Terminological Index

References are by (N.M.K) for items in Chapter 0 and (I, N.M.K) for Chapter I.

A

Adic ring, $J$ -adic ring — (0.7.1.9)
Admissible ring — (0.7.1.2)
Adjacence (closure) of a subprescheme — (I, 9.5.11)
Affine open — (I, 2.1.1)
Affine scheme — (I, 1.7.1)
Affine scheme, ring of — (I, 1.7.1)
Algebra, $A$ - (over an $A$ -Module structure) — (0.4.1.3)
Algebraic prescheme over $K$ — (I, 6.4.1)
Algebraic prescheme, base field — (I, 6.4.1)
Algebraic sheaf on a ringed space — (0.4.1.3)
Allied local rings (apparented, related) — (I, 8.1.4)
Annihilator of an $O_{X}$ -Module — (0.5.3.7)
Artinian prescheme — (I, 6.2.1)

B

Base prescheme of an $S$ -prescheme — (I, 2.5.1)
Birational morphism — (I, 2.2.9), (I, 6.5.6)

C

Canonical injection of an induced ringed space — (0.4.1.2)
Canonical injection of a subprescheme — (I, 4.1.7)
Canonical sections of $O_{X}^{n}$ — (I, 5.5.5)
Chevalley scheme — (I, 8.3)
Closed image of a prescheme by a morphism — (I, 9.5.3)
Closed immersion of preschemes — (I, 4.2.1)
Closed immersion of formal preschemes — (I, 10.14.2)
Closed morphism — (I, 2.2.6)
Closed point — (0.2.1.3)
Closed subprescheme — (I, 4.1.3)
Closed subprescheme defined by a sheaf of ideals — (I, 4.1.2)
Closure of a subprescheme — (I, 9.5.11)
Coherent $O_{X}$ -Module — (0.5.3.1)
Coherent $O_{X}$ -Algebra — (0.5.3.6)
Coherent sheaf of rings — (0.5.3.5)
Complete ring of fractions — (0.7.6.5)
Completed ring of fractions — (0.7.6.5)
Completed tensor product (algebras, modules) — (0.7.7.2), (0.7.7.5)
Completion of an $O_{X}$ -Module / morphism along a closed subset — (I, 10.8.4)
Completion of a prescheme along a closed subset — (I, 10.8.5)
Connected prescheme — (I, 2.1.7)
Continuous homomorphism of topological-ring sheaves — (0.3.1.4)
Cover-by-affine-opens — (I, 2.1.3)

D

Defined-at-a-point (rational map) — (I, 7.2.1)
Diagonal morphism of $X \times_{S} X$ — (I, 5.3.1)
Di-homomorphism of modules — (0.1.0.2)
Direct image of an $O_{X}$ -Module — (0.4.2.1)
Direct image of a presheaf — (0.3.4.1)
Domain of definition of a rational map — (I, 7.2.1)
Dominant $S$ -prescheme — (I, 2.5.1)
Dominant morphism — (I, 2.2.6)
Double homomorphism ( $ϕ$ -homomorphism of modules) — (0.1.0.2)
Dual of an $O_{X}$ -Module — (0.4.1.4)

E

Element-of-finite-type (algebra) — (0.1.0.5)
Element of a prescheme with values in a ring — (I, 3.4.4)
Element of a prescheme with values in a prescheme — (I, 3.4.1)
Element of an $A$ -prescheme with values in an $A$ -algebra — (I, 3.4.4)
Element of an $S$ -prescheme over $s \in S$ — (I, 2.5.1)
Element of an $S$ -prescheme with values in an $S$ -prescheme — (I, 3.4.3)
Equivalent morphisms (for rational maps) — (I, 7.1.1)
Étale morphism — (introduced in EGA IV)
Exterior power of an $O_{X}$ -Module — (0.4.1.4)
Extension of a morphism to completions — (I, 10.9.1)
Extension of a quasi-coherent sheaf — (I, 9.4.1)

F

Faithfully flat module / morphism — (0.6.4.1), (0.6.7.8)
Field, base — (I, 6.4.1)
Finite algebra — (0.1.0.5)
Finite-type module — (0.1.0.5)
Finite-type $O_{X}$ -Module — (0.5.2.1)
Finite-type morphism (between preschemes) — (I, 6.3.1)
Finite-type morphism (between formal preschemes) — (I, 10.13.3)
Finite $K$ -scheme — (I, 6.4.5)
Flat module — (0.6.1.1)
Flat morphism of ringed spaces — (0.6.7.1)
Formal affine open — (I, 10.4.1)
Formal affine open, adic / Noetherian — (I, 10.4.1)
Formal completion (of a prescheme, sheaf, morphism) — (I, 10.8.4), (I, 10.8.5)
Formal prescheme — (I, 10.4.2)
Formal prescheme, adic — (I, 10.4.2)
Formal prescheme, locally Noetherian — (I, 10.4.2)
Formal prescheme, Noetherian — (I, 10.4.2)
Formal prescheme over $A$ ( $A$ -prescheme), over $S$ ( $S$ -prescheme) — (I, 10.4.7)
Formal prescheme, finite type over $S$ — (I, 10.13.3)
Formal scheme, affine — (I, 10.1.2)
Formal scheme over $S$ ( $S$ -scheme) — (I, 10.15.1)
Formal series, restricted — (0.7.5.1)
Formal spectrum of an admissible ring — (I, 10.1.2)
Fundamental system of ideals of definition — (I, 10.3.6), (I, 10.5.1)

G

Generic point — (0.2.1.3)
Generization of a point — (0.2.1.3)
Geometric point of a prescheme — (I, 3.4.5)
Geometric point above $s \in S$ — (I, 3.4.5)
Geometric point localized at $x$ — (I, 3.4.5)
Geometric number of points of a prescheme — (I, 6.4.8)
Germ of a section — (0.3.1.6)
Graded $O_{X}$ -Module — (0.4.1.3)
Graph of a morphism — (I, 5.3.11)
Gluing of preschemes (formal/non-formal) — (I, 2.3.1)
Gluing condition for sheaves / ringed spaces / preschemes — (0.3.3.1), (0.4.1.6), (I, 2.3.1)

H

Homomorphism, continuous, of topological ring sheaves — (0.3.1.4)
Homomorphism defined by a section — (0.5.1.1)
Homomorphism, di-, of modules — (0.1.0.2)
Homomorphism, local, of local rings — (0.1.0.7)
Homomorphism, $ϕ$ - of modules — (0.1.0.2)

I

Ideal of definition of an admissible ring — (0.7.1.2)
Ideal, $O_{X}$ - — (0.4.1.3)
Ideal, sheaf of, of definition of a formal prescheme — (I, 10.3.3), (I, 10.5.1)
Ideal, prime — (0.1.0.6)
Image, direct, of an $O_{X}$ -Module — (0.4.2.1)
Image, direct, of a presheaf — (0.3.4.1)
Image of an $S$ -section — (I, 5.3.11)
Image, closed, of a prescheme by a morphism — (I, 9.5.3)
Inverse image of an $O_{X}$ -Module — (0.4.3.1)
Inverse image of an $S$ -morphism — (I, 3.3.7)
Inverse image of an $S$ -prescheme — (I, 3.3.6)
Inverse image of a presheaf — (0.3.5.3)
Inverse image of a subprescheme — (I, 4.4.1)
Immersion, of preschemes — (I, 4.2.1)
Immersion, closed, of preschemes — (I, 4.2.1)
Immersion, open, of preschemes — (I, 4.2.1)
Immersion, closed, of formal preschemes — (I, 10.14.2)
Immersion, local, of preschemes — (I, 4.5.1)
Induced affine open (canonical injection) — (0.4.1.2)
Induced subprescheme (canonical injection) — (I, 4.1.7)
Induced subprescheme, formal — (I, 10.14.2)
Injection, canonical, of an induced ringed space — (0.4.1.2)
Injection, canonical, of a subprescheme — (I, 4.1.7)
Injection, canonical, of a formal subprescheme — (I, 10.14.2)
Injection, geometric — (I, 3.5.4)
Integral, integral-and-finite (algebra) — (0.1.0.5)
Integral ring — (0.1.0.6)
Integral prescheme — (I, 2.1.7)
Inverse of an invertible $O_{X}$ -Module — (0.5.4.3)
Invertible $O_{X}$ -Module — (0.5.4.1)
Irreducible component — (0.2.1.5)
Irreducible space — (0.2.1.1)
Irreducible prescheme — (I, 2.1.7)
Isomorphism associated with an immersion — (I, 4.2.1)
Isomorphism, local, of preschemes — (I, 4.5.1)

K

Kolmogorov space — (0.2.1.2)

L

Linearly topologized ring — (0.7.1.1)
Local-at-Y ring of a subprescheme — (I, 2.1.6)
Local homomorphism of local rings — (0.1.0.7)
Local immersion of preschemes — (I, 4.5.1)
Local isomorphism of preschemes — (I, 4.5.1)
Local ring — (0.1.0.7)
Local ring, dominant — (I, 8.1.1)
Local ring of $X$ along $Y$ ; of $Y$ in $X$ — (I, 2.1.6)
Local scheme — (I, 2.4.1)
Local scheme of a prescheme at a point — (I, 2.4.1)
Localized point with values in a local ring — (I, 3.4.5)
Locally free $O_{X}$ -Module — (0.5.4.1)
Locally integral prescheme — (I, 2.1.7)
Locally Noetherian prescheme — (I, 6.1.1)
Locally simple sheaf — (0.3.6.1)

M

Maximal ideal of definition — (0.7.1.6)
Meromorphic function — (I, 7.3.1)
Module, $O_{X}$ - — (0.4.1.3)
Module of fractions — (0.1.2.2)
Module, faithfully flat — (0.6.4.1)
Module, flat — (0.6.1.1)
Module, $A$ -flat — (0.6.2)
Module, quasi-finite — (0.7.4.1)
Module, of finite presentation — (0.1.0.5), (0.5.2.5)
Module, of finite type — (0.5.2.1)
Module, simple, pseudo-discrete — (0.3.6.1), (0.3.8.1)
Morphism of ringed spaces / topologically ringed spaces — (0.4.1.1)
Morphism, faithfully flat — (0.6.7.8)
Morphism, flat — (0.6.7.1)
Morphism of presheaves / sheaves on a basis of opens — (0.3.2.3)
Morphism of preschemes — (I, 2.2.1)
Morphism of $S$ -preschemes, $A$ -preschemes — (I, 2.5.2)
Morphism, birational — (I, 6.5.6)
Morphism, of finite type — (I, 6.3.1)
Morphism, diagonal — (I, 5.3.1)
Morphism, dominant — (I, 2.2.6)
Morphism, closed — (I, 2.2.6)
Morphism, graph of a morphism — (I, 3.3.14)
Morphism, locally of finite type — (I, 6.6.2)
Morphism, bounded by another (dominated) — (I, 4.1.8)
Morphism, open — (I, 2.2.6)
Morphism, quasi-compact — (I, 6.6.1)
Morphism, radicial — (I, 3.5.4)
Morphism, reduced — (I, 5.1.5)
Morphism, separated — (I, 5.4.1)
Morphism, structure (of an $S$ -prescheme) — (I, 2.5.1)
Morphism, surjective — (I, 2.2.6)
Morphism, universally injective (radicial) — (I, 3.5.4)
Morphisms, equivalent (for rational maps) — (I, 7.1.1)
$A$ -morphism, $S$ -morphism of preschemes — (I, 2.5.2)
Morphism of formal preschemes — (I, 10.4.5)
Morphism, adic, of formal preschemes — (I, 10.12.1)
Morphism, diagonal, of formal preschemes — (I, 10.15.1)
Morphism, of finite type, of formal preschemes — (I, 10.13.3)
Morphism, separated, of formal preschemes — (I, 10.15.1)
Morphism, structure (of an $S$ -formal prescheme) — (I, 10.4.7)
$A$ -morphism, $S$ -morphism of formal preschemes — (I, 10.4.7)
$ϕ$ -morphism of presheaves — (0.3.5.1)
$Ψ$ -morphism of an $O_{X}$ -Module into an $O_{Y}$ -Module — (0.4.4.1)

N

Nilradical of a ring — (0.1.1.1)
Nilradical of an $O_{X}$ -Algebra — (I, 5.1.1)
Noetherian induction (principle) — (0.2.2.2)
Noetherian prescheme — (I, 6.1.1)
Noetherian space — (0.2.2.1)
Normal ringed space / sheaf of rings — (0.4.1.3)

O

Open immersion — (I, 4.2.1)
Open morphism — (I, 2.2.6)
Open subset, affine — (I, 2.1.1)
Open subset, formal affine — (I, 10.4.1)

P

Partition, multiplicative, of a ring — (0.1.2.1)
Partition, saturated multiplicative — (0.1.4.3)
$f$ -flat ( $O_{X}$ -Module) — (0.6.7.1)
Point of a prescheme over $s \in S$ — (I, 2.5.1)
Point of a prescheme with values in a ring — (I, 3.4.4)
Point of a prescheme with values in a prescheme — (I, 3.4.1)
Point of an $A$ -prescheme with values in an $A$ -algebra — (I, 3.4.4)
Point, closed — (0.2.2.6)
Point, generic — (0.2.1.3)
Point, geometric, of a prescheme — (I, 3.4.5)
Point, geometric, above $s$ — (I, 3.4.5)
Point, geometric, localized at $x$ — (I, 3.4.5)
Point, rational, over $K$ — (I, 3.4.5)
Preadic ring, $J$ -preadic — (0.7.1.9)
Preadmissible ring — (0.7.1.2)
Presheaf, constant — (0.3.6.1)
Presheaf on a basis of open sets — (0.3.2.1)
Prescheme — (I, 2.1.2)
Prescheme, Artinian — (I, 6.2.1)
Prescheme, connected — (I, 2.1.7)
Prescheme over a base prescheme $S$ — (I, 2.5.1)
Prescheme, deduced by reduction mod $J$ — (I, 3.7.1)
Prescheme, induced on an open — (I, 2.1.8)
Prescheme, integral — (I, 2.1.7)
Prescheme, irreducible — (I, 2.1.7)
Prescheme, locally integral — (I, 2.1.7)
Prescheme, locally Noetherian — (I, 6.1.1)
Prescheme, reduced — (I, 5.1.3)
Prescheme, reduced associated with a prescheme — (I, 5.1.3)
$A$ -prescheme, prescheme over a ring $A$ — (I, 2.5.1)
$K$ -prescheme, algebraic — (I, 6.4.5)
$S$ -prescheme, prescheme over a prescheme $S$ — (I, 2.5.1)
$S$ -prescheme, dominant — (I, 2.5.1)
Prescheme of finite type over $S$ , $S$ -prescheme of finite type — (I, 6.3.1)
Prescheme obtained by base extension — (I, 3.3.6)
Prescheme separated over $S$ — (I, 5.4.1)
Prescheme, formal — (I, 10.4.2)
Prescheme, formal adic — (I, 10.4.2)
Prescheme, formal locally Noetherian — (I, 10.4.2)
Prescheme, formal Noetherian — (I, 10.4.2)
Prescheme, formal, over $A$ ( $A$ -prescheme), over $S$ ( $S$ -prescheme) — (I, 10.4.7)
Prescheme, formal of finite type over $S$ — (I, 10.13.3)
Prescheme, formal separated over $S$ — (I, 10.15.1)
Presentation, finite (module) — (0.1.0.5)
Presentation, finite ( $O_{X}$ -Module) — (0.5.2.5)
Principle of Noetherian induction — (0.2.2.2)
Product of $S$ -preschemes — (I, 3.2.1)
Product of $S$ -formal preschemes — (I, 10.7.1)
Product, fiber, of sets — (I, 3.4.2)
Product, tensor, of sheaves on distinct preschemes — (I, 9.1.2)
Product, tensor, completed (algebras) — (0.7.7.5)
Product, tensor, completed (modules) — (0.7.7.2)
Projection, canonical, of a product — (I, 3.2.1)
Prolongation, canonical, of a sub-Module — (I, 9.4.1)
Prolongation of a morphism to completions — (I, 10.9.1)
Power, exterior $p$ -th of an $O_{X}$ -Module — (0.4.1.4)

Q

Quasi-coherent $O_{X}$ -Module — (0.5.1.3)
Quasi-coherent $O_{X}$ -Algebra — (0.5.1.3)

R

Radical of an ideal — (0.1.1.1)
Radical of a ring — (0.1.1.2)
Rank of a locally free $O_{X}$ -Module — (0.5.4.1)
Rank of a torsion-free $O_{X}$ -Module — (I, 7.4.2)
Rank of a finite $K$ -scheme — (I, 6.4.5)
Rank, separable, of a finite $K$ -scheme — (I, 6.4.8)
Rational map (rational $S$ -map) — (I, 7.1.2)
Rational map defined at a point — (I, 7.2.1)
Rational map induced on an open — (I, 7.1.2)
Rational map induced on $Spec (...)$ — (I, 7.2.8)
Rational function — (I, 7.1.2)
Reduced ring — (0.1.1.1)
Reduced prescheme — (I, 5.1.3)
Reduced subprescheme structure — (I, 5.2.1)
Regular ringed space / sheaf of rings — (0.4.1.3)
Restricted formal series — (0.7.5.1)
Restriction of a ringed space to an open — (0.4.1.2)
Restriction of a morphism to an open — (0.4.1.2)
Restriction of a morphism of preschemes to a subprescheme — (I, 4.1.7)
Restriction of a prescheme to an open — (I, 2.1.8)
Restriction of a rational map to an open — (I, 7.1.2)
Ring of an affine scheme — (I, 1.7.1)
Ring, adic, preadic — (0.7.1.9)
Ring, admissible — (0.7.1.2)
Ring, completed, of fractions — (0.7.6.5)
Ring of fractions — (0.1.2.2)
Ring of rational functions on $X$ — (I, 7.1.3)
Ring, integral — (0.1.0.6)
Ring, linearly topologized — (0.7.1.1)
Ring, local — (0.1.0.7)
Ring, local, of $X$ along $Y$ (of $Y$ in $X$ ) — (I, 2.1.6)
Ring, preadmissible — (0.7.1.2)
Ring, reduced — (0.1.1.1)
Ring, regular — (0.4.1.3)
Rings, allied (related) local — (I, 8.1.4)

S

Saturated multiplicative subset — (0.1.4.3)
Scheme — (I, 5.4.1) [in EGA: separated prescheme]
Scheme, affine — (I, 1.7.1)
Scheme, Chevalley — (I, 8.3)
Scheme, local — (I, 2.4.1)
Scheme, local, at a point of a prescheme — (I, 2.4.1)
$K$ -scheme, algebraic — (I, 6.4.1)
Scheme, finite over $K$ , $K$ -scheme finite — (I, 6.4.5)
$S$ -scheme — (I, 5.4.1)
Scheme, formal affine — (I, 10.1.2)
$S$ -formal scheme — (I, 10.15.1)
Section of a sheaf — (0.3.1.6)
$S$ -section of an $S$ -prescheme — (I, 2.5.5), (I, 5.3.11)
Rational $S$ -section of an $S$ -prescheme — (I, 7.1.2)
Section, unit, of $O_{X}$ — (0.4.1.1)
Sheaf, algebraic, on a ringed space — (0.4.1.3)
Sheaf associated with a presheaf — (0.3.5.7)
Sheaf associated with an $A$ -module on $Spec (A)$ — (I, 1.3.4)
Sheaf with values in a category — (0.3.1.1)
Sheaf, coherent, of rings — (0.5.3.5)
Sheaf, normal at a point, normal, reduced at a point, reduced, regular at a point, regular — (0.4.1.3)
Sheaf, graded, of rings — (0.4.1.3)
Sheaf, of rational functions — (I, 7.3.1)
Sheaf, of torsion (torsion sheaf) — (I, 7.4.1)
Sheaf of ideals — (0.4.1.3)
Sheaf of ideals of definition — (I, 10.3.3), (I, 10.5.1)
Sheaf, induced — (0.3.7.1)
Sheaf, locally simple — (0.3.6.1)
Sheaf, obtained by gluing — (0.3.3.1)
Sheaf, pseudo-discrete — (0.3.8.1)
Sheaf, simple — (0.3.6.1)
Sheaf, structure, of a ringed space — (0.4.1.1)
Sheaf, structure, of an affine scheme — (I, 1.3.4)
Sheaf on a basis of opens — (0.3.2.2)
Spectrum of a ring — (I, 1.1.1)
Spectrum, formal, of an admissible ring — (I, 10.1.2)
Specialization of a point — (0.2.1.2)
Sum of preschemes — (I, 3.1.1)
Sub- $O_{X}$ -Algebra generated by a sub-Module — (0.4.1.3)
Subprescheme — (I, 4.1.3)
Subprescheme associated with an immersion — (I, 4.2.1)
Subprescheme, closed — (I, 4.1.3)
Subprescheme, closed, defined by a sheaf of ideals — (I, 4.1.2)
Subprescheme, closed, of formal preschemes — (I, 10.14.2)
Support of a module — (0.1.7.1)
Support of a sheaf of groups — (0.3.1.6)
System, fundamental, of ideals of definition — (I, 10.3.6), (I, 10.5.1)
System, adic inductive, of preschemes — (I, 10.12.2)

T

Topologically nilpotent element — (0.7.1.1)
Topology, $J$ -adic — (0.7.1.9), (0.7.2.3)
Topology, $J$ -preadic — (0.7.1.9)
Topology, spectral — (I, 1.1.2)
Topology, Zariski — (I, 1.1.2) (footnote)
Trivial invertible $O_{X}$ -Module — (I, 2.4.8)
Type, finite ( $O_{X}$ -Module of) — (0.5.2.1)

U

Unit section of $O_{X}$ — (0.4.1.1)
Universally injective morphism — (I, 3.5.4)

V

Value of a section at a point — (0.5.5.1)
Variety (algebraic) — see Chevalley scheme; algebraic prescheme

Z

Zariski topology — (I, 1.1.2) (footnote)

Translator's note. The EGA terminology uses "rare" for what modern English mathematics calls "nowhere dense," "préschéma" for what modern usage calls "scheme," and "schéma" for what modern usage calls "separated scheme." We have preserved this historical vocabulary with bracketed glosses on first occurrence per section. The list above retains EGA's term as the primary entry.

Éléments de Géométrie Algébrique — English