetical clause”), 26.17 Fibre products of schemes (“Parenthetical clause”), 26.18 Base change in algebraic geometry (“Parenthetical clause”), 65.4 Lists of useful properties of morphisms of schemes (“Parenthetical clause”)
(4.4): 7.45 Pullback maps (“more general”)
(5.1): 59.91 The proper base change theorem (“(iii)”)
(5.2): 59.91 The proper base change theorem (“(iii), special case”), 84.4 Proper base change (“more general”)
(5.3): 59.92 Applications of proper base change (“special case”)
(5.3bis): 59.55 Vanishing of finite higher direct images (“special case”)
(5.4): 59.90 Applications of smooth base change (“special case”), 84.4 Proper base change (“more general”)
(5.5): 84.4 Proper base change (“more general”)
(5.6): 59.23 Examples of sheaves (“special case”)
(5.9): 58.9 Finite étale covers of proper schemes
(5.9bis): 58.8 Topological invariance of the fundamental group (“X = S affine”), 58.9 Finite étale covers of proper schemes (“special case”)
(6.5): 58.8 Topological invariance of the fundamental group (“(ii) b')⇒b'') π₀”; “(ii) b')⇒b'') π₁”)
(6.13): 59.32 Henselian rings (“special case”), 59.82 Affine analog of proper base change (“more general”; “special case”), 59.91 The proper base change theorem (“partial, special case”; “special case”), 15.11 Henselian pairs, 58.9 Finite étale covers of proper schemes (“more general”)
(7.1): 30.18 Chow's Lemma (“Noetherian case”), 32.12 Variants of Chow's Lemma (“Chow diagram”; “finite irred. components”)
(8.6): 59.24 Picard groups (“more general”)

Exposé XIII

(1.1): 58.9 Finite étale covers of proper schemes (“partial”)
(1.5): 10.20 Nakayama's lemma (“special case”)

Exposé XIV

(1.1): 59.73 More on constructible sheaves (“special case”)
(1.2): 59.83 Cohomology of torsion sheaves on curves (“special case”)
(1.5): 59.73 More on constructible sheaves (“special case”)
(1.11): 59.45 Topological invariance of the small étale site (“more general”), 59.55 Vanishing of finite higher direct images (“partial”), 59.73 More on constructible sheaves (“partial”), 59.74 Constructible sheaves on Noetherian schemes (“special case”), 59.92 Applications of proper base change (“partial”), 63.13 A constructibility result (“partial”), 37.27 Irreducible components of fibres (“special case”), 33.43 Curves (“special case”)
(3.1): 59.95 Cohomological dimension (“more general”)
(3.2): 59.95 Cohomological dimension (“more general”; “special case”)

Exposé XV

(1.3): 5.7 Connected components

Exposé XVI

(1.5): 59.39 Comparing topologies (“partial”)
(1.6): 59.90 Applications of smooth base change (“Abelian L-torsion clause”; “partial”), 84.4 Proper base change (“Abelian clause”)
(2.1): 63.15 Applications (“special case”), 84.4 Proper base change (“more general”)
(2.2): 63.15 Applications (“Abelian torsion case”), 84.4 Proper base change (“more general”)
(2.3): 59.94 The cospecialization map (“special case”)
(3.1): 111.33 Schemes (“more general”), 26.10 Immersions of schemes (“more general”)
(5.2): 59.83 Cohomology of torsion sheaves on curves (“partial”)

Exposé XVII

(0.2.1): 6.21 Continuous maps and sheaves (“(i), special case”; “(ii), special case”), 85.7 Morphisms of ringed simplicial sites (“ringed simplicial sites”)
(0.3): 13.3 The definition of a triangulated category (“One variable with ε = +1”)
(0.4): 12.5 Abelian categories (“special case”), 12.18 Double complexes and associated total complexes (“case |I| = 2”)
(0.7): 26.9 Schemes, 26.21 Separation axioms
(0.9): 26.17 Fibre products of schemes (“n = 2”)
(0.10): 59.4 The étale topology (“partial”), 59.15 The fpqc topology (“partial”), 59.27 Étale coverings (“partial”), 34.4 The étale topology (“big étale site”; “partial”; “étale pretopology on Sch”; “étale topology on Sch”), 34.7 The fppf topology (“big fppf site”; “partial”), 34.9 The fpqc topology (“fpqc topology on Sch”)
(1.1.1): 13.9 Cones and termwise split sequences (“both final assertions”), 13.10 Distinguished triangles in the homotopy category (“distinguished <=> split”; “distinguished triangles”; “every antidist. arises”; “triangle of split seq”), 22.9 Cones and distinguished triangles (“DG modules over a DGA”), 22.27 Obtaining triangulated categories (“DG cat. with cone situat”; “DG category setting”)
(1.1.3): 13.3 The definition of a triangulated category
(1.1.4): 12.18 Double complexes and associated total complexes (“double complex, ⊕ case”)
(1.1.4.2.1): 12.14 Homotopy and the shift functor (“special case”), 12.18 Double complexes and associated total complexes (“I = {1”)
(1.1.4.7): 12.18 Double complexes and associated total complexes (“special case”)
(1.1.6): 22.26 Differential graded categories (“one covariant variable”), 21.48 Duals (“special case”)
(1.1.10): 22.3 Differential graded algebras (“(i)”), 22.26 Differential graded categories (“(ii)”), 15.73 Hom complexes (“(iii), special case”)
(1.1.11): 22.24 Linear categories, 90.11 Tangent spaces of functors
(1.2.1): 13.14 Derived functors in general (“(ii)”; “(iii)”)
(1.2.2): 13.14 Derived functors in general (“(i), special case”; “(ii)”), 13.20 Right derived functors and injective resolutions (“(ii), special case”)
(1.2.2.1): 13.5 Localization of triangulated categories
(1.2.3): 13.15 Derived functors on derived categories (“(ii)”)
(1.2.4): 13.15 Derived functors on derived categories (“Definedness clause”; “Value clause”), 51.5 More general supports (“special case”)
(1.2.5): 13.15 Derived functors on derived categories (“(iv)”; “(iv), single functor”), 64.6 Derived categories (“(ii), single functor”)
(1.2.6): 13.15 Derived functors on derived categories
(1.2.7): 13.16 Higher derived functors (“enough injectives (RF)”), 13.20 Right derived functors and injective resolutions (“enough injectives (RF)”)
(2.1.1): 13.11 Derived categories (“more general”)
(2.2.6): 17.3 The abelian category of sheaves of modules (“partial”), 18.14 The abelian category of sheaves of modules (“partial”)
(2.2.7): 42.2 Periodic complexes and Herbrand quotients (“special case”), 12.5 Abelian categories (“special case”), 12.13 Complexes (“special case”), 12.14 Homotopy and the shift functor (“partial”; “special case”), 64.7 Filtered derived category (“special case”)
(2.2.8): 22.26 Differential graded categories (“partial”), 64.7 Filtered derived category (“absolute case of K(A/B)”)
(2.2.9): 13.10 Distinguished triangles in the homotopy category (“(i)”; “(i), special case”)
(2.2.10): 13.10 Distinguished triangles in the homotopy category (“(A), special case”), 64.7 Filtered derived category (“special case”)
(2.2.12): 4.36 Presheaves of categories (“partial”), 85.3 Simplicial sites and topoi
(2.2.13.1): 13.3 The definition of a triangulated category, 13.7 Adjoints for exact functors (“special case”), 13.38 Brown representability (“Triangulated case”), 13.39 Brown representability, bis (“Triangulated case”), 22.24 Linear categories, 90.11 Tangent spaces of functors
(2.3.7): 20.28 Cohomology of unbounded complexes (“ringed spaces case”), 13.30 Deriving adjoints (“outer Hom bijection”)
(3.2.2): 38.32 Compactifications (“(v)”), 38.33 Nagata compactification (“more general”)
(3.2.3): 38.33 Nagata compactification (“(iii), special case”)
(3.2.6): 63.11 Derived upper shriek (“(i), special case”)
(3.3.3): 4.33 Fibred categories (“Forward direction”)
(4.1): 6.26 Morphisms of ringed spaces and modules (“ringed spaces”), 18.19 Localization of ringed sites, 18.21 Localization of ringed topoi
(4.1.1): 10.39 Flat modules and flat ring maps (“(i), special case”; “(ii), special case”), 17.17 Flat modules (“(i), ringed space”), 18.28 Flat modules (“(i)”; “(i), presheaves”; “(ii), f = id”; “(ii), special case”)
(4.1.4): 20.28 Cohomology of unbounded complexes (“ringed spaces case”)
(4.1.5): 20.28 Cohomology of unbounded complexes (“ringed spaces”)
(4.1.6): 10.14 Base change (“special case”), 21.17 Flat resolutions (“comm. O, G fixed, left”; “comm. O, G fixed, right”), 21.48 Duals (“⊗ on complexes”)
(4.1.7): 15.60 Derived tensor product (“(i), modules over ring”; “(ii), modules over ring”), 21.17 Flat resolutions (“(i)”)
(4.1.8): 20.26 Flat resolutions (“modules, not complexes”), 21.17 Flat resolutions (“modules, not complexes”)
(4.1.9): 20.48 Tor dimension (“(i), (iii)”; “ringed space, (i)”; “ringed space, module”), 15.68 Tor dimension (“module over a ring”; “over a ring, (i)<=>(iii)”; “ring case, one-sided”), 21.46 Tor dimension (“(i)”; “(i) <=> (iii)”; “module case”)
(4.2): 20.30 Godement resolution (“augmented exactness”; “flasqueness of terms”; “functorial flasque terms”; “stalkwise homotopy”)
(4.2.3): 20.30 Godement resolution (“(i)”; “(ii)”)
(4.2.8): 20.48 Tor dimension (“(iii), special case”; “clause (iii)”), 21.46 Tor dimension (“(iii) f=id, converse”; “(iii) f=id, forward”)
(4.2.9): 20.30 Godement resolution (“special case”)
(4.2.12): 24.28 Derived pullback (“Lg*, special case”; “The case g = id”; “more general”)
(4.3.1): 59.55 Vanishing of finite higher direct images (“special case”), 59.92 Applications of proper base change (“special case”)
(4.3.2): 59.91 The proper base change theorem (“special case”), 59.92 Applications of proper base change (“special case”)
(5.1.1): 6.31 Open immersions and (pre)sheaves (“partial”; “ringed spaces, modules”; “top. spaces, alg. struct”; “topological spaces”)
(5.1.4): 59.92 Applications of proper base change (“special case”)
(5.2.2): 63.9 Derived lower shriek via compactifications (“special case”), 63.10 Properties of derived lower shriek (“special case”), 63.12 Compactly supported cohomology (“special case”)
(5.2.7): 59.91 The proper base change theorem (“(ii), special case”)
(5.2.8.1): 59.83 Cohomology of torsion sheaves on curves (“special case”), 59.92 Applications of proper base change (“special case”)
(5.3.6): 63.14 Complexes with constructible cohomology (“Quasi-finite case”)
(5.3.8): 63.12 Compactly supported cohomology (“special case”), 63.15 Applications (“special case”)
(5.5.1.1): 10.81 Characterizing flatness (“special case”)
(5.5.1.2): 10.39 Flat modules and flat ring maps (“partial”), 10.81 Characterizing flatness (“partial”)
(5.5.3): 42.2 Periodic complexes and Herbrand quotients, 14.5 Cosimplicial objects, 14.25 Dold-Kan for cosimplicial objects (“Dold–Puppe equivalence”), 14.28 Homotopies and cosimplicial objects (“homotopy preservation”), 64.7 Filtered derived category
(5.5.5): 13.14 Derived functors in general (“left derivable at K”; “right derivable at K”)
(5.5.6.3): 10.5 Finite modules and finitely presented modules, 10.8 Colimits (“A = ℤ”), 10.12 Tensor products, 10.39 Flat modules and flat ring maps (“partial”), 10.81 Characterizing flatness (“partial”), 11.2 Noncommutative algebras, 41.9 Flat morphisms, 9.7 Finite extensions, 15.75 Derived hom (“more general”), 64.10 Perfectness (“more general”)
(5.5.25.2): 13.9 Cones and termwise split sequences (“partial”)
(5.5.25.4): 59.73 More on constructible sheaves (“more general”)
(5.5.29): 38.2 Lemmas on étale localization (“Flatness clause”), 17.10 Quasi-coherent modules (“partial”)
(6.1.1): 30.26 Being proper over a base (“Clause (i)”)
(6.1.2): 63.3 Sections with compact support (“abelian sheaves”)
(6.1.4): 63.3 Sections with compact support (“more general”)
(6.2.3): 59.66 Méthode de la trace (“Var 4 (normalization)”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“more general”)
(6.2.4): 37.75 Weightings (“base change”; “composite”; “condition (*)”)
(6.2.5): 63.5 Weightings and trace maps for locally quasi-finite morphisms (“more general”; “special case”)
(6.2.6): 37.75 Weightings (“n(x) is a weighting”), 28.16 Local irreducibility (“more general”)
(6.2.7): 63.12 Compactly supported cohomology (“more general”; “partial”; “special case”)
(6.2.11): 59.70 Extension by zero (“f_! ⊣ f^ (abelian)”; “f_! ⊣ f^ (Λ-modules)”), 63.5 Weightings and trace maps for locally quasi-finite morphisms (“more general”)
(6.3.4): 31.4 Embedded points (“special case”)
(6.3.8): 29.35 Smooth morphisms (“partial”), 44.3 Moduli of divisors on smooth curves (“special case”)
(6.3.15): 9.20 Trace and norm (“(v)”)
(6.3.16): 63.3 Sections with compact support (“f_! as subsheaf of f_*”), 63.4 Sections with finite support (“colimit formula”)
(7.0.1): 48.16 Upper shriek functors, 6.10 Sheaves of modules (“more general”), 18.10 Sheaves of modules (“more general”)
(7.0.6): 30.18 Chow's Lemma (“Noetherian base”), 32.12 Variants of Chow's Lemma (“(i), (iii)”; “finite irred. components”)
(7.3.3): 20.26 Flat resolutions (“special case”), 13.10 Distinguished triangles in the homotopy category (“special case”)
(7.3.7): 63.12 Compactly supported cohomology (“special case”)

Exposé XVIII

(0.1): 63.12 Compactly supported cohomology (“special case”)
(1.2.1): 95.14 Classifying torsors (“partial”), 39.13 Groupoids (“partial”)
(1.3.3): 31.32 Closed subschemes of relative proj (“partial”), 21.4 First cohomology and torsors (“partial”)
(1.5.10.3): 44.3 Moduli of divisors on smooth curves (“special case”)
(1.6.1): 59.28 Kummer theory (“m = O_X (Y empty)”)
(2.3): 10.130 Openness of Cohen-Macaulay loci (“affine case”), 37.22 Cohen-Macaulay morphisms (“fibrewise density”)
(2.4): 37.22 Cohen-Macaulay morphisms
(2.9): 59.66 Méthode de la trace (“Var 4 (I)”)
(2.14.2): 13.12 The canonical delta-functor (“partial”)
(3.1.5): 63.7 Derived upper shriek for locally quasi-finite morphisms (“special case”)
(3.1.8): 63.5 Weightings and trace maps for locally quasi-finite morphisms (“(iii), special case”)
(3.1.12.1): 6.20 Sheafification of presheaves of modules

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