EGA 0

The items of EGA 0 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.

§1

(1.0.4): 9.6 Field extensions
(1.0.5): 10.5 Finite modules and finitely presented modules (“partial”), 10.6 Ring maps of finite type and of finite presentation, 10.31 Noetherian rings (“more general”), 10.36 Finite and integral ring extensions (“more general”; “partial”; “partial, special case”), 10.51 More Noetherian rings (“more general”), 11.2 Noncommutative algebras, 9.7 Finite extensions, 38.25 Variants of a lemma (“more general”)
(1.0.6): 10.17 The spectrum of a ring (“more general”)
(1.0.7): 10.18 Local rings (“partial”)
(1.1.1): 10.17 The spectrum of a ring (“partial”), 10.31 Noetherian rings (“Minimal primes over (0)”)
(1.2.1): 10.9 Localization
(1.2.2): 10.9 Localization (“special case”), 9.3 Examples of fields (“special case”)
(1.2.3): 10.9 Localization, 10.112 Homomorphisms and dimension (“A_p is a local ring”)
(1.2.4): 10.9 Localization (“uniqueness of v*”), 17.27 Localizing sheaves of rings (“more general”), 18.44 Localizing sheaves of rings (“more general”)
(1.2.5): 10.12 Tensor products
(1.2.6): 10.9 Localization (“partial”), 10.17 The spectrum of a ring, 110.22 Underlying space Noetherian not Noetherian
(1.2.7): 10.9 Localization, 10.112 Homomorphisms and dimension (“A_p is local”)
(1.3.2): 10.9 Localization (“more general”), 10.23 Glueing properties (“more general”)
(1.3.4): 10.12 Tensor products (“more general”)
(1.3.5): 10.10 Internal Hom (“more general”), 10.31 Noetherian rings (“more general”), 10.51 More Noetherian rings (“more general”), 38.25 Variants of a lemma (“more general”)
(1.4.6): 10.9 Localization (“special case”)
(1.5.3): 10.10 Internal Hom (“special case”)
(1.5.4): 10.12 Tensor products (“partial”)
(1.5.5): 10.10 Internal Hom (“special case”)
(1.5.8): 10.25 Zerodivisors and total rings of fractions (“reduced ring”), 10.30 More on images
(1.7.1): 10.23 Glueing properties (“M = 0 ⇔ Supp M = ∅”; “partial”), 10.40 Supports and annihilators (“M = 0 ⇔ Supp M = ∅”)
(1.7.2): 10.40 Supports and annihilators (“partial”)
(1.7.4): 10.17 The spectrum of a ring (“two ideals, special case”)

§2

(2.1.2): 5.19 Specialization (“specialization relation”)
(2.1.3): 5.8 Irreducible components, 5.9 Noetherian topological spaces (“special case”), 5.12 Quasi-compact spaces and maps (“partial”)
(2.1.4): 5.15 Constructible sets (“partial”)
(2.1.5): 10.30 More on images (“special case”)
(2.1.7): 5.8 Irreducible components (“first assertion”)
(2.2.1): 111.6 The Spectrum of a ring, 5.9 Noetherian topological spaces (“opens max ⇔ closeds min”)
(2.2.3): 5.9 Noetherian topological spaces (“converse”)
(2.2.4): 5.12 Quasi-compact spaces and maps (“first assertion”)
(2.2.5): 10.31 Noetherian rings (“affine case”; “ring-theoretic form”)

§3

(3.1.2): 59.11 Sheaves (“C = Set”; “Set-valued coefficients”), 26.15 A representability criterion, 6.7 Sheaves (“C = Set”; “concrete category C”), 6.8 Abelian sheaves, 7.6 Sites (“C = Set”), 7.7 Sheaves (“C = Set”), 7.47 Topologies, 85.2 Simplicial topological spaces, 34.9 The fpqc topology (“(F)”; “Set-valued coefficients”)
(3.1.3): 59.11 Sheaves (“special case”)
(3.1.4): 59.11 Sheaves (“more general”), 26.15 A representability criterion, 6.7 Sheaves, 6.8 Abelian sheaves, 6.9 Sheaves of algebraic structures (“failure for top. spaces”), 7.6 Sites, 7.47 Topologies, 65.6 Algebraic spaces, 85.2 Simplicial topological spaces, 5.30 Topological groups, rings, modules
(3.1.5): 111.33 Schemes, 6.31 Open immersions and (pre)sheaves
(3.1.6): 6.13 Stalks of presheaves of algebraic structures
(3.2.1): 6.30 Bases and sheaves (“definition”; “set-valued definition”)
(3.2.2): 26.11 Zariski topology of schemes (“affine basis of a scheme”), 6.30 Bases and sheaves
(3.2.3): 6.30 Bases and sheaves (“morphism definition”; “special case”), 65.6 Algebraic spaces (“more general”)
(3.2.4): 5.5 Bases (“more general”)
(3.2.5): 6.30 Bases and sheaves (“1st para: F ≅ F₁'”; “second paragraph”; “special case”)
(3.2.6): 87.2 Formal schemes à la EGA (“more general”; “partial”), 7.10 Sheafification (“more general”), 18.3 Abelian sheaves (“more general”; “partial”)
(3.3.1): 6.33 Glueing sheaves (“special case”)
(3.4.1): 6.21 Continuous maps and sheaves
(3.4.2): 6.21 Continuous maps and sheaves
(3.4.3): 6.21 Continuous maps and sheaves (“special case”)
(3.4.4): 6.21 Continuous maps and sheaves (“construction of ψ_x”; “ψ''_x = ψ_x ∘ ψ'_{ψ(x)}”)
(3.4.5): 6.32 Closed immersions and (pre)sheaves (“more general”)
(3.4.6): 6.32 Closed immersions and (pre)sheaves (“(F)”)
(3.5.1): 6.21 Continuous maps and sheaves, 6.25 Ringed spaces, 6.30 Bases and sheaves (“special case”)
(3.5.2): 6.21 Continuous maps and sheaves (“category of pairs”; “special case”), 6.25 Ringed spaces (“category of pairs”; “partial”)
(3.5.3): 6.17 Sheafification (“partial, special case”)
(3.5.5): 6.21 Continuous maps and sheaves (“partial”)
(3.5.6): 6.17 Sheafification (“more general”), 6.19 Sheafification of presheaves of algebraic structures (“algebraic structures”), 7.10 Sheafification (“partial”), 7.12 Representable sheaves (“more general”), 7.49 Sheafification in a topology (“more general”), 18.5 Free abelian sheaves (“more general”)
(3.6.1): 59.23 Examples of sheaves, 59.64 Locally constant sheaves (“locally simple”), 6.3 Presheaves, 6.7 Sheaves (“simple sheaf”), 7.10 Sheafification, 18.43 Locally constant sheaves (“locally simple”)
(3.7.1): 6.17 Sheafification, 6.31 Open immersions and (pre)sheaves (“special case”)

§4

(4.1.1): 6.21 Continuous maps and sheaves (“morphism definition”), 6.25 Ringed spaces (“composition”; “morphism definition”; “psi''=psi'o psi”; “ringed space def”; “theta''=psi_*(th)o th'”)
(4.1.2): 111.33 Schemes (“special case”), 26.3 Open immersions of locally ringed spaces (“special case”), 6.31 Open immersions and (pre)sheaves (“special case”)
(4.1.3): 6.10 Sheaves of modules (“more general”), 18.10 Sheaves of modules (“more general”)
(4.1.4): 26.12 Reduced schemes
(4.1.5): 10.12 Tensor products (“special case”), 20.50 Duals (“partial”), 51.22 A bit of uniformity, III, 17.21 Symmetric and exterior powers (“more general”), 17.22 Internal Hom (“special case”), 6.20 Sheafification of presheaves of modules (“special case”), 18.27 Internal Hom (“Injectivity of Hom(F”; “partial”)
(4.1.7): 6.33 Glueing sheaves (“partial”)
(4.2.1): 6.24 Continuous maps and sheaves of modules (“more general”), 6.26 Morphisms of ringed spaces and modules (“Ψ_* functor”), 18.21 Localization of ringed topoi (“more general”)
(4.2.5): 17.6 Closed immersions and abelian sheaves (“Ψ_M is zero outside X”), 6.32 Closed immersions and (pre)sheaves (“zero stalks outside X”; “Ψ_M is zero outside X”)
(4.2.6): 6.26 Morphisms of ringed spaces and modules
(4.3.1): 6.26 Morphisms of ringed spaces and modules (“stalk formula”; “Ψ(G) definition”), 18.13 Morphisms of ringed topoi and modules (“Ψ functor”), 18.36 Stalks of modules (“more general”)
(4.3.2): 10.12 Tensor products (“special case”), 17.3 The abelian category of sheaves of modules (“special case”)
(4.3.6): 6.26 Morphisms of ringed spaces and modules
(4.4.1): 6.10 Sheaves of modules (“special case”), 6.30 Bases and sheaves (“basis data suffices”), 18.10 Sheaves of modules (“special case”)
(4.4.2): 6.21 Continuous maps and sheaves
(4.4.3): 6.21 Continuous maps and sheaves (“partial”), 6.24 Continuous maps and sheaves of modules, 6.26 Morphisms of ringed spaces and modules (“partial”)
(4.4.6): 18.27 Internal Hom (“Functoriality of γ♯”)
(4.4.8): 6.21 Continuous maps and sheaves (“definition of v''”)

§5

(5.1.1): 17.4 Sections of sheaves of modules (“(1)”; “more general”; “parenthetical”), 18.17 Global types of modules
(5.1.3): 17.10 Quasi-coherent modules (“direct sum remark”)
(5.2.1): 10.12 Tensor products (“special case”), 10.14 Base change (“special case”), 11.4 Lemmas on algebras (“special case”), 110.33 A finite flat module which is not projective (“special case”), 55.4 The Picard group of a numerical type (“special case”), 17.9 Modules of finite type, 18.23 Local types of modules
(5.2.2): 10.40 Supports and annihilators (“special case”), 30.9 Coherent sheaves on locally Noetherian schemes (“support closed”), 17.5 Supports of modules and sections (“special case”), 17.9 Modules of finite type (“generating sections”; “support closed”)
(5.2.5): 17.11 Modules of finite presentation (“more general”), 17.12 Coherent modules (“fin. pres. ⇒ quasi-coh.”), 18.23 Local types of modules (“more general”)
(5.2.6): 17.22 Internal Hom (“partial”)
(5.2.7): 17.11 Modules of finite presentation (“free case G = O^r”)
(5.3.2): 10.90 Coherent rings (“submodule clause, ring”), 17.12 Coherent modules (“first assertion”)
(5.3.3): 17.12 Coherent modules (“Extension clause”; “Quotient clause”)
(5.3.4): 10.90 Coherent rings (“special case”), 17.12 Coherent modules (“Coker(u) coherent”; “Ker(u) coherent”)
(5.3.5): 30.9 Coherent sheaves on locally Noetherian schemes (“tensor product”), 17.16 Tensor product (“tensor product”), 18.26 Tensor product (“tensor product”)
(5.3.7): 10.40 Supports and annihilators (“special case”), 10.90 Coherent rings (“special case”), 17.12 Coherent modules, 17.23 The annihilator of a sheaf of modules (“annihilator def”), 18.23 Local types of modules
(5.3.12): 6.24 Continuous maps and sheaves of modules, 18.21 Localization of ringed topoi
(5.4.1): 10.78 Finite projective modules (“affine case”; “affine case, rank n”), 111.40 Invertible sheaves (“invertible”), 17.14 Locally free sheaves (“finite rank”; “loc. free ⇒ qcoh”; “more general”; “n(x) locally constant”; “rank n”), 17.26 Rank and determinant (“partial”), 28.21 Locally free modules (“partial, special case”), 18.23 Local types of modules (“more general”), 18.32 Invertible modules (“rank n”)
(5.4.2): 21.48 Duals (“more general”)
(5.4.3): 17.25 Invertible modules (“more general”), 18.32 Invertible modules (“more general”)
(5.4.4): 17.25 Invertible modules (“partial”)
(5.4.5): 17.14 Locally free sheaves (“locally free case”), 18.18 Intrinsic properties of modules (“partial”), 18.23 Local types of modules (“locally free clause”)
(5.4.6): 17.25 Invertible modules (“graded ring Γ•(L)”; “partial”; “Γ•(L) carrier”)
(5.4.7): 20.6 First cohomology and invertible sheaves (“special case”), 17.25 Invertible modules (“special case”)
(5.5.1): 10.18 Local rings, 26.2 Locally ringed spaces
(5.5.2): 17.25 Invertible modules (“(b) => (c)”)
(5.5.4): 10.13 Tensor algebra (“free case”)
(5.5.5): 15.129 Splitting off a free module (“affine case, closed pt”)

§6

(6.1.1): 10.12 Tensor products, 10.39 Flat modules and flat ring maps (“more general”; “special case”), 10.75 Tor groups and flatness, 10.83 Descent for finite projective modules (“more general”), 41.9 Flat morphisms, 56.3 Functors between categories of modules (“more general”)
(6.1.2): 10.39 Flat modules and flat ring maps (“M flat implies M' flat”; “M' flat implies M flat”; “exactness”; “more general”), 10.81 Characterizing flatness (“special case”), 10.83 Descent for finite projective modules (“more general”)
(6.1.3): 10.39 Flat modules and flat ring maps (“second formula”)
(6.2.1): 10.39 Flat modules and flat ring maps (“M⊗_A N is B-flat”; “more general”)
(6.2.2): 15.67 Pseudo-coherent modules, II
(6.2.3): 10.8 Colimits (“special case”), 10.39 Flat modules and flat ring maps (“constant ring system”; “special case”), 10.81 Characterizing flatness (“special case”)
(6.3.1): 10.9 Localization (“more general”), 10.12 Tensor products, 10.23 Glueing properties (“more general”)
(6.3.3): 10.12 Tensor products (“partial”), 10.23 Glueing properties (“more general”)
(6.3.4): 10.39 Flat modules and flat ring maps (“partial”), 10.81 Characterizing flatness (“partial”), 41.8 Examples of unramified morphisms (“last assertion, A a DVR”), 15.22 Torsion free modules (“finite, Dedekind domain”; “more general”)
(6.4.1): 10.39 Flat modules and flat ring maps (“(a) <=> (c)”; “(a) <=> (d)”; “(a) ⇔ (b), (a) ⇔ (d)”; “(a) ⇔ (c)”; “a)”), 41.9 Flat morphisms (“(a) <=> (c)”; “(a) ⇔ (c)”)
(6.4.3): 10.39 Flat modules and flat ring maps (“M is flat”)
(6.4.4): 10.39 Flat modules and flat ring maps (“last assertion”)
(6.4.5): 10.39 Flat modules and flat ring maps (“first assertion”)
(6.5.1): 10.82 Universally injective module maps (“B faithfully flat”)
(6.5.2): 10.82 Universally injective module maps (“φ⁻¹(aB) = a”), 10.164 Descending properties (“special case”)
(6.6.1): 10.39 Flat modules and flat ring maps (“(e) ⇒ (a), special case”; “more general”), 10.82 Universally injective module maps (“(a)”; “(a) ⇒ φ⁻¹(aB) = a in (d)”), 41.9 Flat morphisms (“more general”), 67.30 Flat morphisms (“(e) ⇒ (a), special case”)
(6.6.2): 10.39 Flat modules and flat ring maps (“partial, special case”), 67.30 Flat morphisms (“partial, special case”)
(6.6.3): 10.39 Flat modules and flat ring maps (“necessity”), 10.82 Universally injective module maps (“first part, M = A”), 35.4 Descent for universally injective morphisms (“more general”)
(6.6.4): 10.39 Flat modules and flat ring maps (“ff maps compose”; “flat maps compose”; “necessity, ff case”; “necessity, flat case”; “sufficiency, flat case”)
(6.7.1): 41.9 Flat morphisms, 39.4 Group schemes, 17.20 Flat morphisms of ringed spaces (“F f-flat (all x)”; “F f-flat at x”; “f flat”; “f flat at x”), 29.26 Flat morphisms
(6.7.2): 6.26 Morphisms of ringed spaces and modules (“special case”)
(6.7.3): 10.39 Flat modules and flat ring maps (“Algebraic reduction”)
(6.7.4): 10.82 Universally injective module maps (“partial”)
(6.7.5): 10.39 Flat modules and flat ring maps (“Affine schemes”; “special case”), 42.14 Flat pullback (“Schemes (EGA preschemes)”), 29.26 Flat morphisms (“Schemes (EGA preschemes)”), 65.4 Lists of useful properties of morphisms of schemes (“Schemes (EGA preschemes)”)
(6.7.6): 17.22 Internal Hom (“partial”), 18.27 Internal Hom (“Left exactness of Hom(-”), 18.31 Flat morphisms (“partial, special case”; “special case”)
(6.7.8): 17.20 Flat morphisms of ringed spaces (“f* exact”)

§7

(7.1.1): 87.4 Topological rings and modules, 15.37 Topological rings and modules
(7.1.2): 10.59 Noetherian local rings (“special case”), 15.37 Topological rings and modules (“admissible”)
(7.1.3): 87.4 Topological rings and modules
(7.1.6): 87.4 Topological rings and modules
(7.1.9): 15.37 Topological rings and modules (“preadic (J^n a basis)”)
(7.1.10): 87.4 Topological rings and modules
(7.1.12): 10.32 Locally nilpotent ideals (“special case”)
(7.1.14): 10.20 Nakayama's lemma (“sufficiency”)
(7.2.3): 10.96 Completion (“special case”)
(7.2.4): 10.96 Completion (“Case M = A”)
(7.2.6): 10.160 The Cohen structure theorem (“special case”)
(7.2.12): 10.96 Completion (“special case”), 33.41 Normalization of one dimensional schemes (“special case”)
(7.3.1): 10.96 Completion (“more general”; “v̂ surjective”), 10.97 Completion for Noetherian rings (“Noetherian, finite mods”; “more general”; “special case”; “v̂ surjective”)
(7.3.2.1): 10.51 More Noetherian rings (“more general”), 30.10 Coherent sheaves on Noetherian schemes
(7.3.3): 10.96 Completion (“partial”), 10.97 Completion for Noetherian rings (“partial”), 15.28 Completion and flatness (“more general”)
(7.3.4): 15.102 Systems of modules (“Hom^ ≅ Hom(M^,N^)”)
(7.3.5): 10.51 More Noetherian rings (“(a) => (c)”), 10.97 Completion for Noetherian rings (“(a) => (b)”)
(7.4.2): 10.51 More Noetherian rings (“more general”)
(7.5.3): 87.28 Restricted power series
(7.5.5): 88.11 Finite type morphisms (“(a) <=> (c)”)
(7.6.4): 10.9 Localization (“Discrete topology on A”)
(7.6.6): 10.9 Localization
(7.6.8): 10.9 Localization (“partial”)
(7.6.11): 10.97 Completion for Noetherian rings (“more general”)
(7.6.15): 10.9 Localization (“special case”)
(7.6.17): 10.112 Homomorphisms and dimension (“special case”)
(7.6.18): 10.97 Completion for Noetherian rings (“partial”)
(7.7.1): 87.4 Topological rings and modules (“lim M/V ⊗ N/W”; “limit identification”)
(7.7.2): 87.4 Topological rings and modules (“(M/V)⊗(N/W) identif.”; “completed tensor product”; “lim (M⊗N)/(...) identif.”)
(7.7.5): 87.4 Topological rings and modules (“completed tensor product”; “existence of open J”; “lim (M⊗N)/(Mμ⊗N+M⊗Nν)”; “lim M/Mμ ⊗ N/Nν”)
(7.7.7): 87.4 Topological rings and modules (“admissibility”)
(7.7.8): 10.97 Completion for Noetherian rings (“B = Â”)
(7.8.2): 15.102 Systems of modules (“canonical isomorphism”)

§12

(12.1.2): 20.11 Čech cohomology and cohomology (“A-linear Č^p → H^p maps”; “A-linear Ȟ^p → H^p”), 20.23 The alternating Čech complex (“alternating Čech complex”), 21.2 Cohomology of sheaves (“more general”)
(12.1.3): 18.42 Constant sheaves (“special case”)
(12.1.4): 20.14 Functoriality of cohomology (“inj. resolution constr.”; “inj. resolution descr.”), 20.15 Refinements and Čech cohomology (“(12.1.4.2)”), 13.18 Injective resolutions (“partial”)
(12.1.7): 20.29 Cohomology of filtered complexes (“boundedness”), 13.21 Cartan-Eilenberg resolutions (“partial”)
(12.2.1): 20.2 Cohomology of sheaves (“more general”), 20.12 Flasque sheaves (“partial”), 20.13 The Leray spectral sequence (“H^p(U,F) comparison”), 6.26 Morphisms of ringed spaces and modules, 21.2 Cohomology of sheaves (“more general”), 21.7 Locality of cohomology (“partial”)
(12.2.3): 20.54 Projection formula (“more general”; “partial”)
(12.2.4): 20.13 The Leray spectral sequence (“abutment”; “abutment, complexes”), 21.14 The Leray spectral sequence (“special case”), 103.11 Pushforward of quasi-coherent modules (“more general”)
(12.3.1): 20.43 Ext sheaves (“more general”), 12.6 Extensions (“partial”)
(12.3.2): 13.27 Ext groups (“E~E' ⇒ δE=δE'”; “canonical map δ”; “surjectivity of δ”; “δE=δE' ⇒ E~E'”), 12.5 Abelian categories, 12.6 Extensions, 15.56 Injective modules (“more general”)
(12.4.1): 20.29 Cohomology of filtered complexes (“''E2, absolute case”; “'E abutment, absolute”; “abutment, absolute case”), 13.21 Cartan-Eilenberg resolutions (“bounded below complexes”)
(12.4.3): 20.7 Locality of cohomology (“special case”), 21.7 Locality of cohomology (“degree-0 complex”)
(12.4.8): 20.14 Functoriality of cohomology (“(12.4.8.1)”)

§13

(13.1.1): 18.2 Abelian presheaves (“special case”)
(13.1.2): 10.86 Mittag-Leffler systems (“inverse systems of sets”; “modules over a ring”), 12.31 Inverse systems (“ℕ-indexed systems”)
(13.1.3): 10.86 Mittag-Leffler systems
(13.2.1): 12.31 Inverse systems (“(i), systems over ℕ”)
(13.2.2): 10.86 Mittag-Leffler systems, 10.87 Inverse systems (“ℕ-indexed systems”), 12.31 Inverse systems (“partial”; “systems over ℕ”; “ℕ-indexed systems”)
(13.2.3): 42.2 Periodic complexes and Herbrand quotients, 12.31 Inverse systems (“special case”), 64.7 Filtered derived category
(13.2.4): 20.2 Cohomology of sheaves (“more general”), 21.2 Cohomology of sheaves (“more general”)
(13.6.1): 20.2 Cohomology of sheaves, 13.16 Higher derived functors, 13.20 Right derived functors and injective resolutions (“existence of R^pT”), 21.2 Cohomology of sheaves
(13.6.2): 13.26 Filtered derived category and injective resolutions (“existence of resolution”)
(13.6.3): 13.26 Filtered derived category and injective resolutions (“existence of v”; “special case”)
(13.6.4): 13.26 Filtered derived category and injective resolutions (“E_1, abutment, biregular”; “E_r indep. of choices”; “partial”), 12.19 Filtrations, 12.24 Spectral sequences: filtered complexes (“partial”)
(13.7.3): 20.29 Cohomology of filtered complexes (“ringed space, bdd below”), 19.13 Additional remarks on Grothendieck abelian categories (“Ext functor case”)

§14

(14.1.2): 28.10 Dimension, 5.10 Krull dimension (“dim_x formula”; “dim_x(X)=inf_U dim(U)”)
(14.1.3): 5.10 Krull dimension
(14.1.6): 5.10 Krull dimension
(14.1.9): 10.60 Dimension (“affine case: X = Spec R”)
(14.1.10): 10.122 Quasi-finite maps (“(1)⇔(4) for fibres”; “(1)⇔(4), Spec f.t. alg”), 28.10 Dimension (“loc. Noetherian scheme”)
(14.1.12): 10.17 The spectrum of a ring
(14.2.1): 5.11 Codimension and catenary spaces (“codim of irreducible Z”)
(14.2.3): 5.11 Codimension and catenary spaces (“Y irreducible”)
(14.3.2): 5.11 Codimension and catenary spaces

§15

(15.1.1): 10.69 Quasi-regular sequences
(15.1.3): 10.69 Quasi-regular sequences (“partial”)
(15.1.4): 10.69 Quasi-regular sequences (“M-quasi-regular”)
(15.1.5): 10.69 Quasi-regular sequences (“special case”)
(15.1.7): 10.68 Regular sequences (“M-regular sequence”; “nonzero quotient added”), 10.69 Quasi-regular sequences (“quasi-regular half”)
(15.1.9): 10.69 Quasi-regular sequences (“converse, local”; “first assertion”; “first part, M = R”), 15.31 Koszul regular sequences (“special case”)
(15.1.10): 10.69 Quasi-regular sequences (“necessity”; “sufficiency”), 15.31 Koszul regular sequences (“special case”)
(15.1.11): 10.69 Quasi-regular sequences (“converse, local case”), 15.31 Koszul regular sequences (“(1) ⇔ (4), local case”)
(15.1.13): 10.68 Regular sequences (“special case”)
(15.1.14): 10.68 Regular sequences (“converse (ff case)”)
(15.1.16): 10.99 Criteria for flatness (“c) => a), M = B”), 10.128 More flatness criteria (“c)=>a), ess fin pres”)
(15.1.17): 10.39 Flat modules and flat ring maps (“more general”), 10.128 More flatness criteria (“(f_i) is M-regular”)
(15.1.20): 10.68 Regular sequences (“first assertion”)
(15.2.1): 10.69 Quasi-regular sequences (“special case”), 59.23 Examples of sheaves (“partial”), 7.10 Sheafification (“partial”)
(15.2.2): 10.68 Regular sequences (“special case”), 10.69 Quasi-regular sequences (“converse: q-reg ⇒ reg”; “special case”), 15.31 Koszul regular sequences (“converse: q-reg ⇒ reg”)
(15.2.4): 10.68 Regular sequences (“special case”)
(15.2.5): 10.68 Regular sequences (“affine, faithfully flat”)

§16

(16.1.1): 10.60 Dimension (“definition of dim A”; “partial”)
(16.1.3): 10.60 Dimension (“dim(A_p) = ht(p)”), 111.18 Catenary rings (“(16.1.3.2)”; “dim(A_p) = ht(p)”), 5.11 Codimension and catenary spaces (“codim(V(p)”)
(16.1.4): 10.105 Catenary rings (“catenary ring def”; “special case”)
(16.1.5): 10.112 Homomorphisms and dimension (“dim(B) ≤ dim(A)”; “ρ injective case”), 29.45 Integral and finite morphisms (“dim(B) ≤ dim(A)”; “equality case”)
(16.1.7): 10.40 Supports and annihilators (“partial”; “special case”)
(16.1.10): 10.60 Dimension (“case M = A”), 10.62 Support and dimension of modules (“local ring case”)
(16.2.1): 10.59 Noetherian local rings (“local case”), 43.15 Algebraic multiplicities (“partial”)
(16.2.2.2): 10.59 Noetherian local rings (“(1) deg bound, (2)”; “special case”)
(16.2.3): 10.59 Noetherian local rings (“special case”), 10.62 Support and dimension of modules (“special case”)
(16.2.3.1): 10.60 Dimension (“(ii), special case”)
(16.2.5): 10.60 Dimension (“local case”; “special case”)
(16.3.1): 10.9 Localization, 10.17 The spectrum of a ring, 10.60 Dimension (“b ⇒ a”; “more general”), 111.18 Catenary rings (“more general”)
(16.3.2): 10.60 Dimension (“more general”)
(16.3.6): 10.60 Dimension (“case M = R, local”)
(16.4.4): 10.72 Depth (“depth = least n Ext≠0”), 47.11 Depth (“Ext^i(k”)
(16.4.5): 10.72 Depth (“(16.4.5.1)”; “more general”; “partial”)
(16.4.6): 10.72 Depth (“(iii), first inequality”)
(16.4.8): 10.72 Depth (“B finite over A”)
(16.5.1): 10.103 Cohen-Macaulay modules (“CM module def”), 10.104 Cohen-Macaulay rings
(16.5.3): 10.103 Cohen-Macaulay modules (“surjective ρ”)
(16.5.4): 10.103 Cohen-Macaulay modules
(16.5.5): 10.103 Cohen-Macaulay modules (“b) ⇒ a)”; “b) ⇒ a), last part”), 10.104 Cohen-Macaulay rings (“M = R; a) ⇔ b)”), 40.12 Slicing groupoids (“special case”)
(16.5.6): 10.103 Cohen-Macaulay modules (“b) ⇒ a)”), 10.104 Cohen-Macaulay rings (“M = R (CM local ring)”)
(16.5.7): 10.103 Cohen-Macaulay modules (“CM: sop ⇒ max reg seq”), 10.104 Cohen-Macaulay rings (“case M = R”)
(16.5.9): 10.104 Cohen-Macaulay rings (“special case”), 47.11 Depth (“r = depth_p(M)”)
(16.5.10): 10.103 Cohen-Macaulay modules (“(ii), Supp M = Spec A”), 10.104 Cohen-Macaulay rings (“(i) for M = A”; “(ii) for M = A”)
(16.5.11): 10.103 Cohen-Macaulay modules, 10.104 Cohen-Macaulay rings (“case M = R”)
(16.5.12): 10.105 Catenary rings (“local CM ring case”)
(16.5.13): 10.103 Cohen-Macaulay modules (“forward direction”), 10.104 Cohen-Macaulay rings (“local ring”; “partial”)

§17

(17.1.1): 10.69 Quasi-regular sequences (“more general”), 10.106 Regular local rings (“(d) ⇒ (c)”; “more general”)
(17.1.3): 10.106 Regular local rings (“CM clause”), 10.135 Local complete intersections (“Cohen–Macaulay”)
(17.1.4): 10.119 Around Krull-Akizuki (“(ii)”), 15.40 Some results on power series rings (“more general”)
(17.1.5): 15.44 Permanence of properties under completion, 54.11 Base change to the completion (“Â regular ⇒ A regular”)
(17.1.7): 10.106 Regular local rings (“b⇒regularity of A”; “r=1: b⇒regularity of A”), 15.40 Some results on power series rings (“special case”)
(17.1.8): 10.106 Regular local rings (“a ⇒ b”)
(17.2.1): 10.77 Projective modules (“n = 0”), 10.109 Rings of finite global dimension (“Necessity”; “dim.proj definition”; “dim.proj resolution char”), 15.71 Injective dimension (“partial”)
(17.2.2): 10.109 Rings of finite global dimension, 13.20 Right derived functors and injective resolutions (“(ii), special case”)
(17.2.5): 10.109 Rings of finite global dimension (“partial”)
(17.2.8): 10.77 Projective modules (“special case”), 10.109 Rings of finite global dimension (“a) <=> a')”)
(17.2.10): 10.109 Rings of finite global dimension (“partial”)
(17.3.1): 10.110 Regular rings and global dimension (“Sufficiency, partial”; “fin. coh. dim ⇔ regular”)
(17.3.4): 10.110 Regular rings and global dimension (“dim.proj ≤ n - prof”), 10.111 Auslander-Buchsbaum (“more general”)
(17.3.5): 10.106 Regular local rings (“(i), sufficiency”), 10.112 Homomorphisms and dimension (“(ii) necessity”), 30.11 Depth (“(i)”)
(17.3.8.1): 15.44 Permanence of properties under completion (“local, J = maximal ideal”)

§18

(18.2.3): 91.4 Modules on first order thickenings of ringed spaces (“special case”), 91.13 Deformations of ringed topoi and the naive cotangent complex (“partial”)
(18.2.5): 91.13 Deformations of ringed topoi and the naive cotangent complex (“more general”)
(18.2.7): 12.5 Abelian categories (“more general”)
(18.4.1): 9.6 Field extensions
(18.5.3): 15.37 Topological rings and modules (“partial”), 5.30 Topological groups, rings, modules (“partial”)

§19

(19.0.1): 10.138 Formally smooth maps (“special case”), 59.9 Presheaves, 37.11 Formally smooth morphisms, 26.15 A representability criterion, 76.13 Formally smooth, étale, unramified transformations (“partial”), 106.8 Formally smooth morphisms (“more general”)
(19.0.2): 10.139 Smoothness and differentials (“special case”), 10.158 Formal smoothness of fields (“necessity”; “sufficiency”), 15.41 Geometric regularity and formal smoothness (“I”; “I, one direction”; “III existence”; “special case”), 15.50 Formal smoothness and regularity (“I, A→A^∧ regular”; “I, complete case”), 33.25 Schemes smooth over fields (“IV, special case”; “more general”)
(19.0.3): 87.4 Topological rings and modules (“𝔍B ⊂ 𝔎 clause”)
(19.1.1): 15.37 Topological rings and modules, 5.30 Topological groups, rings, modules
(19.1.8): 10.74 An application of Ext groups (“partial”)
(19.1.10): 15.3 Stably free modules (“special case”)
(19.1.12): 10.79 Open loci defined by module maps (“P₁,P₂ finite projective”)
(19.3.1): 15.38 Formally smooth maps of topological rings (“linearly topologized”)
(19.3.2): 10.138 Formally smooth maps (“special case”), 15.9 Lifting (“special case”)
(19.3.3): 10.138 Formally smooth maps
(19.3.5): 10.137 Smooth ring maps (“(iii), special case”), 10.138 Formally smooth maps (“(ii), special case”; “(iii), special case”), 15.38 Formally smooth maps of topological rings (“(iii)”)
(19.3.6): 15.38 Formally smooth maps of topological rings (“adic top., f.g. ideals”)
(19.3.8): 15.38 Formally smooth maps of topological rings (“adic topologies on S”; “discrete topologies”)
(19.3.10.1): 10.138 Formally smooth maps (“special case”)
(19.3.10.2): 10.15 Miscellany (“special case”)
(19.4.2): 10.137 Smooth ring maps (“Necessity, special case”), 10.138 Formally smooth maps (“Necessity, special case”)
(19.4.6): 10.137 Smooth ring maps (“special case”), 10.138 Formally smooth maps (“special case”)
(19.5.1): 17.21 Symmetric and exterior powers (“more general”)
(19.5.2): 31.20 The normal cone of an immersion (“more general”)
(19.5.3): 10.139 Smoothness and differentials (“(i), special case”)
(19.5.4): 10.139 Smoothness and differentials (“a) ⇒ c)”)
(19.5.4.1): 10.139 Smoothness and differentials (“a), special case”)
(19.5.4.4): 10.138 Formally smooth maps (“partial”; “special case”), 10.139 Smoothness and differentials (“partial”), 15.39 Formally smooth maps of local rings (“special case”)
(19.5.6.1): 91.2 Deformations of rings and the naive cotangent complex (“partial”)
(19.5.6.2): 98.23 Naive obstruction theories (“special case”)
(19.5.6.4): 10.138 Formally smooth maps (“special case”)
(19.5.7): 10.139 Smoothness and differentials (“Clause (i), special case”)
(19.6.1): 10.158 Formal smoothness of fields (“necessity”; “sufficiency”)
(19.6.3): 10.160 The Cohen structure theorem (“a)⇒c), regular case”), 33.21 Complete local rings (“(a) ⇒ (c), special case”)
(19.6.4): 10.160 The Cohen structure theorem (“special case”), 15.39 Formally smooth maps of local rings (“special case”), 15.41 Geometric regularity and formal smoothness (“char 0: a) ⇔ c)”)
(19.6.5): 10.160 The Cohen structure theorem (“Â ≅ K[[T_1,...,T_n]]”), 90.9 Smooth or unobstructed categories (“complete, K=k: (2)⇔(3)”; “complete, K=k: (2)⇔(5)”), 15.39 Formally smooth maps of local rings (“A regular (k'=k)”), 15.41 Geometric regularity and formal smoothness (“form. smooth ⟹ geom reg”)
(19.6.5.2): 15.41 Geometric regularity and formal smoothness (“existence of examples”), 33.21 Complete local rings (“not k-iso to K[[T]]”)
(19.6.6): 10.45 Perfect fields, 15.41 Geometric regularity and formal smoothness (“a) <=> c)”)
(19.6.7): 15.41 Geometric regularity and formal smoothness (“19.6.7(ii), special case”)
(19.7.1): 90.9 Smooth or unobstructed categories (“classical case”; “complete local case”), 15.41 Geometric regularity and formal smoothness (“b) ⇒ a)”; “more general”), 15.50 Formal smoothness and regularity (“A → A^ regular”; “complete case”)
(19.7.1.3): 15.41 Geometric regularity and formal smoothness (“A complete, B geom reg”)
(19.7.2): 15.41 Geometric regularity and formal smoothness (“existence, J = 𝔪”)
(19.8.2): 15.41 Geometric regularity and formal smoothness (“(ii) existence”)
(19.8.3): 10.9 Localization
(19.8.4): 10.160 The Cohen structure theorem (“partial”)
(19.8.6): 10.160 The Cohen structure theorem (“(ii), special case”)
(19.8.8): 10.160 The Cohen structure theorem (“(i)”; “(i), special case”; “(ii)”), 15.39 Formally smooth maps of local rings (“(i), special case”), 15.40 Some results on power series rings (“more general”), 33.21 Complete local rings (“(i), special case”)
(19.9.3): 10.138 Formally smooth maps (“discrete topology”), 15.38 Formally smooth maps of topological rings (“adic topology case”)
(19.9.4): 15.38 Formally smooth maps of topological rings (“a) <=> b) <=> c)”)
(19.10.2): 10.148 Formally unramified maps (“discrete, unramified”)
(19.10.3): 10.32 Locally nilpotent ideals (“more general”), 10.138 Formally smooth maps, 10.148 Formally unramified maps (“partial”), 10.150 Formally étale maps, 10.158 Formal smoothness of fields (“(iii)”)
(19.10.4): 10.148 Formally unramified maps (“partial, special case”), 37.6 Formally unramified morphisms (“partial”)

§20

(20.1.1): 10.131 Differentials (“special case”), 92.13 Comparison with Lichtenbaum-Schlessinger
(20.1.2): 10.131 Differentials (“Leibniz rule (ii)”; “commutative, 1-sided mod”; “vanishes on image of A”), 92.13 Comparison with Lichtenbaum-Schlessinger
(20.1.3): 91.7 Deformations of ringed spaces and the naive cotangent complex (“more general”), 18.33 Modules of differentials (“action of Der”; “section case (u = id)”)
(20.1.4): 92.16 Deformations of ring maps and the cotangent complex (“Commutative case”)
(20.3.1): 15.37 Topological rings and modules, 5.30 Topological groups, rings, modules (“Hom. cont(A,B)”)
(20.4.1): 10.131 Differentials, 87.4 Topological rings and modules
(20.4.3): 10.131 Differentials (“special case”)
(20.4.8): 10.131 Differentials (“(ii), derivation clause”; “(ii), discrete case”)
(20.4.9): 10.138 Formally smooth maps (“discrete case”)
(20.4.12): 10.131 Differentials (“discrete case”)
(20.4.13): 10.131 Differentials
(20.5.2): 90.3 The base category, 17.28 Modules of differentials (“(20.5.2.3) map u''”), 18.33 Modules of differentials (“(20.5.2.4) & (20.5.2.6)”)
(20.5.4): 90.3 The base category (“special case”)
(20.5.5): 10.131 Differentials (“special case”)
(20.5.7): 10.131 Differentials (“partial”), 10.138 Formally smooth maps (“(ii), sufficiency”), 90.3 The base category
(20.5.9): 10.131 Differentials
(20.5.11): 10.131 Differentials (“formula on classes”), 90.3 The base category
(20.5.12): 10.131 Differentials (“(i)”; “partial”), 10.134 The naive cotangent complex (“(i), special case”), 10.149 Conormal modules and universal thickenings (“kernel of E → C is R/R²”), 90.3 The base category (“partial”)
(20.5.14): 10.138 Formally smooth maps (“injectivity, splitting”), 10.139 Smoothness and differentials (“injectivity, splitting”)
(20.6.3): 10.158 Formal smoothness of fields
(20.6.5): 10.131 Differentials (“H_0 = Ω_{C/B}”), 90.3 The base category (“differential K₁ → K₀”)
(20.6.9): 10.131 Differentials (“partial”)
(20.6.17): 10.131 Differentials (“exact at Ω¹_{C/A}”; “second assertion”)
(20.6.21): 10.39 Flat modules and flat ring maps (“more general”), 35.4 Descent for universally injective morphisms (“partial”)
(20.6.22): 10.131 Differentials (“Case Λ = A”), 10.139 Smoothness and differentials (“partial”), 90.3 The base category (“last three terms”)
(20.6.23): 10.131 Differentials (“partial”), 90.3 The base category (“partial”), 15.41 Geometric regularity and formal smoothness (“residue field, m-adic”)
(20.6.26): 64.7 Filtered derived category
(20.7.2): 10.138 Formally smooth maps (“sufficiency”)
(20.7.3): 10.138 Formally smooth maps (“discrete case, necessity”)
(20.7.4): 10.148 Formally unramified maps (“discrete case”)
(20.7.20): 10.160 The Cohen structure theorem (“special case”)

§21

(21.1.1): 9.6 Field extensions (“more general”)
(21.1.2): 9.5 The characteristic of a field (“special case”)
(21.1.4): 10.14 Base change (“partial”)
(21.1.7): 15.47 Field extensions, revisited (“a)⇒c), special case”)
(21.2.4): 15.47 Field extensions, revisited (“partial”; “special case”)
(21.2.5): 10.131 Differentials (“special case”), 15.47 Field extensions, revisited (“partial”; “special case”)
(21.2.6): 33.36 Frobenii (“more general”)
(21.2.7): 10.138 Formally smooth maps (“special case”)
(21.3.2): 33.36 Frobenii (“more general”)
(21.3.6): 10.131 Differentials (“partial”)
(21.4.2): 15.47 Field extensions, revisited (“more general”)
(21.4.4): 10.158 Formal smoothness of fields (“more general”), 15.47 Field extensions, revisited (“more general”)
(21.4.5): 15.47 Field extensions, revisited (“p-base case”)
(21.4.6): 10.158 Formal smoothness of fields (“(a) ⇔ (b), special case”), 15.47 Field extensions, revisited (“(b) ⇔ (c)”)
(21.5.1): 10.158 Formal smoothness of fields (“more general”)
(21.7.1): 10.131 Differentials (“more general”), 15.35 Cartier's equality and geometric regularity
(21.7.2): 15.35 Cartier's equality and geometric regularity (“special case”)
(21.7.4): 10.148 Formally unramified maps (“b) <=> d)”; “b) ⇔ d)”), 10.158 Formal smoothness of fields (“more general”)
(21.8.1): 15.47 Field extensions, revisited
(21.8.3): 15.47 Field extensions, revisited (“a) => c), k₀=F_p”)
(21.8.5): 15.47 Field extensions, revisited (“k₀=K^p, L/K finite”)
(21.8.8): 10.9 Localization, 9.3 Examples of fields, 15.47 Field extensions, revisited (“∩K_J=K^p, A_J⊂A finite”)
(21.9.2): 10.139 Smoothness and differentials (“special case”), 15.39 Formally smooth maps of local rings (“special case”)
(21.9.5): 11.2 Noncommutative algebras, 9.7 Finite extensions
(21.9.7): 9.26 Transcendence

§22

(22.1.1): 10.139 Smoothness and differentials (“(ii), condition α”), 31.20 The normal cone of an immersion (“more general”)
(22.1.2): 15.41 Geometric regularity and formal smoothness (“base a field, Noetherian”)
(22.1.3): 33.16 Tangent spaces
(22.1.3.2): 33.16 Tangent spaces
(22.1.4): 15.39 Formally smooth maps of local rings
(22.2.1): 10.158 Formal smoothness of fields (“special case”)
(22.2.2): 15.41 Geometric regularity and formal smoothness (“(i)&(ii), char p”)
(22.2.3): 15.41 Geometric regularity and formal smoothness (“a) (char p)”)
(22.2.7): 15.41 Geometric regularity and formal smoothness (“(i)”)
(22.2.9): 90.3 The base category
(22.3.1): 10.31 Noetherian rings (“Noetherianity of A”), 10.160 The Cohen structure theorem (“special case”)
(22.3.4): 15.51 G-rings (“special case”)
(22.4.1): 10.18 Local rings, 11.2 Noncommutative algebras, 9.7 Finite extensions (“(i), special case”)
(22.4.4.2): 10.131 Differentials (“partial”), 10.138 Formally smooth maps (“partial”), 10.139 Smoothness and differentials (“partial”)
(22.4.4.7): 10.131 Differentials (“more general”)
(22.4.6): 10.138 Formally smooth maps (“(22.4.6.1)”)
(22.4.8): 10.39 Flat modules and flat ring maps, 10.82 Universally injective module maps (“more general”), 10.131 Differentials (“more general”; “partial”), 41.9 Flat morphisms, 90.3 The base category (“partial”)
(22.5.3): 10.18 Local rings
(22.5.5): 10.156 Henselization and quasi-finite ring maps (“partial”)
(22.5.8): 10.166 Geometrically regular algebras (“b) ⇔ b')”), 15.36 Geometric regularity (“b) ⇔ b'), char p”), 15.39 Formally smooth maps of local rings (“a) ⇒ A regular”), 15.41 Geometric regularity and formal smoothness
(22.6.4): 10.131 Differentials (“partial”), 10.137 Smooth ring maps (“special case”), 90.3 The base category (“partial”), 15.39 Formally smooth maps of local rings (“c) <=> d)”)
(22.6.7): 10.140 Smooth algebras over fields (“alg. closed k, max ideal”; “separable residue field”), 15.41 Geometric regularity and formal smoothness (“(i) b) ⇔ d)”)
(22.7.2): 15.41 Geometric regularity and formal smoothness (“a) <=> c) char 0”)
(22.7.6): 15.49 Regularity and derivations

§23

(23.1.1): 10.161 Japanese rings (“def. of Japanese”), 10.162 Nagata rings
(23.1.2): 10.161 Japanese rings (“special case”)
(23.1.3): 10.161 Japanese rings
(23.1.4): 10.161 Japanese rings (“case r = 1”)
(23.1.5): 10.162 Nagata rings (“more general”)
(23.1.7): 10.43 Geometrically reduced algebras (“partial”)
(23.2.1): 15.108 Local irreducibility (“def. of geom. unibranch”; “def. of unibranch”)
(23.2.2): 10.46 Universal homeomorphisms (“more general”), 29.55 Normalization (“sufficiency + homeo”)
(23.2.5): 10.31 Noetherian rings (“Noetherian clause”)
(23.2.6): 10.120 Factorization (“residue extensions”), 33.37 Glueing dimension one rings (“K' = K case”)
(23.2.7): 10.120 Factorization (“partial”)
(23.2.8): 10.120 Factorization (“(ii), dim ≤ 1”)
(23.2.9): 10.157 Serre's criterion for normality (“A normal (A = A')”), 15.24 Reflexive modules (“special case”)

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