Index of terminology

Alphabetized by the English head term, merging the four French part-indices of EGA IV into a single English index; section numbers refer to the defining occurrence, with the chapter prefix (0_IV, IV) included to disambiguate Chapter 0 (continued) entries from Chapter IV proper.

A

• $A$-algebra essentially of finite type: IV.1.3.8.
• $A$-algebra of finite presentation: IV.1.4.1.
• $A$-anneau, see $A$-ring.
• $A$-derivation, see Derivation, $A$-.
• $A$-equivalence of $A$-extensions: 0_IV.18.2.2.
• $A$-extension of an $A$-ring by a $B$-bimodule: 0_IV.18.2.2.
• $A$-extension, $A$-trivial: 0_IV.18.2.3, 0_IV.18.3.7.
• $A$-extensions, $A$-equivalent: 0_IV.18.2.2.
• $A$-homomorphism of $A$-rings: 0_IV.18.1.1.
• $A$-ring: 0_IV.18.1.1.
• $A$-ring augmented over $B$: 0_IV.18.1.4.
• Affine envelope (of an $X$-prescheme): IV.21.12.1.
• Affineness defect (of an open in a prescheme): IV.21.12.5.
• Algebra, essentially étale: IV.18.6.1.
• Algebra, étale: IV.17.3.2.
• Algebra, finite and étale (over ${\mathcal{O}}_X$): IV.18.2.3.
• Algebra, formally étale: 0_IV.19.10.2.
• Algebra, formally smooth: 0_IV.19.3.1.
• Algebra, formally smooth relative to a ring: 0_IV.19.9.1.
• Algebra, formally unramified: 0_IV.19.10.2.
• Algebra, of finite presentation: IV.1.4.1.
• Algebra, smooth: IV.17.3.2.
• Algebra, strictly essentially étale: IV.18.6.2.
• Algebra, unramified: IV.17.3.2.
• Algebra, $K$-, unramified over $Y$, over $A$: IV.18.10.11.
• Antifilter of subsets of a set: IV.5.10.8.
• Augmentation: 0_IV.18.1.4.
• Augmentation ideal: 0_IV.18.1.4.
• Augmented algebra of principal parts of order 1: 0_IV.20.4.2.

B

• Base field change: see Change of base field.
• Biequidimensional ring: 0_IV.16.1.4.
• Biequidimensional space: 0_IV.14.3.3.
• Birational morphism of reduced preschemes: IV.6.15.4.

C

• Cartier's equality: see Equality, Cartier's.
• Catenary ring: 0_IV.16.1.4.
• Catenary space: 0_IV.14.3.2.
• Chain (in an ordered set, of irreducible closed subsets of a topological space): 0_IV.14.1.1.
• Chain condition: 0_IV.14.3.2.
• Chain, saturated: 0_IV.14.3.1.
• Change of base field: IV.4.
• Chow's lemma for morphisms of finite presentation: IV.8.10.5.1.
• Codepth of a module over a Noetherian local ring: 0_IV.16.4.9.
• Codepth of a Module at a point, of a Module, of a prescheme: IV.5.7.1.
• Codimension at a point of a closed subset of a topological space: 0_IV.14.2.4.
• Codimension of a cycle: IV.21.6.1.
• Codimension of an arbitrary subset of a prescheme: IV.5.1.3.
• Codimension, combinatorial: 0_IV.14.2.1.
• Codimension, transversal, at a point of a sub-prescheme: IV.19.1.3.
• Codimension, transversal, of an immersion morphism at a point: IV.19.1.3.
• Cohen $A$-algebra: 0_IV.19.8.1.
• Cohen ring: 0_IV.19.8.4.
• Cohen ring, $p$-: 0_IV.19.8.4.
• Cohen-Macaulay module: 0_IV.16.5.1, 0_IV.16.5.13.
• Cohen-Macaulay morphism at a point, Cohen-Macaulay morphism: IV.6.8.1.
• Cohen-Macaulay point of a prescheme: IV.5.7.1.
• Cohen-Macaulay prescheme: IV.5.7.1.
• Cohen-Macaulay ring: 0_IV.16.5.1, 0_IV.16.5.13.
• Cohen-Macaulay ${\mathcal{O}}_X$-Module at a point, Cohen-Macaulay ${\mathcal{O}}_X$-Module: IV.5.7.1.
• Cohomological dimension of a ringed space at a point, cohomological dimension of a ringed space: 0_IV.17.2.14.
• Cohomological dimension, global, of a ring: 0_IV.17.2.8.
• Cohomological dimension: 0_IV.17.2.8.
• Coincidences (prescheme of, of two morphisms): IV.17.4.5.
• Combinatorial codimension, see Codimension, combinatorial.
• Combinatorial dimension, see Dimension, combinatorial.
• Complete intersection morphism: IV.19.3.6.
• Complete intersection ring: IV.19.3.1.
• Complete intersection ring, absolute: IV.19.3.1.
• Complete intersection, prescheme, absolute, at a point: IV.19.3.1.
• Connected morphism: IV.4.5.5.
• Conormal sheaf: IV.16.1.2.
• Constructible property, ind-constructible property: IV.9.2.1, IV.9.2.2 (i)-(ii).
• Constructible subset, see Subset, constructible.
• Constructible topology on a prescheme: IV.1.9.12.
• Couple, henselian: IV.18.5.5.
• Couple, parafactorial: IV.21.13.1.
• Cycle: IV.21.6.1.
• Cycle associated, prime, to a Module, to a prescheme: IV.3.1.1.
• Cycle associated, prime, embedded: IV.3.1.1.
• Cycle, 1-codimensional, with rational coefficients: IV.21.10.9.
• Cycle, $p$-codimensional: IV.21.6.2.
• Cycle linearly equivalent to 0: IV.21.6.7.
• Cycle, locally principal: IV.21.6.7.
• Cycle of poles, cycle of zeros of a meromorphic function: IV.21.6.7.
• Cycle, polar, of a meromorphic function: IV.21.6.7.
• Cycle, prime: IV.21.6.1.
• Cycle, principal: IV.21.6.7.
• Cycle, principal at a point: IV.21.6.7.
• Cycle, purely of codimension $p$: IV.21.6.2.

D

• $D_B(L)$: see Extension, trivial type.
• Decomposition, irredundant, of a Module; reduced irredundant decomposition: IV.3.2.5.
• Decomposition, primary, of 0 in a Module: IV.3.2.5.
• Defect, $k_0$-admissibility, of an extension: 0_IV.21.6.1.
• Defined on a field (subset of a prescheme): IV.4.8.4.
• Defined on a field (sub-${\mathcal{O}}_X$-Module): IV.4.8.4.
• Defined on a field (sub-prescheme): IV.4.8.4.
• Definition, domain of, of a meromorphic function: IV.20.1.4.
• Definition, domain of, of a meromorphic section of a Module: IV.20.1.7.
• Definition, domain of, of a pseudo-morphism: IV.20.2.3.
• Definition, field of: IV.4.8.4.
• Depth, lower semi-continuity of, see Semi-continuity, lower, of depth.
• Depth of a Module at a point, of an ${\mathcal{O}}_X$-Module at a point: IV.5.7.1.
• Depth of a module on a Noetherian local ring: 0_IV.16.4.5.
• Depth, rectified, see Rectified depth.
• Depth, $T$-, of a coherent ${\mathcal{O}}_X$-Module at a point: IV.19.9.1.
• Derivation, $A$-: 0_IV.20.1.2.
• Derivation, $S$-, $(X/S)$-, $f$-: IV.16.5.1.
• Differential of a section of ${\mathcal{P}}_{B/A}^1$: IV.16.3.6.
• Differential, exterior: 0_IV.20.4.6, IV.16.6.3.
• Differentials, $i$-differentials, absolute differentials: 0_IV.20.4.3.
• Differentially smooth morphism: IV.16.10.1.
• Differentially smooth morphism up to order $r$: IV.16.11.3.
• Differentially smooth prescheme over $S$: IV.16.10.1.
• Di-homomorphism of bimodules, see Homomorphism, di-.
• Dimension at a point of a topological space: 0_IV.14.1.2.
• Dimension, cohomological, see Cohomological dimension.
• Dimension, cohomological, of a ringed space at a point, see Cohomological dimension.
• Dimension, combinatorial, of a topological space: 0_IV.14.1.2.
• Dimension formula: IV.5.5.8, IV.14.2.1.
• Dimension, Krull, of a ring: 0_IV.16.1.1.
• Dimension of a cycle: IV.21.6.1.
• Dimension of a module: 0_IV.16.1.7, IV.5.1.12.
• Dimension of a prescheme over a field: IV.4.1.1.
• Dimension, projective / injective, in a point, of a sheaf: 0_IV.17.2.14.
• Dimension, projective / injective, of a module: 0_IV.17.2.1.
• Dimension, relative, of a morphism at a point, of a $Y$-prescheme at a point: IV.17.10.1.
• Discriminant of a $Y$-prescheme finite and locally free: IV.18.2.7.
• Discriminant ideal of a $Y$-prescheme finite and locally free: IV.18.2.7.
• Divisor (on a ringed space): IV.21.1.2.
• Divisor of a meromorphic function: IV.21.1.2.
• Divisor of a meromorphic section of an invertible Module: IV.21.1.4.
• Divisor, positive: IV.21.1.6.
• Divisor, principal: IV.21.3.1.
• Divisor, relative to $S$, transversal to $f$: IV.21.15.2.
• Divisors, linearly equivalent: IV.21.3.1.
• Domain of definition, see Definition, domain of.

E

• ${\mathcal{E}}_{pure}$, see $i$-pure.
• Eisenstein polynomial, see Polynomial, "Eisenstein".
• Element, non-zero-divisor of 0 in an $A$-module: 0_IV.15.1.1.
• Element, $M$-regular, $M$-quasi-regular: 0_IV.15.1.4.
• Embedded associated prime cycle, see Cycle associated, prime, embedded.
• Envelope, affine, see Affine envelope.
• Epimorphism, formal: 0_IV.19.1.2.
• Equality, Cartier's: 0_IV.21.7.1.
• Equicodimensional ring: 0_IV.16.1.4.
• Equicodimensional space: 0_IV.14.2.1.
• Equidimensional at a point (${\mathcal{O}}_X$-Module): IV.5.1.12.
• Equidimensional at a point (morphism, prescheme): IV.13.2.2, IV.13.3.2.
• Equidimensional morphism, equidimensional prescheme over another: IV.13.2.2, IV.13.3.2.
• Equidimensional ring: 0_IV.16.1.4.
• Equidimensional ring, formally: IV.7.1.1.
• Equidimensional ring, strictly: IV.7.2.1.
• Equidimensional space: 0_IV.14.1.3.
• Equivalent morphisms: IV.20.2.1.
• Essentially affine prescheme over another: IV.8.13.4.
• Essentially affine pro-object: IV.8.13.4, IV.8.14.1.
• Essentially proper morphism: IV.18.10.20.
• Étale, see Algebra, étale, Morphism, étale, Prescheme, étale.
• Étale cover, étale cover locally trivial, trivial étale cover: IV.18.2.7.
• Excellent prescheme: IV.7.8.5.
• Excellent ring: IV.7.8.2.
• Exponent, inseparability, of an extension of a field: IV.4.7.4.
• Extension, $k_0$-admissible, of a field for an extension: 0_IV.21.6.1.
• Extension, finite, principle of: IV.9.1.1.
• Extension, Hochschild: 0_IV.18.4.3.
• Extension, primary, of a field: IV.4.3.1.
• Extension, separable algebraic closure of a field in an extension: IV.4.3.4.
• Extension, trivial, type: 0_IV.18.2.3.
• Extension, $A$-, of an $A$-ring $B$ by a $B$-bimodule: 0_IV.18.2.2.
• Extension, $A$-trivial, of a $B$-ring: 0_IV.18.3.7.
• Extension, $B$-, $A$-trivial: 0_IV.18.3.7.
• Extension deduced from another by a homomorphism of bimodules: 0_IV.18.2.8.

F

• $f^{cons}$, see Constructible topology.
• Faithfully flat Module relative to a morphism (${\mathcal{O}}_X$-): IV.2.2.4.
• Family, $p$-free; family, absolutely $p$-free: 0_IV.21.1.9.
• Family of divisors on $X$ relative to $S$, parametrized by $S^{\prime }$: IV.21.15.9.
• Family of morphisms, schematically dominant: IV.11.10.2.
• Family of morphisms, universally schematically dominant: IV.11.10.7.
• Family of sub-preschemes, schematically dense: IV.11.10.2.
• Family of sub-preschemes, transversal at a point: IV.17.13.10.
• Family of sub-preschemes intersecting transversally at a point: IV.17.13.11.
• Family, separating, of homomorphisms of modules: IV.11.9.4.
• Family, separating, of homomorphisms of sheaves of modules: IV.11.9.1.
• Family, universally separating, of homomorphisms of sheaves of modules: IV.11.9.14.
• Fibre, formal, see Formal fibre.
• Field, $k_0$-admissible, see Extension, $k_0$-admissible.
• Field, geometrically regular, see Regular, geometrically.
• Field of definition: see Definition, field of.
• Field of multiplicity, radicial, finite, over a subfield: 0_IV.19.6.6.
• Field, prime local ring of a, see Ring, prime local.
• Field, separably closed: IV.4.3.3.
• Finite étale, see Algebra, finite étale.
• Finite locally free morphism of rank $n$, finite locally free morphism: IV.18.2.7.
• Finite locally free $Y$-prescheme, see Prescheme, $Y$-, finite and locally free.
• Finite saturated part of an algebraic $k$-prescheme: IV.9.8.9.
• Flat fibrewise criterion, see Flatness, fibrewise criterion of.
• Flatness, fibrewise criterion of: IV.11.3.10.
• Flat morphism, see Morphism, flat.
• Formal bimorphism, see Bimorphism, formal.
• Formal epimorphism, see Epimorphism, formal.
• Formal fibre of a ring at a point of its spectrum: IV.7.3.13.
• Formal monomorphism, see Monomorphism, formal.
• Formally catenary ring: IV.7.1.9.
• Formally equidimensional ring: IV.7.1.1.
• Formally étale (algebra): 0_IV.19.10.2.
• Formally étale, formally smooth, formally net, formally unramified morphism: IV.17.1.1.
• Formally left-invertible homomorphism: 0_IV.19.1.5.
• Formally right-invertible homomorphism: 0_IV.19.1.15.
• Formally projective module: 0_IV.19.2.1.
• Formally projective module, strictly: 0_IV.19.2.3.
• Formally smooth (algebra): 0_IV.19.3.1.
• Formally smooth algebra relative to a ring: 0_IV.19.9.1.
• Formally smooth, formally étale, formally unramified prescheme over $S$: IV.17.1.1.
• Formally unramified (algebra): 0_IV.19.10.2.
• Formally principal homogeneous sheaf (or pseudo-torsor) under a sheaf of groups: IV.16.5.15.
• Fractional ideal, see Ideal, fractional.
• Function, meromorphic, on a ringed space: IV.20.1.3.
• Function, meromorphic, defined at a point: IV.20.1.4.
• Function, meromorphic, regular: IV.20.1.8.
• Function, meromorphic, regular relative to $S$: IV.20.6.5.
• Function, meromorphic, relative to $S$: IV.20.6.1.
• Function, pseudo-: IV.20.2.8.
• Function, pseudo-, relative to $S$: IV.20.5.4.
• Fundamental system of open ideals, of open submodules: 0_IV.19.0.2.

G

• Generators, system of $p$-, see System of $p$-generators.
• Generization (of a point): IV.1.10.1.
• Geometric length, see Length, geometric.
• Geometric number of connected components, of irreducible components, for a prescheme over a field: IV.4.5.2.
• Geometric primary type of a Module, see Type, primary, geometric.
• Geometrically connected prescheme (over a field): IV.4.5.2.
• Geometrically integral prescheme (over a field): IV.4.6.2.
• Geometrically integral ${\mathcal{O}}_X$-Module over a field: IV.4.6.19.
• Geometrically irreducible polynomial: IV.9.7.4.
• Geometrically irreducible prescheme (over a field): IV.4.5.2.
• Geometrically isomorphic $k$-preschemes: IV.9.1.4.
• Geometrically normal at a point, geometrically normal prescheme (over a field): IV.6.7.6.
• Geometrically normal, reduced, regular ring, ring possessing the property $(R_k)$ geometric: IV.6.7.6.
• Geometrically pointwise integral ${\mathcal{O}}_X$-Module over a field: IV.4.6.22.
• Geometrically pointwise integral prescheme (over a field): IV.4.6.9.
• Geometrically reduced (or separable) ${\mathcal{O}}_X$-Module over a field: IV.4.6.17.
• Geometrically reduced (or separable) ${\mathcal{O}}_X$-Module over a field at a point: IV.4.6.22.
• Geometrically reduced at a point, geometrically reduced prescheme (over a field): IV.4.6.9.
• Geometrically reduced prescheme (over a field): IV.4.6.2.
• Geometrically regular at a point, geometrically regular prescheme (over a field): IV.6.7.6.
• Geometrically regular ring over a field (local ring): 0_IV.19.6.5.
• Geometrically unibranch point, unibranch point of a prescheme: IV.6.15.1.
• Geometrically unibranch prescheme at a point, geometrically unibranch prescheme: IV.6.15.1.
• Geometrically unibranch ring (local): 0_IV.23.1.7.
• Graph of a pseudo-$S$-morphism: IV.20.4.1.
• Group of classes of 1-codimensional cycles: IV.21.6.7.
• Group of classes of $A$-extensions: 0_IV.18.3.4.

H

• Height of an ideal: 0_IV.16.1.3.
• Henselian couple: IV.18.5.5.
• Henselian local scheme: IV.18.5.8.
• Henselian ring: IV.18.5.8.
• Henselization of a local ring: IV.18.6.5.
• Henselization of a semi-local ring: IV.18.6.7.
• Henselization, strict, of a local ring: IV.18.8.7.
• Henselization, strict, of a semi-local ring: IV.18.8.9.
• Hilbert polynomial, see Polynomial, Hilbert.
• Hilbert-Samuel polynomial: 0_IV.16.2.1.
• Hironaka's lemma, see Lemma, Hironaka's.
• Hochschild extension, see Extension, Hochschild.
• Homeomorphism, universal: IV.2.4.2.
• Homomorphism, $A$-, of $A$-rings: 0_IV.18.1.1.
• Homomorphism associated with the trace: IV.18.2.2.
• Homomorphism, characteristic, of an $A$-algebra relative to a ring and an ideal: 0_IV.20.6.24.
• Homomorphism, characteristic, of a $B$-extension $A$-trivial: 0_IV.20.6.8.
• Homomorphism, characteristic, of an algebra of characteristic $p>0$: 0_IV.22.4.6.
• Homomorphism, di-, of bimodules: 0_IV.18.2.4.
• Homomorphism of Modules defined on a field: IV.4.8.4.
• Homomorphism of Modules universally injective: IV.11.9.14, IV.11.9.18.
• Homomorphism, regular, of ${\mathcal{O}}_X$-Modules: IV.19.4.11.
• Homomorphism, semi-local: IV.18.6.7.
• Homomorphism, trace: IV.18.2.2.

I

• Ideal of denominators of a meromorphic section: IV.20.2.14.
• Ideal, discriminant, of a $Y$-prescheme finite and locally free: IV.18.2.7.
• Ideal, fractional: IV.21.2.1.
• Ideal, fractional, invertible: IV.21.2.1.
• Ideal, integral: IV.21.2.7.
• Ideal, quasi-regular, of a ring: IV.16.9.7.
• Ideal, quasi-regular, of a sheaf of rings: IV.16.9.1.
• Ideal, regular, of a ring: IV.16.9.7.
• Ideal, regular, of a sheaf of rings: IV.16.9.1.
• Ideal, transversally regular, of an algebra: IV.19.2.1.
• Ideal, transversally regular at a point of a prescheme: IV.19.2.1.
• Image, direct, of a 1-codimensional cycle: IV.21.10.14.
• Image, direct, of a divisor: IV.21.5.5.
• Image, inverse, of a 1-codimensional cycle: IV.21.10.3, IV.21.10.11.
• Image, inverse, of a divisor: IV.21.4.2.
• Image, inverse, of a meromorphic function: IV.20.1.11.
• Image, inverse, of a meromorphic function relative to $S$: IV.20.6.7.
• Image, inverse, of an $A$-extension: 0_IV.18.2.5.
• Image, inverse, of an augmented ring: 0_IV.18.1.5.
• Immersion, quasi-regular: IV.16.9.2.
• Immersion, regular: IV.16.9.2.
• Immersion, regular at a point: IV.16.9.10.
• Immersion, transversally regular, at a point relative to $S$, transversally regular relative to $S$: IV.19.2.2.
• Imperfection module, see Module, imperfection.
• Ind-constructible subset, see Subset, ind-constructible.
• Inseparability exponent, see Exponent, inseparability.
• Infinitesimal neighbourhood, $n$-th: IV.16.1.2.
• Intersection complete at a point relative to $S$, intersection complete relative to $S$ (for a flat prescheme locally of finite presentation): IV.19.3.6.
• Intersection complete ring, see Complete intersection ring.
• Invariant, $n$-th normal, of a morphism: IV.16.1.2.
• Invariant, normal, of infinite order, of a morphism: IV.16.1.11.
• Irreducible morphism: IV.4.5.5.
• Irredundant decomposition, see Decomposition, irredundant.
• Irredundant ${\mathcal{O}}_X$-Module: IV.3.2.4.
• Irredundant prescheme: IV.3.2.4.

J

• Jacobson, prescheme of: IV.10.4.1.
• Jacobson, space of: IV.10.3.1.
• Japanese ring: 0_IV.23.1.1.
• Japanese ring, universally: 0_IV.23.1.1.

L

• Lemma, Chow's, for morphisms of finite presentation: IV.8.10.5.1.
• Lemma, Hironaka's: IV.5.12.8.
• Length, geometric, of a Module at a point: IV.4.7.5.
• Length of a chain: 0_IV.14.1.1.
• Linear tangent map, see Tangent, linear map.
• Linearly equivalent cycles, see Cycle linearly equivalent to 0.
• Linearly equivalent divisors, see Divisors, linearly equivalent.
• Local ring, geometrically unibranch, see Geometrically unibranch ring.
• Local ring, unibranch, see Unibranch ring.
• Locally factorial prescheme: IV.21.6.9.

M

• "Main theorem" of Zariski for morphisms of finite presentation: IV.8.12.6.
• Map, open, at a point: IV.1.10.2.
• Map, rational, strict: IV.20.2.1.
• Maximal point of a prescheme: IV.1.1.4.
• Maximal spectrum, see Spectrum, maximal, of a Jacobson ring.
• Meromorphic function, see Function, meromorphic.
• Meromorphic section, see Section, meromorphic.
• Module, augmented algebra of principal parts of order 1, see Augmented algebra of principal parts.
• Module, biequidimensional, catenary, equicodimensional, equidimensional: 0_IV.16.1.7.
• Module, Cohen-Macaulay: 0_IV.16.5.1, 0_IV.16.5.13.
• Module, $\mathcal{F}$-fidelily flat (relative to a morphism), see Faithfully flat Module.
• Module, geometrically integral over a field, see Geometrically integral ${\mathcal{O}}_X$-Module.
• Module, geometrically pointwise integral over a field, see Geometrically pointwise integral ${\mathcal{O}}_X$-Module.
• Module, geometrically reduced (or separable) over a field, see Geometrically reduced ${\mathcal{O}}_X$-Module.
• Module, $A$-, possessing the property $(S_k)$: IV.5.7.2.
• Module, imperfection: 0_IV.20.6.1.
• Module, intègre, see ${\mathcal{O}}_X$-Module, integral.
• Module, irredundant, see Irredundant ${\mathcal{O}}_X$-Module.
• Module, $n$-th invariant normal, see Invariant, $n$-th normal.
• Module of 1-differentials of $f$ (or of $X$ relative to $S$, or of the $S$-prescheme $X$): IV.16.3.1.
• Module, reduced, over a ring: IV.3.2.2.
• Module, torsion-free, relative to $S$ (${\mathcal{O}}_X$-): IV.20.6.2.
• Module, torsion-free (${\mathcal{O}}_X$-): IV.20.1.5.
• Monomorphism, formal: 0_IV.19.1.2.
• Morphism, birational, of reduced preschemes: IV.6.15.4.
• Morphism, Cohen-Macaulay at a point, Cohen-Macaulay morphism: IV.6.8.1.
• Morphism, codepth $\le n$, at a point: IV.6.8.1.
• Morphism, complete intersection, see Complete intersection morphism.
• Morphism, connected: IV.4.5.5.
• Morphism, defined on a field: IV.4.8.4.
• Morphism, differentially smooth: IV.16.10.1.
• Morphism, differentially smooth up to order $r$: IV.16.11.3.
• Morphism, equidimensional, see Equidimensional morphism.
• Morphism, essentially proper: IV.18.10.20.
• Morphism, étale: IV.17.3.1.
• Morphism, étale at a point: IV.17.3.7.
• Morphism, finite and locally free, of rank $n$, see Finite locally free morphism.
• Morphism, finite presentation (Module), see Module of finite presentation.
• Morphism, formally étale, formally smooth, formally net, formally unramified: IV.17.1.1.
• Morphism, irreducible: IV.4.5.5.
• Morphism, lisse, see Morphism, smooth.
• Morphism, locally of finite presentation: IV.1.4.2.
• Morphism, normal at a point, normal morphism: IV.6.8.1.
• Morphism of $A$-extensions: 0_IV.18.2.4.
• Morphism of finite presentation: IV.1.6.1.
• Morphism of finite presentation at a point: IV.1.4.2.
• Morphism of pro-objects: IV.8.13.3.
• Morphism of ultra-preschemes: IV.10.9.5.
• Morphism (plat) of complete intersection: IV.19.3.6.
• Morphism, possessing the property $(R_k)$ at a point, possessing the property $(R_k)$: IV.6.8.1.
• Morphism, possessing the property $(S_k)$ at a point, possessing the property $(S_k)$: IV.6.8.1.
• Morphism, proper at a point: IV.15.7.1.
• Morphism, pseudo-finite: IV.8.12.3.
• Morphism, pseudo-, see Pseudo-morphism.
• Morphism, quasi-compact, see Quasi-compact morphism.
• Morphism, quasi-faithfully flat, quasi-flat: IV.2.3.3.
• Morphism, quasi-separated: IV.1.2.1.
• Morphism, radicial at a point: IV.6.15.3.
• Morphism, reduced at a point, reduced morphism: IV.6.8.1.
• Morphism, regular at a point, regular morphism: IV.6.8.1.
• Morphism, resolving: IV.7.9.1.
• Morphism, smooth at a point, smooth morphism: IV.6.8.1, IV.17.3.1, IV.17.3.7.
• Morphism, submersive, universally submersive: IV.15.7.8.
• Morphism, transversal to $Y$ at the point $x$ relative to $S$: IV.17.13.3.
• Morphism, universal homeomorphism, see Homeomorphism, universal.
• Morphism, universally bicontinuous, universally open: IV.2.4.2.
• Morphism, universally open at a point: IV.14.3.3.
• Morphism, unramified: IV.17.3.1.
• Morphism, unramified at a point: IV.17.3.7.
• Morphisms, equivalent: IV.20.2.1.
• Morphisms, family of, schematically dominant, see Family of morphisms, schematically dominant.
• Morphisms, family of, transversal at a point: IV.17.13.10.
• Morphisms, pair of, transversal at a point: IV.17.13.6.
• Multiplicity of a cycle at a point: IV.21.6.1.
• Multiplicity, radicial, of a point for an ${\mathcal{O}}_X$-Module ($X$ prescheme over a field): IV.4.7.5.
• Multiplicity, radicial, of a prime cycle maximal associated to an ${\mathcal{O}}_X$-Module ($X$ prescheme over a field): IV.4.7.8.
• Multiplicity, radicial, separable, total, of an extension of a field: IV.4.7.4.
• Multiplicity, radicial, separable, total, of a prescheme over a field: IV.4.7.4.
• Multiplicity, total, of a point (of a prime cycle maximal associated) for an ${\mathcal{O}}_X$-Module ($X$ prescheme over a field): IV.4.7.12.

N

• Neighbourhood, formal, of a sub-prescheme: IV.16.1.11.
• Neighbourhood, $n$-th infinitesimal: IV.16.1.2.
• Net, formally, morphism, see Formally étale, formally smooth, formally net, formally unramified morphism.
• Norm of a divisor: IV.21.5.5.
• Norm of a meromorphic section: IV.21.5.3.
• Normal at a point, normal morphism, see Morphism, normal.
• Normal at a point, normal prescheme, see Geometrically normal.
• Normal invariant, see Invariant, normal.
• Normal ring: IV.5.13.5.
• Normally flat ${\mathcal{O}}_Y$-Module along a sub-prescheme: IV.6.10.1.

O

• Open ultra-affine, see Ultra-affine open.
• Operator, differential, relative to $S$: IV.16.8.1.
• Order of a differential operator: IV.16.8.1.
• Order of a meromorphic function at a point $x\in X^{(1)}$: IV.21.6.7.
• ${\mathcal{O}}_X$-Module, see Module.
• ${\mathcal{O}}_X$-Module, integral, integral at a point: IV.3.2.4.
• ${\mathcal{O}}_X$-Module, irredundant: IV.3.2.4.
• ${\mathcal{O}}_X$-Module, normally flat along a sub-prescheme: IV.6.10.1.
• ${\mathcal{O}}_X$-Module, possessing the property $(S_k)$, possessing the property $(S_k)$ at a point: IV.5.7.2.
• ${\mathcal{O}}_X$-Module, reduced, reduced at a point: IV.3.2.2.
• ${\mathcal{O}}_X$-Module, $Z$-closed, $Z$-pure: IV.5.9.9.
• ${\mathcal{O}}_X$-Module, $Z$-closed, $Z$-pure at a point: IV.5.9.9.

P

• $p$-anneau de Cohen, see Cohen ring, $p$-.
• $p$-base, absolute $p$-base: 0_IV.21.1.9.
• Parafactorial couple: IV.21.13.1.
• Parafactorial local ring: IV.21.13.7.
• Part, finite saturated, of an algebraic $k$-prescheme: IV.9.8.9.
• Part, principal, of order $n$, of order infinity, of a section of ${\mathcal{O}}_X$: IV.16.3.6.
• Part, proper at a point (of an $S$-prescheme): IV.15.7.1.
• Part, purely of codimension $p$ (closed): IV.21.6.2.
• Part, quasi-constructible, locally quasi-constructible, of a topological space: IV.10.1.1.
• Part, saturated, see Saturated part.
• Part, very dense, of a topological space: IV.10.1.3.
• Picard group: see EGA III index, "Picard group" (cf. IV.21.3.2).
• Poles, cycle of, see Cycle of poles.
• Polynomial, "Eisenstein": 0_IV.22.4.3.
• Polynomial, geometrically irreducible: IV.9.7.4.
• Polynomial, Hilbert-Samuel: 0_IV.16.2.1.
• Polynomial, separable, over a ring: IV.18.4.3.
• Prescheme, see EGA I, EGA II; the EGA-IV constructions specific to this volume are indexed individually below.
• Prescheme of characteristic $p$: IV.16.12.1.
• Prescheme of coincidences of two morphisms: IV.17.4.5.
• Prescheme of finite presentation over another: IV.1.6.1.
• Prescheme of Jacobson: IV.10.4.1.
• Prescheme, algebraic, over $k$, $k$-algebraic space, pre-algebraic over $k$, $k$-pre-algebraic: IV.10.10.2.
• Prescheme, biequidimensional, see Biequidimensional space (transferred via $\mathrm{Spec}(A)$).
• Prescheme, Cohen-Macaulay: IV.5.7.1.
• Prescheme, complete intersection at a point, see Complete intersection, prescheme.
• Prescheme, connected in codimension $d-1$: IV.5.10.8.
• Prescheme, differentially smooth over $S$: IV.16.10.1.
• Prescheme, equidimensional over another at a point, equidimensional over another: IV.13.2.2, IV.13.3.2.
• Prescheme, essentially affine over another: IV.8.13.4.
• Prescheme, étale over $S$: IV.17.3.1.
• Prescheme, étale over $S$ at a point: IV.17.3.7.
• Prescheme, excellent: IV.7.8.5.
• Prescheme, formally étale, formally smooth, formally unramified over $S$: IV.17.1.1.
• Prescheme, geometrically connected, see Geometrically connected prescheme.
• Prescheme, geometrically integral, see Geometrically integral prescheme.
• Prescheme, geometrically irreducible, see Geometrically irreducible prescheme.
• Prescheme, geometrically pointwise integral, see Geometrically pointwise integral prescheme.
• Prescheme, geometrically normal, see Geometrically normal.
• Prescheme, geometrically reduced, see Geometrically reduced prescheme.
• Prescheme, geometrically regular, see Geometrically regular.
• Prescheme, geometrically unibranch, see Geometrically unibranch prescheme.
• Prescheme, integral at a point: IV.3.2.4.
• Prescheme, intersection complete, absolute, at a point: IV.19.3.1.
• Prescheme, irredundant: IV.3.2.4.
• Prescheme, $k$-pre-algebraic reduced: IV.10.10.4.
• Prescheme, lisse, see Prescheme, smooth.
• Prescheme, locally factorial: IV.21.6.9.
• Prescheme, locally immersible in a regular scheme: IV.5.11.1.
• Prescheme, net over $S$, unramified over $S$: IV.17.3.1.
• Prescheme, net over $S$ at a point, unramified over $S$ at a point: IV.17.3.7.
• Prescheme, pointwise integral: IV.4.6.9.
• Prescheme, possessing the property $(R_k)$, possessing the property $(R_k)$ at a point: IV.5.8.2.
• Prescheme, possessing the property $(R_k)$ geometric at a point, possessing the property $(R_k)$ geometric (over a field): IV.6.7.6.
• Prescheme, possessing the property $(S_k)$, possessing the property $(S_k)$ at a point: IV.5.7.2.
• Prescheme, quasi-separated: IV.1.2.1.
• Prescheme, regular in codimension $k$: IV.5.8.2.
• Prescheme, reduced (over a field), see Geometrically reduced prescheme.
• Prescheme, separable (over a field): IV.4.6.2.
• Prescheme, separable (over a field) at a point: IV.4.6.9.
• Prescheme, smooth over $S$: IV.17.3.1.
• Prescheme, smooth over $S$ at a point: IV.17.3.7.
• Prescheme, unibranch, see Unibranch prescheme.
• Prescheme, universally catenary: IV.5.6.3.
• Prescheme, universally reduced (over a field): IV.4.6.2.
• Prescheme, $Y$-, finite and locally free: IV.18.2.7.
• Principal cycle, see Cycle, principal.
• Principal divisor, see Divisor, principal.
• Principal part, see Part, principal.
• Principal sheaf, formally homogeneous (or pseudo-torsor), see Formally principal homogeneous sheaf.
• Principal sheaf, homogeneous (or torsor), see Sheaf, principal homogeneous.
• Principle of the finite extension, see Extension, finite, principle of.
• Pro-constructible part, see Subset, pro-constructible.
• Pro-object, morphism of pro-objects: IV.8.13.3.
• Pro-object, essentially affine: IV.8.13.4, IV.8.14.1.
• Product, fibre, of $A$-rings: 0_IV.18.1.2.
• Property $(R_k)$: see Prescheme, possessing the property $(R_k)$.
• Property $(R_k)$ geometric: see Prescheme, possessing the property $(R_k)$ geometric.
• Property $(S_k)$: see ${\mathcal{O}}_X$-Module, possessing the property $(S_k)$.
• Property $\mathbf{P}$ geometric: IV.7.3.6.
• Property $\mathbf{P}$ of the first type (of the second type) associated to a property $\mathbf{R}$ of Noetherian local rings: IV.7.3.10.
• Property, constructible, ind-constructible: IV.9.2.1, IV.9.2.2 (i)-(ii).
• Pseudo-fonction, see Function, pseudo-.
• Pseudo-morphism, pseudo-$S$-morphism: IV.20.2.1.
• Pseudo-morphism composed of a morphism and a pseudo-morphism: IV.20.3.1.
• Pseudo-morphism composed of a pseudo-morphism and a morphism: IV.20.3.2.
• Pseudo-morphism, defined at a point: IV.20.2.3.
• Pseudo-morphism, relative to $S$: IV.20.5.1.
• Pseudo-torsor, see Sheaf, formally principal homogeneous.

Q

• Quasi-compact morphism: see EGA I.
• Quasi-constructible part, see Part, quasi-constructible.
• Quasi-faithfully flat morphism, see Morphism, quasi-faithfully flat.
• Quasi-flat morphism: IV.2.3.3.
• Quasi-homeomorphism of topological spaces: IV.10.2.2.
• Quasi-isomorphism of ringed spaces: IV.10.2.8.
• Quasi-regular immersion, see Immersion, quasi-regular.
• Quasi-regular system of generators of an Ideal, see System, quasi-regular, of generators.
• Quasi-separated morphism: IV.1.2.1.
• Quasi-separated prescheme: IV.1.2.1.

R

• $R$-ultra-prescheme: IV.10.10.2.
• Radicial extension, see Extension, radicial.
• Radicial morphism at a point: IV.6.15.3.
• Radicial multiplicity, see Multiplicity, radicial.
• Rational map, strict, see Map, rational, strict.
• Rectified depth: IV.10.8.1.
• Reduced ${\mathcal{O}}_X$-Module, reduced module over a ring: IV.3.2.2.
• Regular at a point, regular morphism, see Morphism, regular.
• Regular element, see Element, $M$-regular.
• Regular homomorphism of ${\mathcal{O}}_X$-Modules: IV.19.4.11.
• Regular immersion, see Immersion, regular.
• Regular ring: 0_IV.17.1.1, 0_IV.17.3.6.
• Regular system of parameters, see System of parameters, regular.
• Regularly immersed sub-prescheme, see Sub-prescheme, regularly immersed.
• Relative dimension, see Dimension, relative.
• Resolution of singularities: IV.7.9.1.
• Resolving morphism, see Morphism, resolving.
• Restriction of a cycle to an open: IV.21.6.3.
• Restriction of a pseudo-morphism to a local scheme: IV.20.3.6.
• Restriction of a pseudo-morphism to an open: IV.20.2.2.
• Ring, biequidimensional: 0_IV.16.1.4.
• Ring, catenary: 0_IV.16.1.4.
• Ring, characteristic, of: 0_IV.21.1.1.
• Ring, Cohen, see Cohen ring.
• Ring, Cohen-Macaulay: 0_IV.16.5.1, 0_IV.16.5.13.
• Ring, complete intersection, see Complete intersection ring.
• Ring, equicodimensional, equidimensional: 0_IV.16.1.4.
• Ring, excellent: IV.7.8.2.
• Ring, formally catenary: IV.7.1.9.
• Ring, formally equidimensional: IV.7.1.1.
• Ring, geometrically normal, geometrically reduced, geometrically regular, possessing property $(R_k)$ geometric: IV.6.7.6.
• Ring, geometrically regular over a field (local ring): 0_IV.19.6.5.
• Ring, geometrically unibranch (local): 0_IV.23.1.7.
• Ring, henselian: IV.18.5.8.
• Ring, Japanese: 0_IV.23.1.1.
• Ring, local complete prime: 0_IV.19.8.3.
• Ring, local, parafactorial: IV.21.13.7.
• Ring, local prime: 0_IV.19.8.3.
• Ring, local, strictly: IV.18.8.2.
• Ring, local unibranch: 0_IV.23.1.7.
• Ring, normal: IV.5.13.5.
• Ring of restricted formal series over a complete valued field: IV.7.4.8.
• Ring, $\mathbf{P}$-: IV.7.3.13.
• Ring, possessing property $(R_k)$: IV.5.8.2.
• Ring, regular: 0_IV.17.1.1, 0_IV.17.3.6.
• Ring, strictly equidimensional: IV.7.2.1.
• Ring, strictly formally catenary: IV.7.2.6.
• Ring, universally catenary: IV.5.6.2.
• Ring, universally Japanese: 0_IV.23.1.1.

S

• Saturated chain, see Chain, saturated.
• Saturated, of a finite part of an algebraic $k$-prescheme: IV.9.8.9.
• Schematically dense family of sub-preschemes, see Family of sub-preschemes, schematically dense.
• Schematically dominant family of morphisms, see Family of morphisms, schematically dominant.
• Schematically dominant family, universally, see Family of morphisms, universally schematically dominant.
• Scheme, local henselian: IV.18.5.8.
• Scheme, strictly local: IV.18.8.2.
• Section, meromorphic, of an ${\mathcal{O}}_X$-Module: IV.20.1.3.
• Section, meromorphic, of an ${\mathcal{O}}_X$-Module defined at a point: IV.20.1.7.
• Section, meromorphic, of an ${\mathcal{O}}_X$-Module inversible regular relative to $S$: IV.20.6.5.
• Section, meromorphic, of an ${\mathcal{O}}_X$-Module relative to $S$: IV.20.6.2.
• Section, meromorphic, regular, of an invertible ${\mathcal{O}}_X$-Module: IV.20.1.8.
• Section, $\mathcal{F}$-regular, see Family, $\mathcal{F}$-regular section.
• Semi-continuity, lower, of depth: IV.19.9.4.
• Separable algebra over a field: IV.4.6.2.
• Separable extension, see Extension, separable.
• Separable polynomial, see Polynomial, separable.
• Separable prescheme over a field, see Prescheme, separable.
• Separably closed field, see Field, separably closed.
• Separating family of homomorphisms, see Family, separating.
• Sequence, $M$-quasi-regular, see Suite, $M$-quasi-regular.
• Sequence, $M$-regular, see Suite, $M$-regular.
• Sequence of sections, transversally regular relative to $S$: IV.19.1.2.
• Sequence, quasi-regular, of generators, see System, quasi-regular, of generators.
• Sheaf, conormal: IV.16.1.2.
• Sheaf of divisors on a ringed space: IV.21.1.2.
• Sheaf of divisors relative to $S$, or transversal to $f$: IV.21.15.2.
• Sheaf of germs of 1-codimensional cycles with coefficients in $A$: IV.21.10.9.
• Sheaf of germs of meromorphic functions on $X$: IV.20.1.4.
• Sheaf of germs of meromorphic functions relative to $S$: IV.20.6.1.
• Sheaf of germs of meromorphic sections of a Module: IV.20.1.3.
• Sheaf of graded rings associated with a morphism: IV.16.1.2.
• Sheaf of modules of fractions with denominators in $\mathcal{S}$: IV.20.1.2.
• Sheaf of $S$-derivations of ${\mathcal{O}}_X$: IV.16.5.7.
• Sheaf of $S$-derivations of ${\mathcal{O}}_X$ into $\mathcal{F}$: IV.16.5.4.
• Sheaf of $n$-th differentials of $X$ relative to $S$: IV.16.6.1.
• Sheaf of principal parts of order $n$ of the $S$-prescheme $X$: IV.16.3.1.
• Sheaf of rings of fractions with denominators in $\mathcal{S}$: IV.20.1.1.
• Sheaf, principal homogeneous (or torsor), under a sheaf of groups: IV.16.5.15.
• Sheaf, principal homogeneous (or torsor), trivial, under a sheaf of groups: IV.16.5.15.
• Sheaf, tangent, of $X$ relative to $S$: IV.16.5.7.
• Skeleton, primary, of a Module; virtual skeleton: IV.9.8.9.
• Space, algebraic, over $k$; $k$-algebraic space, pre-algebraic over $k$, $k$-pre-algebraic: IV.10.10.2.
• Space, biequidimensional: 0_IV.14.3.3.
• Space, catenary: 0_IV.14.3.2.
• Space, equicodimensional: 0_IV.14.2.1.
• Space, equidimensional: 0_IV.14.1.3.
• Space, Jacobson: IV.10.3.1.
• Space, $k$-pre-algebraic reduced: IV.10.10.4.
• Space, tangent, to a prescheme at a point: IV.16.5.13.
• Spectrum, maximal, of a Jacobson ring: IV.10.9.3.
• Strict rational map, see Map, rational, strict.
• Strictly equidimensional ring, see Ring, strictly equidimensional.
• Strictly formally catenary ring, see Ring, strictly formally catenary.
• Strictly formally projective module, see Formally projective module, strictly.
• Strictly local ring, see Ring, local, strictly.
• Strictly local scheme: IV.18.8.2.
• Sub-prescheme, family of, schematically dense, see Family of sub-preschemes, schematically dense.
• Sub-prescheme, regularly immersed, quasi-regularly immersed: IV.16.9.2.
• Sub-prescheme, defined on a field: IV.4.8.4.
• Sub-prescheme, closed, primary, of a prescheme: IV.3.2.4.
• Sub-${\mathcal{O}}_X$-Module, defined on a field: IV.4.8.4.
• Sub-${\mathcal{O}}_X$-Module, primary, of an ${\mathcal{O}}_X$-Module: IV.3.2.4.
• Sub-preschemes intersecting transversally at a point relative to $S$: IV.17.13.7.
• Subset of an $S$-prescheme submersive, universally submersive on its image: IV.15.7.8.
• Subset of a prescheme defined on a field: IV.4.8.4.
• Subset, constructible: see EGA III (Constructible (subset)); EGA-IV (1.9.x) adds the relative version.
• Subset, ind-constructible: IV.1.9.4.
• Subset, pro-constructible: IV.1.9.4.
• Subset, very dense, see Part, very dense.
• Suite $\mathcal{F}$-quasi-regular: 0_IV.15.2.2.
• Suite $\mathcal{F}$-regular, see Sequence, $\mathcal{F}$-regular.
• Suite $M$-quasi-regular, $M$-regular, see Sequence, $M$-quasi-regular, $M$-regular.
• Suite, $M$-regular, suite, $M$-quasi-regular: 0_IV.15.1.7.
• Suite, regular, suite, quasi-regular: 0_IV.15.2.2.
• Sum, amalgamated, of bimodules: 0_IV.18.2.7.
• Support of a cycle: IV.21.6.1.
• Support of a divisor: IV.21.1.2.
• Symmetry, canonical, of $X{\times }_{S}X$: IV.16.3.3.
• Symmetry, canonical, of ${\mathcal{P}}_{X/S}^{\bullet }$, of ${\Omega }_{X/S}^{\bullet }$: IV.16.3.4.
• System, fundamental, of open ideals, of open submodules: 0_IV.19.0.2.
• System of generators, quasi-regular (regular), of an Ideal: IV.16.9.1.
• System of $p$-generators, absolute system of $p$-generators: 0_IV.21.1.9.
• System of parameters for a module on a Noetherian semi-local ring: 0_IV.16.3.6.
• System of parameters, regular, of a regular local ring: 0_IV.17.1.6.

T

• Tangent fibre of $X$ relative to $S$: IV.16.5.12.
• Tangent, linear map: IV.16.5.13.
• Tangent, sheaf, of $X$ relative to $S$: IV.16.5.7.
• Tangent space, see Space, tangent.
• Topology, constructible, on a prescheme: IV.1.9.12.
• Topology on an $A$-module deduced from the topology of $A$: 0_IV.19.0.2.
• Torsion-free ${\mathcal{O}}_X$-Module, see Module, torsion-free.
• Torsor, see Sheaf, principal homogeneous.
• Total multiplicity, see Multiplicity, total.
• Transversal codimension, see Codimension, transversal.
• Transversal morphism, see Morphism, transversal.
• Transversally regular immersion, see Immersion, transversally regular.
• Transversally regular Ideal of an algebra, see Ideal, transversally regular.
• Transversally regular at a point of a prescheme (Ideal), see Ideal, transversally regular at a point.
• Transversally regular sequence of sections relative to $S$, see Sequence of sections, transversally regular.
• Trace form: IV.18.2.1.
• Trace homomorphism: IV.18.2.2.
• Trivial (augmented $A$-ring): 0_IV.18.1.4.
• Trivial ($A$-extension): 0_IV.18.2.3, 0_IV.18.3.7.
• Trivial extension, type: 0_IV.18.2.3.
• Type, primary, of a Module; geometric primary type of a Module: IV.9.8.9.

U

• Ultra-affine open: IV.10.9.5.
• Ultra-prescheme, morphism of ultra-preschemes: IV.10.9.5.
• Unibranch local ring: 0_IV.23.1.7.
• Unibranch prescheme at a point, unibranch prescheme: IV.6.15.1.
• Universal homeomorphism, see Homeomorphism, universal.
• Universally bicontinuous, see Morphism, universally bicontinuous.
• Universally catenary prescheme: IV.5.6.3.
• Universally catenary ring: IV.5.6.2.
• Universally injective homomorphism of Modules: IV.11.9.14, IV.11.9.18.
• Universally Japanese ring: 0_IV.23.1.1.
• Universally open morphism, see Morphism, universally bicontinuous; Morphism, universally open at a point.
• Universally reduced prescheme (over a field): IV.4.6.2.
• Universally schematically dominant family of morphisms, see Family of morphisms, universally schematically dominant.
• Universally separating family of homomorphisms of sheaves of modules: IV.11.9.14.
• Universally submersive morphism, see Morphism, submersive.

V

• Very dense subset, see Part, very dense.
• Virtual skeleton, see Skeleton, primary.

Z

• $Z$-closure of an ${\mathcal{O}}_X$-Module: IV.5.9.11.
• Zariski's "Main theorem" for morphisms of finite presentation: IV.8.12.6.
• Zeros, cycle of, see Cycle of zeros.