Translation ledger — EGA IV

Running French↔English term ledger for the EGA IV translation. Seeded from zz-index-terminologique-part-{1,2,3,4}.md and zz-index-notations-part-{1,2,3,4}.md of the OCR'd French source. Extends the EGA III ledger at $d ocs / b oo k s / e g a / iii / t r an s l a t i o n - l e d g er . m d$ (which in turn extends EGA II).

Terms inherited from EGA II and EGA III

The EGA III ledger transfers unchanged. In particular: préschéma → prescheme, schéma → scheme, morphisme structural → structure morphism, morphisme affine / propre / projectif / entier / fini / quasi-fini / $u ni v erse ll e m e n t f er m \overset{e}{ˊ}$ / séparé, the (TF) / (TN) conditions, condition (ML), retrocompact, constructible set, generization, the spectral-sequence vocabulary, the cohomology / hypercohomology / Cartan-Eilenberg-resolution vocabulary, formal-prescheme vocabulary, and the citation forms for (M), (G), (T), (FAC). The EGA II terminology table (affine morphism, very ample sheaf, blow-up, projecting cone, projective closure, homogenization, etc.) also transfers unchanged.

EGA IV additions

Chapter 0 (continued) — §§14-23 preliminaries

These extend the EGA III preliminaries (§§8-13) with the commutative-algebra preparation Chap IV will use. The table grows as each Chap 0_IV section lands.

French	English	First appearance	Note
dimension combinatoire (d'un espace topologique)	combinatorial dimension (of a topological space)	0_IV.14.1.1	Krull-style: the supremum of lengths of strictly increasing chains of irreducible closed subsets. Rendered $dim_{c} (X)$ or $dim (X)$ .
chaîne (de parties fermées irréductibles)	chain (of irreducible closed subsets)	0_IV.14.1.1	Finite strictly increasing sequence $i_{0} < \dots < i_{n}$ in an ordered set; length is $n$ .
longueur (d'une chaîne)	length (of a chain)	0_IV.14.1.1	Defined as $n$ for a chain $i_{0} < \dots < i_{n}$ .
dimension combinatoire en $x$	combinatorial dimension at $x$	0_IV.14.1.2	`dim_x(X) = inf_U dim(U)` over open neighbourhoods of $x$ .
espace équidimensionnel	equidimensional space	0_IV.14.1.3	All irreducible components have the same dimension.
codimension combinatoire	combinatorial codimension	0_IV.14.2.1	$co d im (Y, X)$ ; supremum of chain lengths of irreducible closed subsets of $X$ having $Y$ as smallest element.
codimension de $Y$ dans $X$ au point $x$	codimension of $Y$ in $X$ at the point $x$	0_IV.14.2.4	`codim_x(Y, X) = sup_U codim(Y ∩ U, U)` over open neighbourhoods of $x$ .
espace équicodimensionnel	equicodimensional space	0_IV.14.2.1	All minimal irreducible closed subsets have the same codimension in $X$ .
chaîne saturée	saturated chain	0_IV.14.3.1	No irreducible closed subset fits strictly between consecutive members.
condition des chaînes	chain condition	0_IV.14.3.2	Synonym for catenarity at the topological-space level.
espace topologique caténaire	catenary topological space	0_IV.14.3.2	Any two irreducible closed subsets $Y \subset Z$ admit a saturated chain of fixed codimension.
espace biéquidimensionnel	biequidimensional space	0_IV.14.3.3	Noetherian Kolmogorov space of finite dimension satisfying the equivalent conditions of $(0_{I V}, 14.3.3)$ .
suite $M$ -régulière	$M$ -regular sequence	0_IV.15.1.1	$(f_{1}, \dots, f_{n})$ with each $f_{i}$ a non-zero-divisor on $M / (f_{1}, \dots, f_{i - 1}) M$ and $(f_{1}, \dots, f_{n}) M \neq = M$ .
suite $F$ -régulière	$F$ -regular sequence	0_IV.15.1	Sheaf-of-modules analog of an $M$ -regular sequence.
non diviseur de zéro (dans un module)	not a zero-divisor (in a module)	0_IV.15.1.1	$f \in A$ such that the homothety $x \mapsto f x$ of $M$ is injective.
élément $M$ -régulier	$M$ -regular element	0_IV.15.1.4	Single-element case of an $M$ -regular sequence; equivalent to "not a zero-divisor in $M$ ".
élément $M$ -quasi-régulier	$M$ -quasi-regular element	0_IV.15.1.4	$f \in A$ for which the canonical homomorphism $(M / f M) [T] \to g r_{∙} (M)$ is bijective.
suite $M$ -quasi-régulière	$M$ -quasi-regular sequence	0_IV.15.1.7	Sequence for which the canonical homomorphism `(15.1.1.1)` is bijective.
suite $F$ -quasi-régulière	$F$ -quasi-regular sequence	0_IV.15.2.2	Sheaf-of-modules analog of an $M$ -quasi-regular sequence.
suite régulière / quasi-régulière	regular sequence / quasi-regular sequence	0_IV.15.1.7	The case $M = A$ . (J.-P. Serre's term " $M$ -suite" `[17]` is the special case where $A$ is Noetherian local and the $f_{i}$ lie in the maximal ideal.)
$M$ -suite (terminologie de J.-P. Serre)	$M$ -sequence (Serre's terminology)	0_IV.15.1.12	Synonym for $M$ -regular sequence in the local Noetherian setting; cited as `[17]`.
filtration $J$ -préadique (sur un module)	$J$ -preadic filtration (on a module)	0_IV.15.1.1	Standard filtration $(J^{k} M)$ associated with an ideal $J = \sum f_{i} A$ .
profondeur (d'un module)	depth (of a module)	0_IV.16.4.1	Rendered $p ro f (M)$ , $p ro f_{m} (M)$ in formulas; English "depth" in running prose.
coprofondeur (d'un module)	codepth (of a module)	0_IV.16.4.9	`coprof(M) = dim(M) − prof(M)`; rendered $co p ro f (M)$ , $co p ro f_{m} (M)$ in formulas, "codepth" in prose.
dimension de Krull (d'un anneau)	Krull dimension (of a ring)	0_IV.16.1.1	$dim (A)$ ; the combinatorial dimension of $Spec (A)$ .
hauteur (d'un idéal premier)	height (of a prime ideal)	0_IV.16.1.3	`ht(𝔭) = dim(A_𝔭) = codim(V(𝔭), Spec(A))`. Generalized to arbitrary ideals as $h t (a) = co d im (V (a), Spec (A))$ .
anneau caténaire	catenary ring	0_IV.16.1.4	$Spec (A)$ is catenary; characterized by `dim(A_𝔮) + dim(A_𝔭/𝔮 A_𝔭) = dim(A_𝔭)` for $q \subset p$ .
anneau équidimensionnel / équicodimensionnel / biéquidimensionnel	equidimensional / equicodimensional / biequidimensional ring	0_IV.16.1.4	Topological conditions on $Spec (A)$ , transferred from $(0_{I V}, 14)$ .
dimension (d'un module)	dimension (of a module)	0_IV.16.1.7	`dim_A(M) = dim(A/Ann(M)) = dim(Supp(M))` for $M$ of finite type.
anneau semi-local noethérien	Noetherian semi-local ring	0_IV.16.2.1	Noetherian ring with finitely many maximal ideals; radical $r$ .
idéal de définition (d'un anneau semi-local)	ideal of definition (of a semi-local ring)	0_IV.16.2.1	Ideal $q \subset r$ containing a power of $r$ ; equivalently $A / q$ Artinian.
polynôme de Hilbert-Samuel	Hilbert-Samuel polynomial	0_IV.16.2.1	$P_{q} (M, n)$ such that $l o n g (M / q^{n} M) = P_{q} (M, n)$ for $n$ large. Degree denoted $d (M)$ , independent of $q$ .
filtration $q$ -bonne	$q$ -good filtration	0_IV.16.2.2.1	Bourbaki's terminology preserved; $M_{n + 1} = q M_{n}$ for $n$ large.
Krull-Chevalley-Samuel (théorème de)	Krull-Chevalley-Samuel (theorem of)	0_IV.16.2.3	$d (M) = s (M) = dim (M)$ for $M$ of finite type over a Noetherian semi-local ring.
système de paramètres (pour un module)	system of parameters (for a module)	0_IV.16.3.6	$(x_{i})_{1 \leq i \leq n}$ with $n = dim (M)$ and $M / (\sum x_{i} M)$ of finite length.
Hauptidealsatz	Hauptidealsatz	0_IV.16.3.2	Krull's principal-ideal theorem; preserved in German as in EGA.
homomorphisme local	local homomorphism	0_IV.16.3.9	$ϕ : A \to B$ between local rings with $ϕ (m_{A}) \subset m_{B}$ .
module de Cohen-Macaulay	Cohen-Macaulay module	0_IV.16.5.1	$co p ro f (M) = 0$ ; equivalently $M = 0$ or $dim (M) = p ro f (M)$ . Global notion via `(16.5.13)`.
anneau régulier (local)	(local) regular ring	0_IV.17.1.1
système régulier de paramètres	regular system of parameters	0_IV.17.1.6	System of parameters that generates $m$ ; equivalent to $A$ being regular.
dimension projective (d'un module)	projective dimension (of a module)	0_IV.17.2.1	Rendered $dim . p ro j (M)$ , $dim . p ro j_{A} (M)$ .
dimension injective (d'un module)	injective dimension (of a module)	0_IV.17.2.1	Rendered $dim . inj (M)$ , $dim . in j_{A} (M)$ .
dimension cohomologique globale	global cohomological dimension	0_IV.17.2.8	Rendered $dim . co h (A)$ ; "cohomological dimension" in running prose.
dimension projective / injective ponctuelle	pointwise projective / injective dimension	0_IV.17.2.14	For an $O_{X}$ -Module on a ringed space: `sup_x dim. proj(ℱ_x)` (resp. inj).
dimension cohomologique ponctuelle	pointwise cohomological dimension	0_IV.17.2.14	For a ringed space $X$ : `sup_x dim. coh(𝒪_x) = dim. coh(X)`.
théorème des syzygies (Hilbert)	syzygy theorem (Hilbert)	0_IV.17.3.1	Cited as the source of `dim. coh(A) = dim(A)` for $A$ regular local.
anneau noethérien régulier	regular Noetherian ring	0_IV.17.3.6	$A_{p}$ regular for every prime $p$ ; equivalently $A_{m}$ regular for every maximal $m$ .
anneau de Cohen-Macaulay	Cohen-Macaulay ring	0_IV.16.5.1	Abbreviation `(CM)` preserved.
$A$ -anneau	$A$ -ring	0_IV.18.1.1	Pair $(B, ρ)$ with $ρ : A \to B$ a ring homomorphism; $A$ not assumed commutative.
$A$ -homomorphisme (d'anneaux)	$A$ -homomorphism (of rings)	0_IV.18.1.1	Morphism in the category of $A$ -rings.
homomorphisme structural	structural homomorphism	0_IV.18.1.1	The map $ρ : A \to B$ defining the $A$ -ring structure on $B$ .
produit fibré (de $A$ -anneaux)	fibre product (of $A$ -rings)	0_IV.18.1.2	$E \times_{B} F$ ; sub-ring of $E \times F$ of pairs $(x, y)$ with $f (x) = g (y)$ .
$A$ -anneau augmenté (sur $B$ )	augmented $A$ -ring (over $B$ )	0_IV.18.1.4	$A$ -ring $E$ with surjective $A$ -homomorphism $f : E \to B$ ; from `(M, VIII)`.
augmentation	augmentation	0_IV.18.1.4	The surjective map $f : E \to B$ .
idéal d'augmentation	augmentation ideal	0_IV.18.1.4	The kernel $J$ of the augmentation $f$ .
anneau augmenté trivial	trivial augmented ring	0_IV.18.1.4	Augmented ring admitting a right inverse $s : B \to E$ of the augmentation.
image réciproque (d'un anneau augmenté)	inverse image (of an augmented ring)	0_IV.18.1.5	The fibre product $E \times_{B} F$ viewed as augmented $A$ -ring over $F$ .
$A$ -extension (d'un anneau par un bimodule)	$A$ -extension (of a ring by a bimodule)	0_IV.18.2.2	Exact sequence $0 \to L \to E \to B \to 0$ with $j (L)$ of square zero in $E$ .
$A$ -équivalence (de $A$ -extensions)	$A$ -equivalence (of $A$ -extensions)	0_IV.18.2.2	Isomorphism of $A$ -rings $u : E \sim E^{'}$ compatible with the extension diagrams.
$A$ -extension $A$ -triviale	$A$ -trivial $A$ -extension	0_IV.18.2.3	Extension whose underlying augmented ring is trivial.
extension triviale type	trivial type extension	0_IV.18.2.3	The canonical model $D_{B} (L)$ on $B \times L$ with product $(x, s) (y, t) = (x y, x t + sy)$ .
$D_{B} (L)$	$D_{B} (L)$	0_IV.18.2.3	Notation for the trivial type extension of $B$ by $L$ .
di-homomorphisme (de bimodules)	di-homomorphism (of bimodules)	0_IV.18.2.4	Pair $(v, w)$ with $w (b z) = v (b) w (z)$ , $w (z b) = w (z) v (b)$ for $z \in L$ , $b \in B$ .
image réciproque (d'une extension)	inverse image (of an extension)	0_IV.18.2.5	$F = E^{'} \times_{B^{'}} B$ viewed as $A$ -extension of $B$ by $L^{'}$ .
somme amalgamée (de $A$ -bimodules)	amalgamated sum (of $A$ -bimodules)	0_IV.18.2.7	$Y \oplus_{X} Z$ ; quotient of $Y \times Z$ by the image of $x \mapsto (f (x), - g (x))$ .
extension déduite (au moyen d'un homomorphisme)	extension deduced (by means of a homomorphism)	0_IV.18.2.8	$H = E \oplus_{L} L^{'}$ ; the pushout extension along $w : L \to L^{'}$ .
groupe des classes de $A$ -extensions	group of classes of $A$ -extensions	0_IV.18.3.4	$E x a n_{A} (B, L)$ ; commutative-group structure from the addition $L \times L \to L$ .
$E x a n_{A} (B, L)$	$E x a n_{A} (B, L)$	0_IV.18.3.4	Group of $A$ -equivalence classes of $A$ -extensions of $B$ by $L$ .
$E x a n_{A / A^{'}} (B, L)$	$E x a n_{A / A^{'}} (B, L)$	0_IV.18.3.7	Kernel of $u^{*} : E x a n_{A} (B, L) \to E x a n_{A^{'}} (B, L)$ ; classes $A^{'}$ -trivial under $u$ .
$E x a l_{A} (B, L)$	$E x a l_{A} (B, L)$	0_IV.18.4.1	Subgroup of $E x a n_{A} (B, L)$ formed by classes that are $A$ -algebras (with $A$ commutative).
$E x a l co m_{A} (B, L)$	$E x a l co m_{A} (B, L)$	0_IV.18.4.2	Subgroup of $E x a l_{A} (B, L)$ formed by classes that are commutative $A$ -algebras.
extension de Hochschild	Hochschild extension	0_IV.18.4.3	$A$ -algebra extension $E$ of $B$ by $L$ split as $A$ -modules; classified by $H_{A}^{2} (B, L)$ .
2-cocycle de Hochschild	Hochschild 2-cocycle	0_IV.18.4.3	$A$ -bilinear $f : B \times B \to L$ satisfying `(18.4.3.1)`; symmetric in the commutative case.
2-cobord de Hochschild	Hochschild 2-coboundary	0_IV.18.4.3	$f (x, y) = xg (y) - g (x y) + g (x) y$ for some $A$ -linear $g : B \to L$ .
cohomologie de Hochschild	Hochschild cohomology	0_IV.18.4.3	$H_{A}^{i} (B, L) = H^{i} (Hom_{B} (P_{∙}, L))$ ; classifies Hochschild extensions in degree 2.
$H_{A}^{2} (B, L)^{s}$	$H_{A}^{2} (B, L)^{s}$	0_IV.18.4.3	Image of the symmetric-2-cocycle subgroup; classifies commutative Hochschild extensions.
$E x an t o p_{A} (B, L)$	$E x an t o p_{A} (B, L)$	0_IV.18.5.1	Inductive limit $lim E x a n_{A / J} (B / K, L)$ over open-ideal pairs $(J, K)$ .
$E x a lt o p_{A} (B, L)$ / $E x a l co t o p_{A} (B, L)$	$E x a lt o p_{A} (B, L)$ / $E x a l co t o p_{A} (B, L)$	0_IV.18.5.1	Topological algebra (resp. commutative algebra) analogues of `Exantop`.
$Hom . co n t_{C} (M, N)$	$Hom . co n t_{C} (M, N)$	0_IV.18.5.3	Additive group of continuous $C$ -homomorphisms $M \to N$ .
épimorphisme formel	formal epimorphism	0_IV.19.1.2	Continuous $A$ -homomorphism $u : M \to N$ with $u (M)$ dense in $N$ ; equivalently $M \to N \to N / W_{λ}$ surjective for every $λ$ .
monomorphisme formel	formal monomorphism	0_IV.19.1.2	Continuous $A$ -homomorphism $u : M \to N$ whose source topology coincides with the inverse image of the target topology; $\overset{u}{^} : \hat{M} \to \overset{u}{^} (\hat{M})$ an isomorphism.
bimorphisme formel	formal bimorphism	0_IV.19.1.2	At once a formal monomorphism and a formal epimorphism; equivalently $\overset{u}{^} : \hat{M} \to \hat{N}$ is a topological isomorphism. (Compare $(0_{I V}, 20.7.6)$ .)
formellement inversible à gauche	formally left-invertible	0_IV.19.1.5	Continuous $A$ -homomorphism $u : M \to N$ such that every continuous $v : M \to E$ (with $E$ discrete) factors through $u$ ; locked at §0_IV.19.1.5.
formellement inversible à droite	formally right-invertible	0_IV.19.1.15	Dual notion; for every open $W \subset N$ , $N / W^{''}$ factors through some $M / V^{'}$ ; implies formal epimorphism.
module formellement projectif	formally projective module	0_IV.19.2.1	Lifting property for surjections of discrete $(A / J)$ -modules; weaker than strict formal projectivity.
module strictement formellement projectif	strictly formally projective module	0_IV.19.2.3	There exist matched fundamental systems $(J_{λ})$ , $(V_{λ})$ with $M / V_{λ}$ projective over $A / J_{λ}$ ; coincides with formal projectivity when topology of $M$ is deduced from that of $A$ `(19.2.4)`.
anneau formellement lisse / étale / non ramifié	formally smooth / étale / unramified ring	0_IV.19.3.1	For a given topology (usually $J$ -preadic or discrete). Always preserve the topology qualifier.
algèbre formellement lisse / étale / non ramifiée	formally smooth / étale / unramified algebra	0_IV.19.3.1
algèbre symétrique $S_{A}^{∙} (V)$	symmetric algebra $S_{A}^{∙} (V)$	0_IV.19.3.2	$S_{A}^{∙} (V) = \oplus_{n ⩾ 0} S_{A}^{n} (V)$ ; formally smooth $A$ -algebra over discrete $A$ when $V$ is projective.
anneau de séries formelles $A [[T_{α}]]$	formal power series ring $A [[T_{α}]]$	0_IV.19.3.4	Broad algebra over $A$ of the monoid $N^{(I)}$ ; equipped with the product topology, is formally smooth.
algèbre topologique $S_{C}^{∙} (V)$	topological symmetric algebra $S_{C}^{∙} (V)$	0_IV.19.5.1	Symmetric algebra equipped with the canonical linear topology generated by $a \cdot S_{C}^{∙} (V) + U \cdot S_{C}^{∙} (V)$ .
premiers critères de lissité formelle	first criteria for formal smoothness	0_IV.19.4	Title of §19.4; collects the `Exalcotop`-criterion and the Hochschild-cocycle reformulation.
anneau gradué associé	associated graded ring	0_IV.19.5.1	$g r_{J}^{∙} (B) = \oplus_{n} J^{n} / J^{n + 1}$ ; receives the canonical homomorphism $ϕ : S_{C} (J / J^{2}) \to g r_{J}^{∙} (B)$ of `(19.5.2)`.
morphisme de transition	transition homomorphism	0_IV.19.5.6.2	Standard projective-system terminology; carried over verbatim.
extension formellement lisse de corps	formally smooth extension of fields	0_IV.19.6.1	Cohen's theorem: a field extension $k \subset K$ is formally smooth iff $K$ is separable over $k$ .
corps de représentants	field of representatives	0_IV.19.6.2	Subfield $K^{'} \subset A$ of a separated complete local ring such that $K^{'} \sim A / m$ ; existence under formal-smoothness of the residue extension.
anneau géométriquement régulier (sur $k$ )	geometrically regular ring (over $k$ )	0_IV.19.6.5	Noetherian local $k$ -algebra $A$ such that $A \otimes_{k} k^{'}$ is regular for every finite extension $k^{'} / k$ . Cross-ref `(IV, 6.7.6)`.
multiplicité radicielle finie	finite radicial multiplicity	0_IV.19.6.6	Of a field extension $K / k$ : there exists a finite radicial $k_{0} / k$ with $(K \otimes_{k} k_{0})_{re d}$ separable over $k_{0}$ . Cross-ref `(IV, 4.7.4)`.
algèbre de Cohen	Cohen $A$ -algebra	0_IV.19.8.1	Complete flat local $A$ -algebra $B$ with $B \otimes_{A} k$ a field and a separable extension of $k$ .
anneau local premier	prime local ring	0_IV.19.8.3	Local ring of the form $Z_{p Z}$ for a prime ideal $p Z$ of $Z$ .
anneau local complet premier	complete prime local ring	0_IV.19.8.3	Completion of a prime local ring: $Z_{p}$ (for $p$ prime) or $Q$ (for $p = 0$ ).
anneau de Cohen	Cohen ring	0_IV.19.8.5	Complete unramified DVR of mixed characteristic with prescribed residue field; in $p > 0$ characteristic, a $p$ -Cohen ring.
$p$ -anneau de Cohen	$p$ -Cohen ring	0_IV.19.8.5	Cohen ring of mixed characteristic `(0, p)`: complete DVR with $p A = m$ and $Z \to A$ injective.
anneau local complet	complete local ring	0_IV.19.8.8	Local ring complete for its maximal-ideal-preadic topology; the central object of Cohen's structure theorem.
caractéristique mixte / inégale	mixed / unequal characteristic	0_IV.19.8.5	A local ring $(A, m, k)$ with $c ha r (A) = 0$ and $c ha r (k) = p > 0$ .
caractéristique égale	equal characteristic	0_IV.19.8.5	A local ring $(A, m, k)$ with $c ha r (A) = c ha r (k)$ .
algèbre formellement lisse relativement à $Λ$	formally smooth algebra relatively to $Λ$	0_IV.19.9.1	Relative-formal-smoothness: factor only through $Λ$ -homomorphisms; $E x a l co t o p_{A /Λ} (B, L) = 0$ for discrete $B$ -modules $L$ `(19.9.8)`.
$Ω_{B / A}$ (module de différentielles relatives)	$Ω_{B / A}$ (module of relative differentials)	0_IV.20.1.1	Sometimes $Ω_{B / A}^{1}$ . Sheaf version: $Ω_{X / Y}^{1}$ .
$Der_{A} (B, M)$ ( $A$ -dérivations de $B$ à valeurs dans $M$ )	$Der_{A} (B, M)$ ( $A$ -derivations of $B$ with values in $M$ )	0_IV.20.1.1
$A$ -dérivation (de $B$ dans $L$ )	$A$ -derivation (of $B$ into $L$ )	0_IV.20.1.2	Map $D : B \to L$ satisfying `(i)` $A$ -bimodule homomorphism and `(ii)` Leibniz rule $D (f g) = f \cdot D (g) + D (f) \cdot g$ .
$Z$ -dérivation	$Z$ -derivation	0_IV.20.1.2	Additive map satisfying Leibniz; "derivation" in EGA's casual usage.
extension d'une dérivation	extension of a derivation	0_IV.20.1.3	Used informally for the affine-action picture of `(20.1.3)` and `(20.1.4)`.
produit semi-direct trivial	trivial semi-direct product	0_IV.20.1.5	The trivial-type extension $D_{C} (L)$ realised as a semi-direct product (cf. `(18.2.3)`).
$Der . co n t_{A} (B, L)$	$Der . co n t_{A} (B, L)$	0_IV.20.3.1	Continuous-derivation module; subgroup of $Der_{A} (B, L)$ (sub- $A$ -module when $A$ , $B$ commutative and $L$ a $B$ -module). EGA's literal notation kept.
dérivation continue	continuous derivation	0_IV.20.3.1	$A$ -derivation continuous for the given topologies on $B$ and $L$ .
$E x an t o p_{B} (C, L)$ (suite des $Der . co n t$ )	$E x an t o p_{B} (C, L)$ (in the $Der . co n t$ sequence)	0_IV.20.3.6	Six-term exact sequence for continuous derivations, obtained from `(20.2.2.1)` by passage to the inductive limit.
partie principale d'ordre $n$	principal part of order $n$	0_IV.20.4.2	Element of $P_{B / A}^{n}$ ; the case $n = 1$ is the basic object of §20.4.
algèbre augmentée des parties principales d'ordre `1`	augmented algebra of principal parts of order `1`	0_IV.20.4.2	$P_{B / A}^{1} = (B \otimes_{A} B) / J^{2}$ , equipped with $B$ -algebra structure via $j_{1}$ and augmentation $ϵ$ deduced from $p$ .
$P_{B / A}^{1}$ (OCR $P_{B / A}^{1}$ / $P g^{1}$ )	$P_{B / A}^{1}$	0_IV.20.4.2	Augmented $B$ -algebra of principal parts of order `1`; OCR's $P g^{1}$ / $P_{B / A}^{1}$ rendered with script $P$ .
$P_{B / A}^{n}$	$P_{B / A}^{n}$	0_IV.20.4.14	Higher-order analogue $(B \otimes_{A} B) / (J_{B / A})^{n + 1}$ ; basis of "differential calculus of order $n$ ".
$J_{B / A}$ (noyau de $B \otimes_{A} B \to B$ )	$J_{B / A}$ (kernel of $B \otimes_{A} B \to B$ )	0_IV.20.4.1	Diagonal ideal; written $J$ when unambiguous.
différentielle (de $x$ relativement à $A$ )	differential (of $x$ relative to $A$ )	0_IV.20.4.6	$d_{B / A} (x) = (p_{2} - p_{1}) (x)$ , also written $d (x)$ or `dx`.
différentielle extérieure (de $B$ relative à $A$ )	exterior differential (of $B$ relative to $A$ )	0_IV.20.4.6	$d_{B / A} : B \to Ω_{B / A}$ ; the universal $A$ -derivation.
différentielle universelle	universal differential	0_IV.20.4.8	The exterior differential $d_{B / A}$ viewed via its universal property `(20.4.8.2)`.
module des différentielles absolues	module of absolute differentials	0_IV.20.4.3	$Ω_{B} = Ω_{B / Z}$ ; the case $A = Z$ .
$\hat{Ω}_{B / A}$	$\hat{Ω}_{B / A}$	0_IV.20.4.3	Separated completion of the topological $B$ -module $Ω_{B / A}$ .
$u_{C / B / A}$ (changement de base sur $Ω$ )	$u_{C / B / A}$ (base-change map on $Ω$ )	0_IV.20.5.2	Canonical $C$ -module homomorphism $Ω_{B / A} \otimes_{B} C \to Ω_{C / A}$ deduced from $u : B \to C$ .
$v_{C / B / A}$ (changement d'algèbre de base sur $Ω$ )	$v_{C / B / A}$ (change-of-base-ring map on $Ω$ )	0_IV.20.5.3	Canonical $C$ -module homomorphism $Ω_{C / A} \to Ω_{C / B}$ deduced from $v : A \to B$ .
$δ_{C / B / A}$ (homomorphisme conormal)	$δ_{C / B / A}$ (conormal homomorphism)	0_IV.20.5.11	Canonical $C$ -homomorphism $K / K^{2} \to Ω_{B / A} \otimes_{B} C$ for $C = B / K$ ; appears in the fundamental sequence `(20.5.12.1)`.
Frobenius (endomorphisme de $A$ en caractéristique $p$ )	Frobenius (endomorphism of $A$ in characteristic $p$ )	0_IV.21.1.4	$F : A \to A$ , $a \mapsto a^{p}$ .
$p$ -base	$p$ -basis	0_IV.21.1.4	A family whose images in $A / A^{p}$ form a basis.
de caractéristique $p$ (anneau)	of characteristic $p$ (ring)	0_IV.21.1.1	Ring admitting (unique) homomorphism from prime field $F_{p}$ (or $Q$ for $p = 0$ ); equivalently $p A = 0$ for prime $p$ .
corps premier	prime field	0_IV.21.1.1	$F_{p}$ or $Q$ ; the prime subfield of a field of characteristic $p$ .
$A^{p}$	$A^{p}$	0_IV.21.1.4	Subring $F_{A} (A) \subset A$ of $p$ -th powers; $A$ is naturally an $A^{p}$ -algebra.
$A^{[p]}$ ( $A$ comme $A$ -algèbre via `F_A`)	$A^{[p]}$ ( $A$ as $A$ -algebra via `F_A`)	0_IV.21.1.4	$A$ -algebra $A_{F_{A}}$ with scalar action $λ \cdot x = λ^{p} x$ . For $A$ -module $E$ , $E^{[p]} = E \otimes_{A} A^{[p]}$ .
$p$ -libre (sur $A$ )	$p$ -free (over $A$ )	0_IV.21.1.9	Family whose degree- $< p$ monomials are linearly independent in the $A [B^{p}]$ -module $B$ .
système de $p$ -générateurs	system of $p$ -generators	0_IV.21.1.9	Family whose degree- $< p$ monomials generate $B$ as an $A [B^{p}]$ -module.
« absolument » $p$ -libre (resp. $p$ -base)	"absolutely" $p$ -free (resp. $p$ -basis)	0_IV.21.1.9	Case $A = F_{p}$ , where $A [B^{p}] = B^{p}$ ; mention of $A$ then omitted.
$Θ_{B / A}$	$Θ_{B / A}$	0_IV.21.3.2	$Ω_{B / A} \otimes_{B, j} A$ ; appears in the exact sequence with $Ξ_{B / A}$ and $Ω_{A}^{1}$ .
$Ξ_{B / A}$	$Ξ_{B / A}$	0_IV.21.3.2	Kernel of $π_{B / A} : Θ_{B / A} \to Ω_{A}^{1}$ in the di-homomorphism setting `(21.3.1)`.
$π_{B / A}$ , $ω_{B / A}$	$π_{B / A}$ , $ω_{B / A}$	0_IV.21.3.2	Canonical homomorphisms attached to the di-homomorphism $(j, F_{A})$ ; $ω_{B / A}$ is the $B$ -module map $(Ω_{B / A})^{[p]} \to Ω_{A}^{1} \otimes_{A} B$ .
corps $k_{0}$ -admissible (pour une extension)	$k_{0}$ -admissible field (for an extension)	0_IV.21.6.1	$k$ with $k_{0} \subset k \subset K \subset L$ such that $Υ_{L / K / k_{0}} \to Υ_{L / K / k}$ is bijective.
défaut de $k_{0}$ -admissibilité	$k_{0}$ -admissibility defect	0_IV.21.6.1	$d (L / K, k / k_{0}) = r g_{L} Δ (L / K, k / k_{0})$ ; zero iff $k$ is $k_{0}$ -admissible for $L / K$ .
égalité de Cartier	Cartier's equality	0_IV.21.7.1	$r g Ω_{L / K}^{1} - r g Υ_{L / K} = d e g . t r_{K} L$ for $L$ of finite type over $K$ .
exposant caractéristique	characteristic exponent	0_IV.21.6.2	$p$ if $c ha r = p > 0$ , else `1`; unifies char-`0` and char- $p > 0$ statements.
corps parfait / imparfait	perfect / imperfect field	0_IV.21.4.4	$K = K^{p}$ (char $p > 0$ ) or char `0`; equivalent to $Ω_{K}^{1} = 0$ (char $p$ ).
module d'imperfection	imperfection module	0_IV.20.6.1	$Υ_{C / B / A}$ in EGA notation; kernel of $v_{C / B / A} : Ω_{B / A}^{1} \otimes_{B} C \to Ω_{C / A}^{1}$ . Also rendered $Υ_{B / A}$ when $A = Z$ . (See also $(0_{I V}, 21.4.2)$ .)
homomorphisme caractéristique	characteristic homomorphism	0_IV.20.6.8	$χ_{E} : Υ_{C / B / A} \to L$ associated to an $A$ -trivial $B$ -extension $E$ of $C$ by $L$ . Also denoted $χ_{B}$ or $χ_{B /Λ}$ in the algebra-with-ideal case $(0_{I V}, 20.6.24)$ .
$Υ_{B / A /Λ}^{C}$ (module d'imperfection ternaire)	$Υ_{B / A /Λ}^{C}$ (ternary imperfection module)	0_IV.20.6.14	$Ker (u_{B / A /Λ} \otimes 1_{C}) = Ker (Ω_{A /Λ}^{1} \otimes_{A} C \to Ω_{B /Λ}^{1} \otimes_{B} C)$ ; written $Υ_{B / A}^{C}$ when $Λ = Z$ .
complexe $K_{∙} (C / B / A)$ (de chaînes)	(chain) complex $K_{∙} (C / B / A)$	0_IV.20.6.5	Two-term $C$ -module complex with $K_{0} = Ω_{C / A}^{1}$ , $K_{1} = Ω_{B / A}^{1} \otimes_{B} C$ , differential $v_{C / B / A}$ ; homology recovers $Ω_{C / B}^{1}$ and $Υ_{C / B / A}$ .
complexe $T_{∙} (C / B / A)$	complex $T_{∙} (C / B / A)$	0_IV.20.6.15	Two-term acyclic $C$ -module complex with $T_{0} = T_{1} = Ω_{B / A}^{1} \otimes_{B} C$ , differential the identity; homotopic to `0`.
complexe $K_{∙}^{'} (C / A /Λ)$	complex $K_{∙}^{'} (C / A /Λ)$	0_IV.20.6.15	Direct sum $K_{∙} (C / A /Λ) \oplus T_{∙} (C / B / A)$ ; same (co)homology as $K_{∙} (C / A /Λ)$ .
complexe $K_{∙}^{C} (B / A /Λ)$	complex $K_{∙}^{C} (B / A /Λ)$	0_IV.20.6.15	The base-changed complex $K_{∙} (B / A /Λ) \otimes_{B} C$ .
complexe $F_{∙} (C / A)$ (fibré conormal)	complex $F_{∙} (C / A)$ (conormal-bundle complex)	0_IV.20.6.26	Two-term $C$ -module complex $L / L^{2} \to Ω_{B / A}^{1} \otimes_{B} C$ for $C = B / L$ a polynomial-algebra presentation. Plays the role of a conormal bundle for $Spec (C)$ over $Spec (A)$ ; reused for duality and Riemann-Roch.
compatible avec une dérivation	compatible with a derivation	0_IV.20.6.x	Standard usage; "this homomorphism is compatible with the derivation $D$ " preserved as in source.
morphisme de transition	transition morphism	0_IV.20.7.14	Standard projective-system terminology; carried over verbatim.
bimorphisme formel	formal bimorphism	0_IV.20.7.6	Formal monomorphism + formal epimorphism; cf. `(19.1.2)`.
formellement inversible à gauche	formally left-invertible	0_IV.20.7.2	Cf. `(19.1.5)`; condition on a continuous homomorphism dual to "formally projective".
$\hat{Ω}_{B / A}^{1}$ (séparé complété du module de différentielles)	$\hat{Ω}_{B / A}^{1}$ (separated completion of the differentials module)	0_IV.20.7.14	Projective limit of the $Ω_{B^{'} / A^{'}}^{1}$ over open-ideal pairs.
$Hom . co n t_{C} (M, N)$ (homomorphismes continus)	$Hom . co n t_{C} (M, N)$ (continuous homomorphisms)	0_IV.20.7.2	Notation for continuous $C$ -homomorphisms between topological $C$ -modules; carried over from §18.5.
$Der . co n t_{A} (B, L)$ (dérivations continues)	$Der . co n t_{A} (B, L)$ (continuous derivations)	0_IV.20.7.2	Continuous $A$ -derivations $B \to L$ for topological $A$ , $B$ and topological $B$ -module $L$ .
$E x a l co t o p_{A} (B, L)$	$E x a l co t o p_{A} (B, L)$	0_IV.20.7.8	Topological-algebra variant of $E x a l co m_{A} (B, L)$ ; defined in $(0_{I V}, 18.5)$ and reused systematically in §20.7.
critère différentiel (de lissité formelle)	differential criterion (for formal smoothness)	0_IV.22.0	EGA's labels `(D_I)`, $(D_{II})$ , $(D_{III})$ preserved verbatim.
critère jacobien de Zariski	Zariski's Jacobian criterion	0_IV.22.6.7	Differential criterion for formal smoothness of a finite-type algebra over a separable extension `(22.6.7)`.
critère jacobien de Nagata	Nagata's Jacobian criterion	0_IV.22.7.3	Analogue of Zariski's criterion for quotients of formal-series rings over a field `(22.7.3)`; key tool for `(22.7.6)`.
relèvement de la lissité formelle	lifting of formal smoothness	0_IV.22.1.1	Theorem characterising when $B$ is $Λ$ -formally-smooth for the $J$ -preadic topology in terms of $J / J^{2}$ , $g r_{J}$ , $χ$ and $Ω_{B /Λ}^{1} \otimes_{B} C$ .
χ_{A/k} (homomorphisme caractéristique de l'algèbre locale $A$ )	χ_{A/k} (characteristic homomorphism of the local algebra $A$ )	0_IV.22.2.1	Map $Υ_{K / k} \to m / m^{2}$ defined when $K = A / m$ , $k$ a field; central to `(22.2.2)`.
$F L (K / k, V)$	$F L (K / k, V)$	0_IV.22.2.4	Set of equivalence classes of triples $(A, α, β)$ with $A$ a complete Noetherian local $k$ -algebra formally smooth; in bijection with $Hom_{K}^{inj} (Υ_{K / k}, V)$ `(22.2.5)`.
extension radicielle finie	finite radicial extension	0_IV.22.5.5	Purely inseparable finite extension $k^{'} / k$ with $k^{' p} \subset k$ ; appears in the criterion `(22.5.7)` and the proof of `(22.5.8)`.
« polynôme d'Eisenstein »	"Eisenstein polynomial"	0_IV.22.4.3	Unitary polynomial $T^{d} + \sum c_{k} T^{d - k}$ with $c_{k} \in m$ ; quotation marks preserved as in source.
« absolument $p$ -libre »	"absolutely $p$ -free"	0_IV.22.4.4	Family $(a_{i})$ $p$ -free over $K^{p}$ ; quotation marks preserved as in source.
χ′_{B/A} (homomorphisme caractéristique de la $A$ -algèbre $B$ )	χ′_{B/A} (characteristic homomorphism of the $A$ -algebra $B$ )	0_IV.22.4.6.2	Map $Ξ_{B_{(K)} / K} \to V$ relative to $i$ , $j$ , $m$ ; analogue of `(22.4.5.3)` in the maximal- $m$ setting.
de multiplicité radicielle finie	of finite radicial multiplicity	0_IV.22.6.9	Cf. $(0_{I V}, 19.6.6)$ ; weaker than separable, generalises `(22.6.7)`.
algèbre de séries formelles $k [[T_{1}, \dots, T_{r}]]$	algebra of formal series $k [[T_{1}, \dots, T_{r}]]$	0_IV.22.7.2	Setting of Nagata's criterion `(22.7.2)`–`(22.7.3)`; $A_{p}$ regular and formally smooth over $k$ when $k (p)$ separable over $k$ .
anneau japonais	Japanese ring	0_IV.23.1.1	Modern: Nagata / pseudo-geometric. Translator's note at first use.
universellement japonais	universally Japanese	0_IV.23.1.1
clôture intégrale	integral closure	0_IV.23.1.1	Of an integral ring in an extension of its field of fractions, or in `(23.2.1)` of $A_{re d}$ .
fermeture intégrale	integral closure	0_IV.23.1.1	EGA uses both $c l \overset{o}{^} t u re in t \overset{e}{ˊ} g r a l e$ (of $A$ ) and $f er m e t u re in t \overset{e}{ˊ} g r a l e$ (of $A$ in $K^{'}$ ); we render both as "integral closure".
extension radicielle	radicial extension	0_IV.23.1.2	Purely inseparable; EGA's term preserved (cf. `radiciel`).
extension quasi-galoisienne	quasi-Galois extension	0_IV.23.1.2	Normal extension (not necessarily separable).
corps des fractions	field of fractions	0_IV.23.1.1
anneau intégralement clos	integrally closed ring	0_IV.23.1.3
anneau unibranche	unibranch ring	0_IV.23.2.1	$A_{re d}$ integral and the integral closure of $A_{re d}$ is a local ring. Generalizes `(III, 4.3.6)`.
anneau géométriquement unibranche	geometrically unibranch ring	0_IV.23.2.1	Unibranch and the residue field of the integral closure is a radicial extension of that of $A$ .
anneau de Krull	Krull ring	0_IV.23.2.7	In the sense of Bourbaki, Alg. comm., chap. VII, §1.
$A_{re d}$	$A_{re d}$	0_IV.23.2.1	Quotient of $A$ by its nilradical. `Spec(A)_red = Spec(A_red)`.

Chapter IV — §§1-21

Locked piecewise as each section lands. §IV.1 establishes the relative-finiteness and constructibility vocabulary.

French	English	First appearance	Note
condition de finitude relative (sur un morphisme)	relative finiteness condition (on a morphism)	IV.1.1.1	Umbrella for: quasi-compact, quasi-separated, locally of finite type, locally of finite presentation, of finite type, of finite presentation.
morphisme quasi-compact	quasi-compact morphism	IV.1.1.1	Already in `(I, 6.6.1)`; the §1 review restates and complements it.
morphisme quasi-séparé	quasi-separated morphism	IV.1.2.1	The diagonal $Δ_{X / Y}$ is quasi-compact.
morphisme localement de type fini	locally of finite type morphism	IV.1.4.2	Already in `(I, 6.6.2)`.
morphisme localement de présentation finie	locally of finite presentation morphism	IV.1.4.2
morphisme de présentation finie	morphism of finite presentation	IV.1.6.1	Locally of finite presentation + quasi-compact + quasi-separated.
morphisme de type fini	morphism of finite type	IV.1.4.2	Already in `(I, 6.3.1)`.
point maximal (d'un préschéma)	maximal point (of a prescheme)	IV.1.1.4	Generic point of an irreducible component; on $Spec (A)$ these are the minimal prime ideals of $A$ .
A-algèbre essentiellement de type fini	$A$ -algebra essentially of finite type	IV.1.3.8	A-isomorphic to $S^{- 1} C$ with $C$ of finite type and $S$ multiplicative.
A-algèbre de présentation finie	$A$ -algebra of finite presentation	IV.1.4.1	Isomorphic to $A [T_{1}, \dots, T_{n}] / a$ with $a$ of finite type.
homomorphisme d'augmentation (d'une algèbre)	augmentation homomorphism (of an algebra)	IV.1.4.3.1	The map $B \otimes_{A} B \to B$ , $x \otimes y \mapsto x y$ ; its kernel is the diagonal ideal.
partie pro-constructible	pro-constructible part	IV.1.9.4	Locally an intersection of locally constructible parts.
partie ind-constructible	ind-constructible part	IV.1.9.4	Locally a union of locally constructible parts.
topologie constructible ( $T_{co n s}$ )	constructible topology ( $T_{co n s}$ )	IV.1.9.13	Topology on $X$ whose opens (resp. closed sets) are the ind-constructible (resp. pro-constructible) parts. $X^{co n s}$ denotes the underlying set with this topology.
$f^{co n s}$	$f^{co n s}$	IV.1.9.14	The map $X^{co n s} \to Y^{co n s}$ underlying a morphism $f : X \to Y$ for the constructible topologies.
morphisme radiciel	radicial morphism	IV.1.8.7.1	Already in EGA I; the §1 lemma reformulates radiciality via the diagonal. Every radicial morphism is separated.
générisation (d'un point)	generization (of a point)	IV.1.10.1	Inherited from $(0_{I}, 2.1.2)$ ; the IV.10 results use it systematically.
application ouverte en un point	open map at a point	IV.1.10.2	$f$ is open at $x$ if it sends neighbourhoods of $x$ to neighbourhoods of $f (x)$ .
point associé (à un Module)	point associated (to a Module)	IV.3.1.1	$x \in X$ with $m_{x} \in A ss (F_{x})$ ; written $A ss (F)$ . EGA's vocabulary follows Bourbaki, Alg. comm., chap. IV.
cycle premier associé (à un Module)	associated prime cycle (of a Module)	IV.3.1.1	Closed irreducible subset whose generic point lies in $A ss (F)$ .
$A ss (F)$	$A ss (F)$	IV.3.1.1	Set of points of $X$ associated to $F$ ; preserved verbatim.
cycle premier immergé	embedded prime cycle	IV.3.1.1	Associated prime cycle strictly contained in another; non-maximal in the family of associated cycles.
cycle premier maximal / non immergé	maximal / non-embedded prime cycle	IV.3.1.1	Synonyms. For locally Noetherian $X$ , these are the irreducible components of $X$ .
section $F$ -régulière	$F$ -regular section	IV.3.1.9	Section $f$ of $O_{X}$ with $F \to H o m_{O_{X}} (f O_{X}, F)$ injective; equivalently $A ss (F) \subset X_{f}$ .
Module réduit	reduced Module	IV.3.2.2	Coherent $F$ with no embedded associated prime cycle and $l o n g (F_{x}) = 1$ at every maximal point of $S u pp (F)$ . Local notion: "reduced at $x$ ".
Module irrédondant	irredundant Module	IV.3.2.4	$A ss (F)$ has a single point. EGA's term; we keep the literal anglicization "irredundant".
Module intègre	integral Module	IV.3.2.4	Of finite type, irredundant, and reduced (so $l o n g (F_{x}) = 1$ at the unique associated point).
sous-Module primaire (dans $F$ )	primary sub-Module (in $F$ )	IV.3.2.4	$G \subset F$ with $F / G$ irredundant.
préschéma irrédondant / intègre	irredundant / integral prescheme	IV.3.2.4	$O_{X}$ irredundant (resp. integral); irredundant implies irreducible.
sous-préschéma fermé primaire	primary closed sub-prescheme	IV.3.2.4	Defined by an Ideal $J \subset O_{X}$ primary in $O_{X}$ .
décomposition irrédondante	irredundant decomposition	IV.3.2.5	Family $(F_{α})$ of irredundant quotients with locally finite $(S u pp (F_{α}))$ and injective $F \to ⨁ F_{α}$ .
décomposition irrédondante réduite	reduced irredundant decomposition	IV.3.2.5	Irredundant decomposition with pairwise distinct $A ss (F_{α})$ and no proper sub-family that is itself irredundant.
décomposition primaire (de `0` dans $F$ )	primary decomposition (of `0` in $F$ )	IV.3.2.5	Family $(G_{α})$ of sub-Modules with $F / G_{α}$ irredundant and $⋂ G_{α} = 0$ .
$F^{(x)}$	$F^{(x)}$	IV.3.2.6	Canonical irredundant quotient at $x \in A ss (F)$ ; uniquely determined as the image of $F \to i_{} (i (F))$ when $\overline{x}$ is non-embedded.
changement du corps de base	base field change	IV.4	Section title; the §4 systematic study of how invariants behave under $k ⇝ K$ .
degré de transcendance	transcendence degree	IV.4.1.1	$d e g . t r_{k} L$ ; finite for $k (x)$ when $X$ is locally of finite type over $k$ .
dimension (d'un préschéma algébrique)	dimension (of an algebraic prescheme)	IV.4.1.1	`dim(X) = sup_x deg.tr_k k(x)` over maximal points. Independent of $k$ `(5.2.2)`; equals topological dimension `(0, 14.1.2)`.
extension primaire	primary extension	IV.4.3.1	Largest separable algebraic sub-extension is $k$ itself.
extension régulière	regular extension	IV.4.3	Separable and primary; locked at §IV.4.3 indirectly via `(4.6.2)` for "geometrically integral".
extension séparable (d'un corps)	separable extension (of a field)	IV.4.3.5	$L \otimes_{k} K$ reduced for every finite radicial $K / k$ . Bourbaki Alg. VIII.
extension radicielle	radicial extension	IV.4.3.5	Purely inseparable.
extension quasi-galoisienne	quasi-Galois extension	IV.4.6.6	Normal, not necessarily separable.
extension galoisienne	Galois extension	IV.4.3.4	Normal and separable.
extension algébriquement close	algebraically closed extension	IV.4.4.4	Standard.
corps séparablement clos	separably closed field	IV.4.3.3	Algebraic closure is radicial.
corps parfait	perfect field	IV.4.3.6	Standard; $K^{p} = K$ in char $p > 0$ .
fermeture algébrique séparable (de $k$ dans $L$ )	separable algebraic closure (of $k$ in $L$ )	IV.4.3.4	Largest separable algebraic sub-extension; denoted $L_{s}$ or $k^{'}$ .
préschéma géométriquement irréductible	geometrically irreducible prescheme	IV.4.5.2	$X \otimes_{k} Ω$ irreducible for one (equivalently every) algebraically closed $Ω/ k$ . Birational criterion via $k (x)$ primary `(4.5.9)`.
préschéma géométriquement connexe	geometrically connected prescheme	IV.4.5.2	$X \otimes_{k} Ω$ connected for one (equivalently every) algebraically closed $Ω/ k$ . No birational criterion.
nombre géométrique de composantes irréductibles	geometric number of irreducible components	IV.4.5.2	$n = C a r d (I rr (X \otimes_{k} Ω))$ ; independent of $Ω/ k$ `(4.5.1)`.
nombre géométrique de composantes connexes	geometric number of connected components	IV.4.5.2	Likewise.
$k$ -irréductible / $k$ -connexe	$k$ -irreducible / $k$ -connected	IV.4.5.12	Abbreviations for "geometrically irreducible (resp. connected) relative to $k$ ". Conflicts with Weil's terminology — locked at §IV.4.5.12.
morphisme irréductible / connexe	irreducible / connected morphism	IV.4.5.5	Geometric fibres $f^{- 1} (y)$ are geometrically irreducible (resp. connected) over $k (y)$ .
$Spec (L \otimes_{k} K)$	$Spec (L \otimes_{k} K)$	IV.4.3.2	Central object of §4.3; its irreducible / connected component structure encodes geometric properties of $L / k$ .
point géométrique	geometric point	IV.4.4	Implicit throughout: a $k$ -morphism $Spec (K) \to X$ for $K$ algebraically closed.
recollement (de préschémas)	gluing (of preschemes)	IV.4.5.20	EGA's term; details deferred to Chap V. Footnote in §IV.4.5.20.
préschéma séparable sur $k$	separable prescheme over $k$	IV.4.6.2	Synonym for "geometrically reduced" or "universally reduced". $X \otimes_{k} K$ reduced for every $K / k$ .
préschéma géométriquement réduit	geometrically reduced prescheme	IV.4.6.2	Same as separable.
préschéma universellement réduit	universally reduced prescheme	IV.4.6.2	Same.
préschéma géométriquement intègre	geometrically integral prescheme	IV.4.6.2	Geometrically reduced + geometrically irreducible.
$k$ -algèbre séparable	separable $k$ -algebra	IV.4.6.2	$Spec (A)$ separable over $k$ ; $A \otimes_{k} K$ reduced for every $K / k$ . Coincides with Bourbaki's definition for $A$ finite over $k$ , not in general.
géométriquement réduit en un point	geometrically reduced at a point	IV.4.6.9	Local form of `(4.6.2)`: $O_{X^{'}, x^{'}}$ reduced for every base change and every $x^{'}$ over $x$ .
géométriquement ponctuellement intègre	geometrically pointwise integral	IV.4.6.9	Local form; $O_{X^{'}, x^{'}}$ integral after every base change.
géométriquement localement intègre	geometrically locally integral	IV.4.6.14	Same as pointwise integral for locally Noetherian preschemes.
Module géométriquement réduit / intègre	geometrically reduced / integral Module	IV.4.6.17	$F \otimes_{k} k^{'}$ reduced (resp. integral) for every finite radicial (resp. finite) $k^{'} / k$ .
multiplicité radicielle (d'un corps sur $k$ )	radicial multiplicity (of a field over $k$ )	IV.4.7.4	$μ_{r} (K ∥ k) = sup l o n g (K \otimes_{k} k_{1})$ over finite radicial $k_{1} / k$ . Power of $p$ when finite.
multiplicité séparable	separable multiplicity	IV.4.7.4	Geometric number of irreducible components, or $+ \infty$ .
multiplicité totale	total multiplicity	IV.4.7.4	Product of radicial and separable multiplicities.
exposant d'inséparabilité	inseparability exponent	IV.4.7.4	$f$ such that $μ_{r} = p^{f}$ in characteristic $p > 0$ .
exposant caractéristique	characteristic exponent	IV.4.7.3	$p$ in char $p > 0$ , else `1`.
longueur géométrique (d'un Module en un point)	geometric length (of a Module at a point)	IV.4.7.5	$λ_{x} (F) = l o n g_{O_{x}} (F_{x}) \cdot μ_{r} (k (x) ∥ k)$ .
multiplicité radicielle de $x$ pour $F$	radicial multiplicity of $x$ for $F$	IV.4.7.5	Synonym for $λ_{x} (F)$ .
multiplicité totale de $x$ pour $F$	total multiplicity of $x$ for $F$	IV.4.7.12	Product of radicial multiplicity by separable multiplicity of $k (x) / k$ .
corps de définition	field of definition	IV.4.8.4	Sub-extension $K^{'}$ of $K$ over which a given object (Module, morphism, sub-prescheme, subset) is defined.
plus petit corps de définition	smallest field of definition	IV.4.8.7	Exists for Modules, morphisms, closed sub-preschemes `(4.8.9-12)`. Of finite type over $k$ when $X$ is `(4.8.13)`. Radicial finite for $(X_{(K)})_{re d}$ `(4.8.14)`.
défini sur $K^{'}$	defined over $K^{'}$	IV.4.8.4	In image of canonical map `(4.8.2.n)`. Locked at §IV.4.8.4.
descente fidèlement plate	faithfully flat descent	IV.2.5	Section titles of §§2.5-2.7; the "passage from $Y^{'}$ to $Y$ along a faithfully flat $Y^{'} \to Y$ " leitmotif. Locked at §IV.2.5.
permanence (d'une propriété)	permanence (of a property)	IV.2.5	Used in section titles to mean "stability under faithfully flat descent". Render literally as "permanence".
propriété stable par descente	property stable under descent	IV.2.5	Running phrase; render literally.
propriété locale pour fpqc	fpqc-local property	IV.2.5	Alternative phrasing used in remarks; render literally.
point maximal de $S u pp (F)$	maximal point of $S u pp (F)$	IV.2.5.5	Generic point of an irreducible component of $S u pp (F)$ ; cf. `(IV, 1.1.4)`.
morphisme quasi-fidèlement plat	quasi-faithfully flat morphism	IV.2.3.3	Weakening of "faithfully flat"; used systematically in `(2.5.4)`, `(2.6.3)`, `(2.7.3, (i))` to obtain partial-permanence statements.
à fibres finies (morphisme)	with finite fibres (morphism)	IV.2.6.1	Set-theoretic property of $f$ : each fibre $f^{- 1} (y)$ is a finite set. Distinct from "quasi-finite morphism", which adds the finite-type requirement.
homéomorphisme universel	universal homeomorphism	IV.2.6.4	Stable under arbitrary base change; both universally open and universally closed and bijective.
universellement bicontinu	universally bicontinuous	IV.2.6.4	Property "(iii bis)"; used as a relaxed alternative to `(iii)` when $g$ is locally of finite presentation.
adhérence (d'un sous-préschéma)	closure (of a sub-prescheme)	IV.2.8	Section title and `(2.8.5)`. $\overset{ˉ}{Z}$ denotes the closure sub-prescheme.
sous-préschéma adhérence (de $Z^{'}$ dans $X$ )	closure sub-prescheme (of $Z^{'}$ in $X$ )	IV.2.8.5	Unique $Y$ -flat closed sub-prescheme $\overset{ˉ}{Z}$ of $X$ with $i^{- 1} (\overset{ˉ}{Z}) = Z^{'}$ . Underlying space is the topological closure of $Z^{'}$ in $X$ .
base régulière de dimension 1	regular base of dimension 1	IV.2.8	Locally Noetherian, regular, irreducible prescheme of dimension 1; the running hypothesis of §IV.2.8 ensuring $A_{m}$ is a DVR for every closed point.
fibre générique	generic fibre	IV.2.8.1	$X_{η} = f^{- 1} (η)$ for $η$ the generic point of an irreducible base. Used systematically in §IV.2.8.
$f$ -plat (en un point, ou tout court)	$f$ -flat (at a point, or simply)	IV.2.1.1	Equivalently $Y$ -flat: $F_{x}$ is a flat $O_{f (x)}$ -module. Carried over from $(0_{I}, 6.7.1)$ .
morphisme plat (en un point)	flat morphism (at a point)	IV.2.1.1	$f : X \to Y$ is flat at $x$ if $O_{X}$ is $f$ -flat at $x$ . Flat tout court = flat at every point.
Module plat (en un point)	flat Module (at a point)	IV.2.1.1	Case $f = 1_{X}$ of $f$ -flat: $F_{x}$ is a flat $O_{x}$ -module.
Module fidèlement plat relativement à $f$	faithfully flat Module relative to $f$	IV.2.2.4	Quasi-coherent $O_{X}$ -Module satisfying the equivalent conditions of `(2.2.1)`; local on $Y$ but not on $X$ . Synonym "faithfully flat relative to $Y$ ".
morphisme fidèlement plat	faithfully flat morphism	IV.2.2.6	$f$ -flat and surjective; equivalently $O_{X}$ is faithfully flat relative to $f$ . Carried over from $(0_{I}, 6.7.8)$ .
morphisme quasi-plat	quasi-flat morphism	IV.2.3.3	There exists a quasi-coherent $O_{X}$ -Module $F$ of finite type that is $f$ -flat with $S u pp (F) = X$ . Every flat morphism is quasi-flat.
morphisme universellement ouvert	universally open morphism	IV.2.4.2	$f_{(Y^{'})}$ is open for every base change $Y^{'} \to Y$ . Equivalent to `(III, 4.3.9)` when $Y$ is locally Noetherian and $f$ of finite type; cf. `(14.3.2)`.
topologie quotient (par $f$ )	quotient topology (by $f$ )	IV.2.3.11	The topology on $f (X) \subset Y$ induced by $Y$ is the quotient of that of $X$ by the equivalence relation defined by $f$ ; `(2.3.11)` and `(2.3.12)`.
image fermée (d'un préschéma par $f$ )	closed image (of a prescheme under $f$ )	IV.2.3.2	The sub-prescheme $Z \subset Y$ defined by the kernel of $θ : O_{Y} \to f_{*} (O_{X})$ ; $f = j \circ g$ with $j : Z \to Y$ the canonical injection. Stable under flat base change.
résolution gauche (d'un Module)	left resolution (of a Module)	IV.2.1.10	Exact complex $\dots \to L_{1} \to L_{0} \to F \to 0$ ending in degree zero. Used with the flatness hypothesis to compute $H_{i}$ of the tensor product.
morphisme de présentation finie (Module)	morphism of finite presentation (Module)	IV.2.1.11	$O_{X}$ -Module locally a cokernel of a map between finite free Modules. Same vocabulary as Bourbaki, Alg. comm., chap. I, §2.
anneau formellement équidimensionnel	formally equidimensional ring	IV.7.1.1	Noetherian local $A$ with `Â` equidimensional. Locked at §IV.7.1.1.
anneau formellement caténaire	formally catenary ring	IV.7.1.9	Noetherian local $A$ satisfying the equivalent conditions of `(7.1.8)`. Entails universally catenary `(7.1.11)`.
anneau biéquidimensionnel	biequidimensional ring	IV.7.1.4	Equidimensional and catenary; cf. `(0, 14.3.3)`.
dimension d'un préschéma algébrique	dimension of an algebraic prescheme	IV.5.2.1	For $X$ irreducible locally of finite type over a field $k$ : $dim (X) = d e g . t r_{k} k (ξ)$ ; biequidimensional.
polynôme de Hilbert (d'un Module cohérent)	Hilbert polynomial (of a coherent Module)	IV.5.3.1	For $F$ coherent on a projective scheme over an Artinian local $A$ : degree of $χ_{A} (F (n))$ equals $dim (S u pp (F))$ .
dimension du support (d'un Module)	dimension of the support (of a Module)	IV.5.1.12	`dim(ℱ) = dim(Supp(ℱ))`; equals $sup_{x} dim (F_{x})$ .
équidimensionnel en un point (Module)	equidimensional at a point (Module)	IV.5.1.12	$F_{x}$ equidimensional as $O_{x}$ -module; equivalently $O_{S, x}$ equidimensional for $S = S u pp (F)$ .
formule des dimensions	dimension formula	IV.5.5.8	Theorem `(5.5.8)`: `dim(A) + deg.tr_A B ≥ dim(B_𝔮) + deg.tr_k k'` for $A$ Noetherian local integral, $B$ integral of finite type over $A$ . Equality is catenary condition.
anneau universellement caténaire	universally catenary ring	IV.5.6.2	Noetherian $A$ satisfying the equivalent conditions of `(5.6.1)`: every $A [T_{1}, \dots, T_{n}]$ catenary, or every finite-type $A$ -algebra catenary, or the equality form of the dimension formula.
préschéma universellement caténaire	universally catenary prescheme	IV.5.6.3	Locally Noetherian $X$ with every $O_{x}$ universally catenary; equivalent to $X_{re d}$ universally catenary.
profondeur (d'un Module en un point)	depth (of a Module at a point)	IV.5.7.1	$p ro f (F_{x})$ ; carried over from $(0_{I V}, 16.4.5)$ .
coprofondeur (d'un Module)	codepth (of a Module)	IV.5.7.1	`coprof(ℱ) = sup_x coprof(ℱ_x)`. EGA's notation; "codepth" in prose.
𝒪_X-Module de Cohen-Macaulay (en un point)	Cohen-Macaulay 𝒪_X-Module (at a point)	IV.5.7.1	$co p ro f (F_{x}) = 0$ ; equivalently $F_{x}$ is a Cohen-Macaulay $O_{x}$ -module.
point de Cohen-Macaulay (d'un préschéma)	Cohen-Macaulay point (of a prescheme)	IV.5.7.1	Point $x \in X$ where $O_{x}$ is a Cohen-Macaulay ring.
préschéma de Cohen-Macaulay	Cohen-Macaulay prescheme	IV.5.7.1	$O_{X}$ is a Cohen-Macaulay $O_{X}$ -Module; equivalently $co p ro f (X) = 0$ .
propriété $(S_{k})$ (Serre)	property $(S_{k})$ (Serre)	IV.5.7.2	`prof(ℱ_x) ≥ inf(k, dim(ℱ_x))` for every $x$ . Introduced for $k = 2$ by Serre to express his normality criterion.
propriété $(S_{k})$ en un point	property $(S_{k})$ at a point	IV.5.7.2	`prof(ℱ_{x'}) ≥ inf(k, dim(ℱ_{x'}))` for every generization $x^{'}$ of $x$ .
sans cycle premier associé immergé	without embedded associated prime cycle	IV.5.7.5	Module has no embedded associated prime cycle (cf. `(IV, 3.1.1)`). Equivalent to property `(S_1)`.
préschéma régulier en codimension $\leq k$	prescheme regular in codimension $\leq k$	IV.5.8.2	Synonym for property $(R_{k})$ : set of non-regular points has codimension $> k$ .
propriété $(R_{k})$	property $(R_{k})$	IV.5.8.2	Locally Noetherian prescheme regular in codimension $\leq k$ ; $(R_{n})$ for every $n$ ⟺ regular.
critère de normalité de Serre	Serre's normality criterion	IV.5.8.6	Theorem `(5.8.6)`: $X$ normal ⟺ `(S_2)` and `(R_1)`.
anneau strictement équidimensionnel	strictly equidimensional ring	IV.7.2.1	Equidimensional and without embedded associated prime ideals; $dim (A / p) = dim (A)$ for every $p \in A ss (A)$ .
anneau strictement formellement caténaire	strictly formally catenary ring	IV.7.2.6	Satisfies the equivalent conditions of `(7.2.5)`: formally catenary and the formal fibres satisfy `(S_1)`.
$A^{(1)}$	$A^{(1)}$	IV.7.2.1	For $A$ integral Noetherian, $⋂_{p} A_{p}$ over primes $p$ of height `1`. Locked at §IV.7.2.1.
$A^{(ω)}$	$A^{(ω)}$	IV.7.2.1	For $A$ integral Noetherian local, $⋂_{p} A_{p}$ over primes $p \neq = m$ . Locked at §IV.7.2.1.
fibre formelle (d'un anneau semi-local)	formal fibre (of a semi-local ring)	IV.7.3.13	Fibre of $Spec (\hat{A}) \to Spec (A)$ ; at $p$ it is $Spec (\hat{A} \otimes_{A} k (p))$ and also the formal fibre of $A / p$ at its generic point.
𝐏-morphisme	𝐏-morphism	IV.7.3.1	Flat morphism $X \to Y$ of locally Noetherian preschemes such that $P (f^{- 1} (y), k (y))$ holds for every $y \in Y$ . Bold $P$ preserved.
𝐏-anneau	𝐏-ring	IV.7.3.13	Semi-local Noetherian ring whose formal fibres satisfy $P$ ; for general Noetherian rings, defined via localizations `(7.4.5)`.
propriété $P$ géométrique	geometric property $P$	IV.7.3.6	$P$ satisfies `(P_IV)`: stable under finite-type extension of the base field.
propriété du premier (resp. second) type	property of the first (resp. second) type	IV.7.3.10	$P$ defined from a base property $R (A)$ not involving $k$ (resp. via finite-type extensions). Locked at §IV.7.3.10.
conditions (P_I), (P_II), (P_III), (P_IV)	conditions `(P_I)`, `(P_II)`, $(P_{III})$ , `(P_IV)`	IV.7.3.4	Transitivity, descent, base-field condition, and finite-type-extension condition for $P$ . EGA letter-pair labels preserved verbatim.
conditions (R_I), (R_II), (R'_I), (R_III)	conditions `(R_I)`, `(R_II)`, $(R_{I}^{'})$ , $(R_{III})$	IV.7.3.10	Corresponding conditions on the underlying property $R (A)$ . Preserved verbatim.
corps des séries formelles restreintes	ring of restricted formal series	IV.7.4.8	$k T_{1}, \dots, T_{n}$ ⊂ $k [[T_{1}, \dots, T_{n}]]$ of series whose coefficients tend to `0`. Open problem in `(7.4.8, B)`.
condition `(R_IV)`	condition `(R_IV)`	IV.7.5.1	For $C$ a local ring at a prime of a complete Noetherian local ring and $t$ $C$ -regular in the maximal ideal: $R (C / tC)$ implies $R (C)$ . Preserved verbatim.
condition `(R_V)`	condition `(R_V)`	IV.7.5.2	For local hom. $ψ : C \to D$ with $^{a} ψ$ a $P$ -morphism: $R^{'} (C)$ implies $R (D)$ . Preserved verbatim.
produit tensoriel complété	completed tensor product	IV.7.5.5	$A \hat{\otimes}_{k} B$ ; cross-ref $(0_{I}, 7.7.5)$ . Used in `(7.5.5)`-`(7.5.7)` for the Chevalley application to complete local rings over a field.
anneau japonais	Japanese ring	IV.7.6.1	Local case treated in §7.6; cross-ref $(0_{I V}, 23.1.1)$ for the definition.
anneau universellement japonais	universally Japanese ring	IV.7.7.1	$A$ Noetherian such that every integral $A$ -algebra of finite type is a Japanese ring; cross-ref `(0, 23.1.1)`. Equivalent formulations in `(7.7.2)` (Nagata).
anneau excellent	excellent ring	IV.7.8.2	Noetherian, universally catenary, formal fibres geometrically regular, and a Nagata-type condition (iii). Term preserved verbatim.
préschéma excellent	excellent prescheme	IV.7.8.5	Locally Noetherian prescheme covered by spectra of excellent rings. Property local on affine open covers.
morphisme résolvant	resolving morphism	IV.7.9.1	For a reduced locally Noetherian $X$ : a proper birational $f : X^{'} \to X$ with $X^{'}$ regular.
résoudre les singularités (de $X$ )	resolve the singularities (of $X$ )	IV.7.9.1	Existence of a resolving morphism for $X$ ; abbreviated "resolve $X$ ".
condition de résolubilité	resolvability condition	IV.7.9.10	Used in `(7.9.10, (ii))` as the hypothesis "for every finite morphism $Y_{1} \to Y$ , one can resolve $(Y_{1})_{re d}$ ".
morphisme plat de préschémas localement noethériens	flat morphism of locally Noetherian preschemes	IV.6.0	Section title of §IV.6; the central object of study. Builds on `(IV, 2)` by adding Noetherian hypotheses.
platitude et dimension	flatness and dimension	IV.6.1	Title of §6.1; relates $dim (B)$ to $dim (A) + dim (B \otimes_{A} k)$ for flat local homomorphisms.
platitude et dimension projective	flatness and projective dimension	IV.6.2	Title of §6.2.
platitude et profondeur	flatness and depth	IV.6.3	Title of §6.3.
platitude et propriété $(S_{n})$	flatness and property $(S_{n})$	IV.6.4	Title of §6.4.
platitude et propriété $(R_{n})$	flatness and property $(R_{n})$	IV.6.5	Title of §6.5.
propriétés de transitivité	transitivity properties	IV.6.6	Title of §6.6; behavior of Cohen-Macaulay, $(S_{n})$ , regular, $(R_{n})$ , normal, reduced under composition and descent of flat morphisms.
application aux changements de base dans les préschémas algébriques	application to base changes in algebraic preschemes	IV.6.7	Title of §6.7; how the listed properties transfer under base-field extensions.
géométriquement régulier	geometrically regular	IV.6.7.6	At a point: regular at every point of $X \otimes_{k} k^{'}$ over $x$ , for every finite extension $k^{'} / k$ . Cross-ref $(0_{I V}, 19.6.5)$ .
propriété $(R_{n})$ géométrique	geometric property $(R_{n})$	IV.6.7.6	Geometric variant of $(R_{n})$ ; quantifies over finite extensions of $k$ . Alternative name: "geometrically regular in codimension $\leq n$ ".
géométriquement normal	geometrically normal	IV.6.7.6	Geometric variant of normality.
anneau géométriquement régulier / normal / réduit / $(R_{n})$	geometrically regular / normal / reduced / $(R_{n})$ ring	IV.6.7.6	$Spec (A)$ has the corresponding property. Cross-ref $(0_{I V}, 19.6.5)$ for the regular case.
morphismes réguliers, normaux, réduits, lisses	regular, normal, reduced, smooth morphisms	IV.6.8	Title of §6.8; one family of morphisms-of-codepth- $\leq n$ , Cohen-Macaulay, $(S_{n})$ , regular, $(R_{n})$ , normal, reduced, smooth.
morphisme de coprofondeur $\leq n$ (en un point)	morphism of codepth $\leq n$ (at a point)	IV.6.8.1	Flat morphism with fibre satisfying the codepth condition. Definition `(6.8.1, (i))`.
morphisme de Cohen-Macaulay (en un point)	Cohen-Macaulay morphism (at a point)	IV.6.8.1	Flat morphism with Cohen-Macaulay fibre. Definition `(6.8.1, (ii))`.
morphisme $(S_{n})$ / $(R_{n})$ (en un point)	$(S_{n})$ / $(R_{n})$ morphism (at a point)	IV.6.8.1	Flat morphism whose fibre has the corresponding property. Definition `(6.8.1, (iii)`, `(v))`.
morphisme régulier (en un point)	regular morphism (at a point)	IV.6.8.1	Flat morphism with geometrically regular fibre. Definition `(6.8.1, (iv))`.
morphisme normal / réduit (en un point)	normal / reduced morphism (at a point)	IV.6.8.1	Flat morphism with geometrically normal / reduced fibre. Definition `(6.8.1, (vi)`, `(vii))`.
morphisme lisse (en un point)	smooth morphism (at a point)	IV.6.8.1	Regular morphism that is locally of finite presentation. Equivalent characterizations in `(6.8.6)`. Open locus `(6.8.7)`.
théorème de platitude générique	generic flatness theorem	IV.6.9.1	For $Y$ locally Noetherian integral, $u : X \to Y$ of finite type, $F$ coherent: there exists a non-empty open $U \subset Y$ with $F ∣_{u^{- 1} (U)}$ flat over $U$ .
normalement plat le long de $Y$	normally flat along $Y$	IV.6.10.1	Hironaka's term. $F$ such that $g r_{J}^{∙} (F)$ is a flat $O_{Y}$ -Module. Equivalently each $j * (J^{k} F)$ is locally free.
$g r_{J}^{∙} (F)$	$g r_{J}^{∙} (F)$	IV.6.10.1	Graded $O_{Y}$ -Module $⨁_{k} j * (J^{k} F / J^{k + 1} F)$ ; $J$ the Ideal defining $Y$ .
$U_{S_{n}} (F)$ , $U_{C_{n}} (F)$	$U_{S_{n}} (F)$ , $U_{C_{n}} (F)$	IV.6.10.4	Sets of $x \in X$ where $F$ satisfies $(S_{n})$ (resp. $co p ro f (F_{x}) \leq n$ ). Cross-ref `(IV, 6.11)` for openness criteria.
$CM (F)$	$CM (F)$	IV.6.11.3	Set of $x \in X$ with $F_{x}$ Cohen-Macaulay; equals $U_{C_{0}} (F)$ .
condition (CMU)	condition (CMU)	IV.6.11.8	Every integral closed sub-prescheme $Y \subset X$ contains a non-empty open $W$ on which the induced prescheme is Cohen-Macaulay. Preserved verbatim.
Reg(X), Sing(X)	Reg(X), Sing(X)	IV.6.12.1	Regular locus and singular locus of a locally Noetherian prescheme. Locked at §IV.6.12.1 (Part B).
lieu singulier	singular locus	IV.6.12.1	$S in g (X)$ ; complement is the regular locus $R e g (X)$ .
critère de Nagata pour que $R e g (X)$ soit ouvert	Nagata's criterion for $R e g (X)$ to be open	IV.6.12.4	Theorem (6.12.4): three equivalent conditions on a Noetherian ring $A$ ensuring $R e g (X^{'})$ is open in every $X^{'}$ locally of finite type over $Spec (A)$ .
$U_{R_{n}} (X)$	$U_{R_{n}} (X)$	IV.6.12.9	Set of $x \in X$ where $X$ satisfies condition $(R_{n})$ ; open whenever $R e g (X)$ is open.
Nor(X)	Nor(X)	IV.6.13.1	Normal locus of a locally Noetherian prescheme; contains $R e g (X)$ and lies inside the open set where $O_{x}$ is integral. Locked at §IV.6.13.1.
morphisme birationnel (préschémas réduits)	birational morphism (reduced preschemes)	IV.6.15.4	Generalization of `(I, 2.2.9)` to reduced preschemes possibly with infinitely many irreducible components.
radiciel en un point	radicial at a point	IV.6.15.3	$f : X \to Y$ with $f^{- 1} (y)$ empty or a single point $x$ and $k (x) / k (y)$ radicial.
préschéma unibranche	unibranch prescheme	IV.6.15.1	$X$ whose every local ring is unibranch (cf. `(0, 23.2.1)`).
préschéma géométriquement unibranche	geometrically unibranch prescheme	IV.6.15.1	$X$ whose every local ring is geometrically unibranch (cf. `(0, 23.2.1)`); $X$ unibranch + the residue field of the integral closure is radicial over that of $O_{x}$ .
unibranche / géométriquement unibranche (en un point)	unibranch / geometrically unibranch (at a point)	IV.6.15.1	Pointwise version: defined via the local ring $O_{x}$ ; equivalent for $X$ and $X_{re d}$ .
étude des fibres (d'un morphisme plat)	study of the fibres (of a flat morphism)	IV.12.0	Section title of §IV.12; the leitmotif is the passage from "locally constructible" to "open" when one adds flatness to the morphism.
morphisme plat de présentation finie	flat morphism of finite presentation	IV.12.0.1	Central protagonist of §IV.12; the hypothesis under which constructible loci of fibre-properties become open.
propriété de dimension de la fibre	fibre-dimension property	IV.12.1.1	Umbrella for the eight assertions of `(12.1.1)` concerning dimensions of associated prime cycles, equidimensionality, $(S_{k})$ , codepth, Cohen-Macaulay, geometrically reduced / integral.
$F$ -régulier (élément, dans une fibre)	$F$ -regular (element, in a fibre)	IV.12.1.1.1	An element $t$ of the local ring at the closed point of a DVR base whose multiplication is injective on the fibre Module; arises from flatness via $(0_{I}, 6.3.4)$ .
stable par générisation (ensemble)	stable under generization (set)	IV.12.0.2	Step C) of the §IV.12 method: from constructible + stable-under-generization deduce open via $(0_{III}, 9.2.5)$ .
restriction équidimensionnelle de $f$	equidimensional restriction of $f$	IV.12.1.1.5	$Z^{'} \to Y^{'}$ from `(12.1.1.5)`: the irreducible components of every fibre have the same dimension as the generic-point fibre.
critère de platitude par fibres	fibrewise flatness criterion	IV.12.3.1	Cited as `(11.3.10)` in `(12.3.1)`; deduces flatness of a finitely presented $B$ -module from flatness on every fibre.
propriétés cohomologiques locales des fibres	local cohomological properties of the fibres	IV.12.3	Section title of §IV.12.3; covers projective dimension, Tor-dimension, $H_{n}$ of a complex of $f$ -flat Modules, and the $E x t^{n}$ / $T o r_{n}$ Corollary `(12.3.4)`.
$T or . dim_{B} (M)$	$T or . dim_{B} (M)$	IV.12.3.2	Tor-dimension of a $B$ -module: smallest $i$ with $T o r_{j}^{B} (M, N) = 0$ for $j > i$ and every $B$ -module $N$ . Equals $dim . p ro j_{B} (M)$ under the flatness hypotheses of `(12.3.2)`.
morphisme équidimensionnel	equidimensional morphism	IV.13.2.2	Locally of finite type morphism $f : X \to Y$ with $dim_{x} (f^{- 1} (f (x))) = dim (f^{- 1} (η))$ at every $x$ ; in the irreducible case, `(13.2.2)`; in general, `(13.3.2)` via the equivalent conditions of `(13.3.1)`. Not stable under arbitrary base change `(13.3.9)`.
équidimensionnel au point $x$	equidimensional at the point $x$	IV.13.2.2	Local form of equidimensionality; equivalent conditions in `(13.3.1)`. Set of points where $f$ is equidimensional is open in $X$ `(13.3.2)`.
théorème de semi-continuité de Chevalley	Chevalley's semi-continuity theorem	IV.13.1.3	$x \mapsto dim_{x} (f^{- 1} (f (x)))$ is upper semi-continuous for $f$ locally of finite type. Locked at §IV.13.1.3.
$F_{n} (X)$ (lieu des fibres de dimension $\geq n$ )	$F_{n} (X)$ (locus of fibres of dimension $\geq n$ )	IV.13.1.3	Closed subset $x \in X : dim_{x} (f^{- 1} (f (x))) \geq n$ for $f$ locally of finite type.
$Y [T_{1}, \dots, T_{e}]$	$Y [T_{1}, \dots, T_{e}]$	IV.13.3.1	Abbreviation for $Y \otimes_{Z} Z [T_{1}, \dots, T_{e}]$ ; the affine $e$ -space over $Y$ appearing in the quasi-finite criterion for equidimensionality.
morphisme ouvert (en un point, partout)	open morphism (at a point, everywhere)	IV.14.1.1	Continuous map $ψ : X \to Y$ open at $x \in X$ : image of every neighbourhood of $x$ is a neighbourhood of $ψ (x)$ . Globally open: open at every point.
universellement ouvert au point $x$	universally open at the point $x$	IV.14.3.3	Pointwise variant of universally open; for every base change $Y^{'} \to Y$ , the morphism $f_{(Y^{'})}$ is open at every point of $X^{'}$ above $x$ .
critère de Chevalley	Chevalley's criterion	IV.14.4.4	Equidimensional + (base) geometrically unibranch implies universally open. Central theorem of §IV.14.4.
critère d'ouverture	openness criterion	IV.14.0	Umbrella name for the §IV.14 results characterizing universally open morphisms via dimension formulas, equidimensionality, or quasi-sections.
morphisme génériquement ouvert	generically open morphism	IV.14.1.3	Dominant locally-of-finite-type morphism between Noetherian irreducible preschemes is open at the generic point of the source (consequence of generic flatness `(6.9.1)`).
quasi-section (d'un morphisme)	quasi-section (of a morphism)	IV.14.5.0	Section-like data on a closed subprescheme. In §IV.14.5: an irreducible part $X^{'}$ of $X$ locally closed in $X$ , containing a prescribed $x \in X$ , such that $X^{'} \to Y$ is quasi-finite and dominant. Existence theorem `(14.5.3)`.
relèvement des générisations	lifting of generizations	IV.14.1.6	The criterion `(1.10.3)` for openness at a point under "locally of finite presentation": every generization $y^{'}$ of $y = f (x)$ lifts to a generization $x^{'}$ of $x$ .
formule des dimensions (pour morphismes ouverts)	dimension formula (for open morphisms)	IV.14.2.1	`dim(𝒪_x) = dim(𝒪_y) + dim_x(f⁻¹(y))` under the hypothesis that $f$ is open at the generic points of the irreducible components of $f^{- 1} (y)$ containing $x$ (locally Noetherian setting). Cf. `(6.1.2)`.
étude des fibres (d'un morphisme universellement ouvert)	study of the fibres (of a universally open morphism)	IV.15.0	Section title of §IV.15. Companion to §IV.14: behaviour of fibre multiplicities, geometric reducedness, flatness, Cohen-Macaulay property, separable rank, and connected components under the universally open hypothesis.
multiplicité des fibres	multiplicity of the fibres	IV.15.1.1	Subsection title `(15.1)`. Concerns the geometric length $λ_{z} (F_{y})$ and the ordinary length $l o n g ((F_{y})_{z})$ at generic points of $S u pp (F_{y})$ ; the inequalities `(15.1.1.1)`-`(15.1.1.2)` compare these to the corresponding lengths on the generic fibre $F_{η}$ .
rang séparable (d'une fibre)	separable rank (of a fibre)	IV.15.5.1	$n (z) =$ geometric number of points of $f^{- 1} (z)$ for $f$ separated and quasi-finite; cf. `(I, 6.4.8)`. Lower semi-continuous at points where $f$ is universally open.
critère valuatif de platitude	valuative criterion of flatness	IV.15.2.2	`(11.8.1)`; in the proof of `(15.2.2)` it reduces flatness of a coherent Module under universally open + geometrically reduced + reduced-base hypotheses to the case of a DVR base.
factorisation de Stein	Stein factorization	IV.15.5.3	$f = g \circ f^{'}$ with $g$ finite and $f^{'}$ surjective with geometrically connected fibres; `(III, 4.3.3)`-`(III, 4.3.4)`. Used systematically in §IV.15.5 to relate geometric connected-component counts of fibres of a proper morphism to point counts of the Stein factor.
composante connexe d'une fibre le long d'une section	connected component of a fibre along a section	IV.15.6.1	$X_{y}^{\circ}$ = connected component of $f^{- 1} (y)$ containing $g (y)$ , for a $Y$ -section $g$ . Central object of §IV.15.6, including the union $X^{\circ}$ over $y \in Y$ .
$X^{\circ}$ (réunion des $X_{y}^{\circ}$ )	$X^{\circ}$ (union of the $X_{y}^{\circ}$ )	IV.15.6.4	The union $⋃_{y \in Y} X_{y}^{\circ}$ in $X$ . Open in $X$ under the hypotheses of `(15.6.4)`-`(15.6.5)` ( $f$ universally open + fibres $X_{y}^{\circ}$ geometrically reduced).
morphisme propre au point $y$	morphism proper at the point $y$	IV.15.7.1	Definition `(15.7.1)`: there exists an open neighbourhood $U$ of $y$ such that $f^{- 1} (U) \to U$ is proper. Section `(15.7)` is independent of the rest of §15.
partie propre sur $Y$ au point $y$	part proper over $Y$ at the point $y$	IV.15.7.1	$Z \subset X$ such that there exists an open neighbourhood $U$ of $y$ with $Z \cap f^{- 1} (U)$ closed in $f^{- 1} (U)$ and proper over $U$ . Cross-ref `(II, 5.4.10)`.
critère valuatif de propreté locale	valuative criterion of local properness	IV.15.7.2	Theorem `(15.7.2)` and corollaries `(15.7.4)`-`(15.7.6)`: properness of $f$ at $y$ characterised by lifting of DVR-valued points dominating $O_{y}$ . Refinement of `(II, 7.3.10)`.
morphisme submersif	submersive morphism	IV.15.7.8	$f$ surjective and the topology of $Y$ is the quotient of the topology of $X$ by the equivalence relation defined by $f$ . Universally submersive: stable under arbitrary base change.
partie submersive sur $f (Z)$	submersive part over $f (Z)$	IV.15.7.8	Subset $Z \subset X$ such that the topology of $f (Z) \subset Y$ is the quotient of that of $Z$ by $f ∣_{Z}$ . Universally submersive: stable under base change. Every $Z$ containing the image of a $Y$ -section is universally submersive over $Y$ .
résolution rationnelle (section)	rational section (of a morphism)	IV.15.7.6	`(I, 7.1.2)`; a $Y$ -section defined only on a dense open of $Y$ . Appears in `(15.7.6, d))` characterising morphisms of finite type that factor through an immersion-then-proper composition.
principe de l'extension finie	principle of finite extension	IV.9.1.1	`(9.1.1)`. Given a set $C$ of extensions of $k$ satisfying (i)-(iii), $C$ non-empty ⟺ $C$ contains a finite extension of $k$ ⟺ $Ω \in C$ for any algebraically closed $Ω$ . Standard tool for descent of properties to a finite extension.
sous-extension de type fini	sub-extension of finite type	IV.9.1.1	Used in condition (ii) of the principle of finite extension: every $K \in C$ admits a sub-extension $K^{'}$ of $K$ of finite type over $k$ with $K^{'} \in C$ .
géométriquement isomorphe (préschémas)	geometrically isomorphic (preschemes)	IV.9.1.4	Two $k$ -preschemes $X$ , $Y$ of finite type such that there exists an extension $K$ of $k$ and a $K$ -isomorphism $X_{(K)} \sim Y_{(K)}$ ; equivalent forms in `(9.1.4)`.
propriété constructible / ind-constructible (de préschémas algébriques)	constructible / ind-constructible property (of algebraic preschemes)	IV.9.2.1	Relation $P (X, F, k)$ invariant under base-field extension and (resp. weakly) constructible on Noetherian integral bases. Conventional language, not a mathematical definition `(9.2.2, (i))`. Negation, finite "or", "and" preserve constructibility `(9.2.2, (iii))`.
propriété constructible / ind-constructible d'une partie constructible	constructible / ind-constructible property of a constructible part	IV.9.2.2	Variant of `(9.2.1)` for relations $P (X, Z, k)$ where $Z \subset X$ is constructible; one must assume $Z$ constructible in $X$ , not merely fibrewise.
propriété stable par changement de base	property stable under base change	IV.9.1.4	Hypothesis on $Q (g)$ ensuring condition (i bis) of `(9.1.3)` holds for the principle of finite extension. Recurrent leitmotif in §9.
propriété constructible de morphismes	constructible property of morphisms	IV.9.3.1	Conjunction property for $f : X \to Y$ derived from a constructible property on fibres `(9.3.1)`. Used for surjective, quasi-finite, radicial, fibre-dimension `(9.3.2)`.
propriété ind-constructible de morphismes	ind-constructible property of morphisms	IV.9.3.5	Existence-of-base-extension form $Q (g)$ over `(k')` for $Q$ one of `(8.10.5, (i)-(xiv))`. Used in `(9.3.5)`.
propriété topologique (constructibilité de)	topological property (constructibility of)	IV.9.5	Section heading. Covers fibre-emptiness, density, closedness, openness, local closedness, and the dimension function on irreducible components of a constructible fibre part `(9.5.1)`-`(9.5.5)`.
propriété d'un morphisme (constructibilité de)	property of a morphism (constructibility of)	IV.9.6	Section heading. Surjective, dominant, separated, proper, radicial, finite, quasi-finite, immersions, isomorphism, monomorphism are locally constructible `(9.6.1)`; affine, quasi-affine, projective, quasi-projective, ampleness are ind-constructible `(9.6.2)`; ampleness becomes locally constructible for proper morphisms `(9.6.3)`.
polynôme géométriquement irréductible	geometrically irreducible polynomial	IV.9.7.4	$F \in k [T_{1}, \dots, T_{n}]$ non-constant such that $F^{K}$ is irreducible for every extension $K / k$ ; equivalently $Spec (k [T_{1}, \dots, T_{n}] / (F))$ is geometrically integral.
$F^{ρ}$ / $F^{B}$ (polynôme transporté)	$F^{ρ}$ / $F^{B}$ (polynomial transported)	IV.9.7.3	For a ring map $ρ : A \to B$ and $F \in A [T_{1}, \dots, T_{n}]$ , the polynomial of $B [T_{1}, \dots, T_{n}]$ with each coefficient replaced by its image under $ρ$ . EGA's literal notation preserved.
géométriquement irréductible (préschéma)	geometrically irreducible (prescheme)	IV.9.7.7	$X_{(k^{'})}$ irreducible for every (equivalently, one algebraically closed) extension $k^{'} / k$ . Property treated as constructible in `(9.7.7, (i))`.
géométriquement connexe	geometrically connected	IV.9.7.7	$X_{(k^{'})}$ connected for every (equivalently, one algebraically closed) extension $k^{'} / k$ .
géométriquement réduit	geometrically reduced	IV.9.7.7	$X_{(k^{'})}$ reduced for every (equivalently, one algebraically closed) extension $k^{'} / k$ . Synonym in case $X$ is of finite type over $k$ : separable over $k$ .
géométriquement intègre	geometrically integral	IV.9.7.7	$X_{(k^{'})}$ integral for every (equivalently, one algebraically closed) extension $k^{'} / k$ . Equivalently geometrically irreducible + geometrically reduced.
nombre géométrique de composantes irréductibles	geometric number of irreducible components	IV.9.7.8	The number of irreducible components of $X_{(Ω)}$ for $Ω$ an algebraically closed extension; invariant under further base-field extension `(4.5.2)`.
nombre géométrique de composantes connexes	geometric number of connected components	IV.9.7.8	Analogous count for connected components; same invariance property.
géométriquement unibranche (en un point)	geometrically unibranch (at a point)	IV.9.7.10	Preserved from §IV.6.15.1; `(9.7.10)` proves the locus is locally constructible when the normalization of $X_{re d}$ is finite over $X$ .
section $S$ (d'un morphisme)	$S$ -section (of a morphism)	IV.9.7.12	Morphism $g : S \to X$ over $S$ with $f \circ g = 1_{S}$ . Used in `(9.7.12)` to produce a locally constructible "connected component containing the section".
décomposition primaire au voisinage d'une fibre générique	primary decomposition near a generic fibre	IV.9.8	Section title of §IV.9.8; the family of constructibility theorems concerning $A ss (F_{s})$ , primary decompositions, dimensions, lengths, and multiplicities of irreducible components of $S u pp (F_{s})$ .
longueur géométrique (de $F$ en un point)	geometric length (of $F$ at a point)	IV.9.8.6	`(4.7.5)`; length of $F_{(k^{'})}$ at a geometric point over $x$ , for $k^{'}$ algebraically closed over $k (x)$ . Invariant for $(Z_{i})_{η}$ geometrically integral `(9.8.6)`.
multiplicité radicielle	radicial multiplicity	IV.9.8.7	`(4.7.8)`; for an irreducible component $Z$ of $S u pp (F)$ , the radicial multiplicity of $Z$ with respect to $F$ at its generic point. Invariant under base-field extension by `(4.7.9)`.
multiplicité totale (pour $F$ )	total multiplicity (for $F$ )	IV.9.8.8	`(4.7.12)`; sum-version weighting the generic points of $S u pp (F)$ by their geometric length × radicial multiplicity; locally constructible by `(9.8.8)`.
partie saturée (d'un préschéma algébrique)	saturated part (of an algebraic prescheme)	IV.9.8.9	Finite $P \subset X$ such that for every $x$ , $y \in P$ , the generic points of the irreducible components of $\overline{x} \cap \overline{y}$ lie in $P$ . Every finite $Q$ has a smallest containing saturated part, its saturation.
squelette primaire (d'un Module cohérent)	primary skeleton (of a coherent Module)	IV.9.8.9	The system $(P, Q, ω, d, m)$ with $Q = A ss (F)$ , $P$ the saturation of $Q$ , $ω$ the inclusion order on closures, $d (x) = dim \overline{x}$ , $m (x) = l o n g_{O_{X, x}} F_{x}$ on maximal elements of $Q$ .
squelette virtuel	virtual skeleton	IV.9.8.9	Abstract system $(P, Q, ω, d, m)$ modelling a primary skeleton; the obvious notion of isomorphism is defined.
type primaire (d'un Module cohérent)	primary type (of a coherent Module)	IV.9.8.9	Isomorphism class of the primary skeleton of $F$ ; depends only on $F$ up to virtual-skeleton isomorphism.
type primaire géométrique	geometric primary type	IV.9.8.9	Primary type of $F \otimes_{k} k^{'}$ for $k^{'} / k$ algebraically closed; independent of the chosen $k^{'}$ . Locally constructible by `(9.8.9.1)`.
propriétés locales des fibres	local properties of fibres	IV.9.9	Section title of §IV.9.9; constructibility of pointwise local properties ( $dim_{x}$ , $co d i m_{x}$ , $dim . p ro j$ , `coprof`, $(S_{k})$ , Cohen-Macaulay, geometric $(R_{k})$ , normal, reduced, integral, regular).
équidimensionnel (anneau local)	equidimensional (local ring)	IV.9.9.1	$O_{Z, x}$ whose irreducible components (i.e. minimal primes) have a common dimension; constructibility of the locus by `(9.9.1, (iii))`.
ponctuellement intègre (géométriquement)	pointwise integral (geometrically)	IV.9.9.4	Preserved from `(4.6.22)`; pointwise variant of geometric integrality. Constructibility of the locus by `(9.9.4, (v))`.
propriété $(R_{k})$ géométrique (en un point)	geometric property $(R_{k})$ (at a point)	IV.9.9.4	Pointwise variant of $(R_{k})$ ; at $y \in X_{s}$ , either $X_{s}$ is geometrically regular at $y$ or every irreducible component of the geometric non-regular locus through $y$ has codimension $> k$ . Locked at §IV.9.9.4.
séparable (préschéma sur $k$ )	separable (prescheme over $k$ )	IV.9.9.4	Synonym of "geometrically reduced" for an algebraic prescheme over a field, recorded explicitly in `(9.9.4, (iv))`. The condition matches the field-extension sense via `(4.6.1)` and Bourbaki Alg. VIII.
ensemble de platitude	flatness locus	IV.11.1.1	The set $U \subset X$ of points where $F$ is $f$ -flat. Open under Noetherian + locally-of-finite-type hypotheses `(11.1.1)`; open in general under locally-of-finite-presentation hypotheses `(11.3.1)`.
critère local de platitude	local criterion of flatness	IV.11.0	Section-title leitmotif of §IV.11. Refines the Noetherian local criterion $(0_{III}, 10.2.1 - 10.2.6)$ to the locally-of-finite-presentation setting via passage to projective limits.
platitude au point $x$	flatness at the point $x$	IV.11.1.1	$F_{x}$ is a flat $O_{f (x)}$ -module. The set of such $x$ is open under either Noetherian + locally-of-finite-type `(11.1.1)` or locally-of-finite-presentation hypotheses `(11.3.1)`.
platitude d'une limite projective	flatness of a projective limit	IV.11.2.6	Theorem `(11.2.6)`: under `(8.5.1)`/`(8.8.1)` finite-presentation hypotheses, $F$ is $f$ -flat at $x \in X = lim X_{λ}$ iff there exists $λ$ with $F_{λ}$ $f_{λ}$ -flat at the projection $x_{λ}$ .
théorème de Raynaud (platitude des $g r_{J}^{∙} (M)$ )	Raynaud's theorem (flatness of $g r_{J}^{∙} (M)$ )	IV.11.2.9	Theorem `(11.2.9)`: graded flatness commutes with filtered inductive limits. Strengthens `(11.2.6.1, (ii))`. Proof is by reduction to the polynomial-algebra case and a Noetherian induction `(11.2.9.5)`.
$F$ normalement plat le long de $Y$ au point $y$	$F$ normally flat along $Y$ at a point $y$	IV.11.3.4	Generalizes `(6.10.1)` to the locally-of-finite-presentation setting: $(g r_{J}^{∙} (F))_{y}$ is a flat $O_{Y, y}$ -module, where $J$ defines $Y \subset X$ . Hironaka's terminology.
ensemble de platitude normale	normal-flatness locus	IV.11.3.5	The set $W \subset Y$ where $F$ is normally flat along $Y$ ; open under the hypotheses of `(11.3.5)` ( $g r_{J}^{∙} (O_{X})$ and $g r_{J}^{∙} (F)$ both $f$ -flat). Compatible with arbitrary base change `(11.3.4, (iv))`.
suite quasi-régulière relative	relative quasi-regular sequence	IV.11.3.8	Theorem `(11.3.8)`: under finite-presentation hypotheses, the equivalence between fibrewise regularity + flatness of the quotient and absolute regularity + flatness, for sequences $(g_{i})$ of sections of $O_{X}$ with $(g_{i})_{x} \in m_{x}$ . Open locus in $Z = V (g_{1}, \dots, g_{n})$ .
critère de platitude par fibres	fibrewise flatness criterion	IV.11.3.10	Theorem `(11.3.10)`: under locally-of-finite-presentation or locally-Noetherian + coherence hypotheses, $F$ is $f$ -flat and $F_{s}$ is $f_{s}$ -flat at $x$ iff $h$ is flat at $y$ and $F$ is $g$ -flat at $x$ . Reduces to lemma `(11.3.10.2)` on the local criterion.
critère par fibres pour $B / A \to C / B$	fibrewise criterion $B / A \to C / B$	IV.11.3.10.1	Lemma `(11.3.10.1)`: refinement of $(0_{III}, 10.2.5)$ — $M$ flat over $A$ and $M \otimes_{A} k$ flat over $B \otimes_{A} k$ is equivalent to $B$ flat over $A$ and $M$ flat over $B$ , for local hom. of Noetherian local rings.
morphisme normal / réduit au point $x$	normal / reduced morphism at the point $x$	IV.11.3.13	Inherited from `(6.8.1)`; under flatness + locally-of-finite-presentation `(11.3.13, (ii))`, normality (resp. reducedness) of $Y$ at $y$ together with normality (resp. reducedness) of $f$ at $x$ implies the same for $X$ at $x$ .
géométriquement unibranche transféré	geometrically unibranch transferred	IV.11.3.14	Corollary `(11.3.14)`: under $f$ locally of finite presentation and normal at $x$ , $Y$ geometrically unibranch at $y = f (x)$ implies $X$ geometrically unibranch at $x$ .
stratification par libres-localisations	stratification by free localizations	IV.11.3.15	Proposition `(11.3.15)`: for $M$ of finite presentation flat over $A$ , there exist $f_{1}, \dots, f_{n}$ generating $A$ such that $M_{f_{i + 1}} / (\sum_{j \leq i} f_{j} M_{f_{i + 1}})$ is free over the corresponding ring. Equivalently, $Spec (A)$ is partitioned into locally-closed sets on each of which $M$ is free.
descente fpqc des conditions de finitude	fpqc descent of finiteness conditions	IV.11.3.16	Proposition `(11.3.16)` / Corollary `(11.3.17)`: faithfully flat morphisms of finite presentation descend "of finite type" and "of finite presentation" for morphisms $g : Y \to Z$ . The algebraic version `(11.3.17)` is the standard fpqc-descent statement for $A$ -algebra finite-type/finite-presentation properties.
élimination des hypothèses noethériennes (§11.3)	elimination of Noetherian hypotheses (§11.3)	IV.11.3.0	Section title of §11.3; programme of transferring the Noetherian-base flatness statements of §§11.1-11.2 to the general locally-of-finite-presentation setting via the projective-limit theorem `(11.2.6)`.

§IV.19 additions (Part B, §§19.6-19.9)

French	English	First appearance	Note
suite régulière relativement à un module filtré quotient	regular sequence relative to a quotient filtered module	IV.19.6.1	Section title `(19.6)`. Concerns the canonical homomorphisms $ψ_{N}$ , $ψ_{M}$ , $ψ_{R}$ `(19.6.2)` from graded-tensor-polynomial algebras to associated graded modules of $(N_{k}^{'})$ , $(M_{k}^{'})$ , $(R_{k}^{''})$ . Equivalences a)-h) of `(19.6.3)` under separation hypotheses.
filtration $J$ -préadique (induite par)	$J$ -preadic filtration (induced by)	IV.19.6.1	Filtration $R_{k} = R \cap N_{k}$ on a submodule $R$ induced from a filtration $(N_{k})$ on $N$ . Used jointly with $(N_{k}^{'})$ , $(R_{k}^{'})$ , $(R_{k}^{''})$ in `(19.6.1)`-`(19.6.3)`.
$g r_{∙}^{'}$ , $g r_{∙}^{''}$	$g r_{∙}^{'}$ , $g r_{∙}^{''}$	IV.19.6.1	Graded modules of the finer filtration $(N_{k}^{'}) = \sum J^{j} N_{k - j}$ and the auxiliary $(R_{k}^{''}) = \sum J^{j} R_{k - j}$ . Compared by the canonical homomorphism $γ : g r_{∙}^{''} (R) \to g r_{∙}^{'} (R)$ in `(19.6.2)`.
critère de platitude normale (de Hironaka)	(Hironaka's) normal flatness criterion	IV.19.7.1	Section title `(19.7)`. Theorem `(19.7.1)`: equivalence of " $g r_{L}^{∙} (M)$ is $(A / L)$ -flat" with the conjunction of $(A / K)$ -flatness on $g r_{J}^{∙} (M)$ and bijectivity of $ψ_{M}$ , under separation / Noetherian hypotheses. Hironaka, Resolution (1964) cited via this criterion.
$g r_{J}^{∙} (M)$ (filtration $J$ -préadique d'un module)	$g r_{J}^{∙} (M)$ ( $J$ -preadic filtration of a module)	IV.19.7.1	$⨁_{n} J^{n} M / J^{n + 1} M$ ; central object of Hironaka's normal flatness theorem.
idéalement séparé (Bourbaki, Alg. comm., III, §5)	ideally separated (Bourbaki, Alg. comm., III, §5)	IV.19.7.1	Separation condition $⋂_{k} J^{k} M = 0$ on every quotient of a submodule; sufficient for equivalence b) ⇔ c) in `(19.7.1)`. Krull-intersection type.
$A s s_{A} (M)$ (idéaux premiers associés)	$A s s_{A} (M)$ (associated prime ideals)	IV.19.7.1.2	Bourbaki Alg. comm. IV, §1. Used in `(19.7.1.2)` to reduce flatness of $g r_{J}^{∙} (M)$ to localizations at prime ideals associated to $A / L$ .
série de Poincaré (d'un module gradué)	Poincaré series (of a graded module)	IV.19.7.3	$P (T) = \sum r an k_{A / m} (g r_{m}^{n} (M)) T^{n}$ . Identity `(19.7.3.1)` $Q (T) = P (T) (1 - T)^{n}$ characterises normal flatness over a regular quotient of dimension $n$ .
propriétés de passage à la limite projective	properties of passage to projective limit	IV.19.8.0	Section title `(19.8)`. Compatibility of $F$ -regularity and (transversal) regularity of immersions with filtered projective limits of preschemes, in the flat-transition-morphism setting of `(8.5.1)`/`(8.8.1)`.
limite projective d'immersions régulières	projective limit of regular immersions	IV.19.8.1	Proposition `(19.8.1, (ii))`: an immersion $j : X \to S$ between limits is regular iff some $j_{λ} : X_{λ} \to S_{λ}$ is regular, under flat transition morphisms and either Noetherian bases or surjective transition morphisms.
limite projective d'immersions transversalement régulières	projective limit of transversally regular immersions	IV.19.8.2	Proposition `(19.8.2)`: the relative-to- $S$ version of `(19.8.1)`. Combined with `(11.3.8)` and `(19.2.4)`.
$T$ -profondeur (de $F$ en un point)	$T$ -depth (of $F$ at a point)	IV.19.9.1	`prof_{T, t}(ℱ) = inf_{z ∈ T ∩ Spec(𝒪_{X, t})} prof(ℱ_z)`. Used in `(19.9.3)`-`(19.9.7)` to relate fibrewise depth to the existence of transversally regular sequences cutting out $Y$ .
semi-continuité inférieure de la profondeur	lower semi-continuity of the depth	IV.19.9.4	Corollary `(19.9.4)`: $y \mapsto p ro f_{Y, y} (F)$ is lower semi-continuous in $Y$ for $X$ locally Noetherian and $F$ coherent. Constructible-base version `(19.9.7)` adds local constructibility.
invariance de la profondeur par morphisme plat	invariance of depth under a flat morphism	IV.19.9.5	Proposition `(19.9.5)`: $p ro f_{Y^{'}, y^{'}} (g^{*} (F)) = p ro f_{Y, y} (F)$ for $g : X^{'} \to X$ flat between locally Noetherian preschemes, with $Y^{'} = g^{- 1} (Y)$ . Consequence of $(0_{I}, 6.3.1)$ and `(2.3.4)`.
transversalement $F$ -régulière (suite, cf. relative)	transversally $F$ -regular (sequence, cf. relative)	IV.19.9.6	Already in `(19.2.1)`; in `(19.9.6, (b))` realizes the fibrewise-depth condition $p ro f_{Y_{f (y)}, y} (F_{f (y)}) \geq n$ by a transversally $F$ -regular sequence cutting out $Y$ near $y$ .
$V_{n}$ (lieu de profondeur fibre $\geq n$ )	$V_{n}$ (locus of fibre-depth $\geq n$ )	IV.19.9.6	Set of $y \in Y$ satisfying the equivalent conditions of `(19.9.6)`; open in $Y$ , retrocompact in $X$ when $Y$ is locally constructible.
suite $F$ -régulière (préfaisceau d'anneaux)	$F$ -regular sequence (sheaf of rings)	IV.19.9.0	Section title `(19.9)`. The sheaf-of-modules analog of an $M$ -regular sequence; cf. $(0_{I V}, 15.1)$ for the algebraic case. Combined with the depth machinery $p ro f_{T} (F)$ of `(5.10.1)`.
$ρ_{G} : H \to j_{} (j^{} (H))$	$ρ_{G} : H \to j_{} (j^{} (H))$	IV.19.9.8	Canonical homomorphism relative to the open immersion $j : X - Z \to X$ . Theorem `(19.9.8)`: injective (resp. bijective) under $p ro f ((F_{f (x)})_{x}) \geq 1$ (resp. $\geq 2$ ) for $x \in Z$ . Generalized to higher cohomology in `(19.9.9)`.
cohomologie locale (Chap. III, 3e partie)	local cohomology (Chap. III, 3rd part)	IV.19.9.9	Cited in `(19.9.9)` for the generalization $H^{i} (X, H) \to H^{i} (X - Z, H ∣_{X - Z})$ bijective for $i \leq k - 2$ , injective for $i = k - 1$ , under $prof_{Z} (H) \geq k$ .

Translator's policy notes

préschéma → prescheme, schéma → scheme: inherited verbatim from EGA II/III. The 1961-1967 distinction is preserved (a prescheme is not yet required to be separated). Modernizing this term would silently rewrite the theorems.
anneau japonais → Japanese ring: EGA's vocabulary, older than Nagata's "pseudo-geometric ring" or Matsumura's "Nagata ring". We keep the EGA term and add a one-line translator's note at the first occurrence in §0_IV.23.
formellement lisse pour la topologie 𝒥-préadique: render fully — "formally smooth for the $J$ -preadic topology" — never silently drop the topology qualifier. In EGA, the topology argument carries content: a discrete $J = 0$ formal smoothness collapses to ordinary "smooth" (in the sense of (4.4.x)), while the $J$ -preadic version with $J \neq = 0$ does not.
profondeur and $p ro f (M)$ : EGA's "depth" is $p ro f (M)$ ; we keep the symbol prof in formulas and use "depth" in running English prose. The notation index entry is $p ro f (M)$ .
$p$ -base → $p$ -basis: standard English form (Matsumura, Bourbaki AC); preserves the historical content.
anneau excellent: render literally as "excellent ring". EGA IV §IV.7 establishes the term; we do not modernize.
module d'imperfection → "imperfection module": EGA's term; $Υ_{B / A}$ notation preserved.
$(D_{I}), (D_{II}), (D_{III})$ differential-criterion labels: preserved verbatim. The English literature sometimes uses different letter pairs; we keep EGA's.
Henselian (capital): proper noun (after Kurt Hensel). We capitalize throughout, including derived terms (strict Henselization, Hensel's lemma, henselization).
Jacobson (capital): proper noun. "Jacobson prescheme", "Jacobson condition (J)".
Cohen (capital): proper noun (after I. S. Cohen). "Cohen ring", "Cohen-Macaulay ring".
(IV, M.N.K) vs $(0_{I V}, M . N . K)$ : EGA writes both (IV, M.N.K) (for cross-section references inside Chap IV) and $(0_{I V}, M . N . K)$ (for Chap 0_IV preliminaries). The two are distinct in the print and we keep both forms; the README documents the citation key.
§11 of Chap IV: published in Part 3 (1966) alongside §§8-10, 12-15. (Earlier internal notes claimed §11 was unpublished; that was incorrect — the 1966 fascicule prints §11 in its expected position.)

§IV.8 additions (Part A, §§8.1-8.8)

French	English	First appearance	Note
limite projective de préschémas	projective limit of preschemes	IV.8.1.1	Section title (8.2). Filtered projective system in the category of `S_0`-preschemes with affine transition morphisms.
système projectif de préschémas	projective system of preschemes	IV.8.1.1	Standard usage throughout §8.
théorie de la réduction modulo $p$	theory of reduction modulo $p$	IV.8.1.2	EGA's quotation marks preserved at first appearance.
« point de vue kroneckérien »	"Kroneckerian point of view"	IV.8.1.2	EGA's quotation marks preserved; reduction to preschemes of finite type over $Z$ .
« Géométrie algébrique absolue »	"absolute algebraic geometry"	IV.8.1.2	EGA's quotation marks preserved; preschemes of finite type over $Z$ .
partie constructible (dans la limite)	constructible part (in the limit)	IV.8.3	Section heading (8.3): "Constructible parts in a projective limit of preschemes".
partie pro-constructible / ind-constructible	pro-constructible / ind-constructible part	IV.8.3.2	Inherited from `(IV, 1.9.4)`; reused systematically in §8.3.
critères d'irréductibilité et de connexion	irreducibility and connectedness criteria	IV.8.4	Section heading (8.4).
module de présentation finie	module of finite presentation	IV.8.5	Section heading (8.5). Inherited from $(0_{I}, 5.2.5)$ ; used for the equivalence-of-categories scholium.
« fonctoriellement »	"functorially"	IV.8.5.3	EGA's quotation marks preserved in the scholium statement.
$P (X)$ / $E (X)$ / $Oc (X)$ / $Fc (X)$ / $LFc (X)$	$P (X)$ / $E (X)$ / $Oc (X)$ / $Fc (X)$ / $LFc (X)$	IV.8.3.9	Parts; constructible; constructible-open; constructible-closed; constructible-locally-closed parts of $X$ .
$Q (F)$ (quotients de présentation finie)	$Q (F)$ (quotients of finite presentation)	IV.8.5.10	Set of quotient Modules of $F$ that are of finite presentation.
$Spr (Y)$ / $Spr_{o} (Y)$ / $Spr_{f} (Y)$	$Spr (Y)$ / $Spr_{o} (Y)$ / $Spr_{f} (Y)$	IV.8.6.1	Sub-preschemes of $Y$ of finite presentation (resp. induced on open sets, resp. closed) of finite presentation.
sous-préscheme de présentation finie	sub-prescheme of finite presentation	IV.8.6	Section heading (8.6).
critère pour une limite projective d'être un préschéma réduit (resp. intègre)	criterion for a projective limit to be a reduced (resp. integral) prescheme	IV.8.7	Section heading (8.7).
morphisme de transition (entre $S_{λ}$ )	transition morphism (between $S_{λ}$ )	IV.8.2.2	Affine morphism $u_{μ λ} : S_{μ} \to S_{λ}$ arising from the Algebra homomorphism $ϕ_{μ λ}$ .
limite projective dans $C_{T}$	projective limit in $C_{T}$	IV.8.2.4	Lemma `(8.2.4)`: limits in a slice category agree with limits in the ambient category.
préschéma en groupes	prescheme in groups	IV.8.8.3	EGA's term preserved literally; cross-ref `(II, 8.3.9)`.
« compatible avec les produits fibres »	"compatible with fibre products"	IV.8.8.3	EGA's quotation marks preserved in the scholium.
« compatible avec la formation des images réciproques de sous-préschémas »	"compatible with the formation of inverse images of sub-preschemes"	IV.8.8.3	EGA's quotation marks preserved in the scholium.
noyau d'un couple de morphismes	kernel of a pair of morphisms	IV.8.8.3	Inverse image of the diagonal sub-prescheme; central to "compatibility with finite projective limits".
« compatible avec les limites projectives finies »	"compatible with finite projective limits"	IV.8.8.3	EGA's quotation marks preserved in the scholium.

§IV.8 additions (Part B, §§8.9-8.14)

French	English	First appearance	Note
élimination des hypothèses noethériennes	elimination of Noetherian hypotheses	IV.8.9	Section heading; standard EGA phrase for the Part 3 program.
théorème de platitude générique	generic flatness theorem	IV.8.9.4	EGA's quotation marks preserved at first appearance.
passage à la limite projective	projective passage to the limit	IV.8.10	Section heading. "Passage to the limit" in running prose.
propriétés de permanence	permanence properties	IV.8.10	Section heading.
lemme de Chow pour les morphismes de présentation finie	Chow's lemma for morphisms of finite presentation	IV.8.10.5.1	EGA's finite-presentation analog of `(II, 5.6.1)`.
`« Main Theorem » de Zariski`	Zariski's Main Theorem	IV.8.12	EGA italicizes the English phrase; we preserve italics with asterisks.
pseudo-fini	pseudo-finite	IV.8.12.3	EGA's term: of finite type and factoring as immersion-then-finite. Necessarily separated.
traduction en termes de pro-objets	translation in terms of pro-objects	IV.8.13	Section heading.
pro-objet (d'une catégorie)	pro-object (of a category)	IV.8.13.3	Filtered projective system viewed as an object of $P ro (C)$ ; full development deferred to chap. V.
pro-variété, pro-schéma	pro-variety, pro-scheme	IV.8.13.3	Cited in connection with Serre's local class field theory and Néron's reduction theory.
essentiellement affine (au-dessus de $S$ )	essentially affine (over $S$ )	IV.8.13.4	Structure morphism factors through an affine morphism followed by one of finite presentation.
$C$ , $C_{0}^{'}$ , $C^{'}$	$C$ , $C_{0}^{'}$ , $C^{'}$	IV.8.13.4	Sub-categories of $(S c h)_{/ S}$ and $P ro ((S c h)_{/ S}^{fp})$ introduced in `(8.13.4)`.
foncteur représenté (par un préschéma)	functor represented (by a prescheme)	IV.8.14.2	$h_{X} (T) = Hom_{S} (T, X)$ ; cited as $(0_{III}, 8.1.8)$ .
groupes pro-algébriques	pro-algebraic groups	IV.8.13.6	Serre [40]. EGA argues for quasi-compact group schemes as a conceptually simpler replacement.

pseudo-fini → "pseudo-finite": EGA-IV neologism (8.12.3). Hyphenated in English to match French.
Main Theorem: EGA prints the phrase in English with French quotation marks ("« Main Theorem »"); we preserve the English wording and render the emphasis as italics ( $* M ain T h eore m *$ ), matching standard English mathematical typography.
pro-objet, pro-variété, pro-schéma → "pro-object", "pro-variety", "pro-scheme": hyphenated forms preserved throughout. EGA defers full development to chap. V.

§IV.10 additions (Jacobson preschemes)

French	English	First appearance	Note
partie quasi-constructible	quasi-constructible subset	IV.10.1.1	Finite union of locally closed subsets of $X$ . Notation $Q c (X)$ ; coincides with "constructible" when $X$ is Noetherian.
partie localement quasi-constructible	locally quasi-constructible subset	IV.10.1.1	Quasi-constructible in some open neighbourhood of every point. Notation $L Q c (X)$ .
partie très dense	very dense subset	IV.10.1.3	$X_{0} \subset X$ satisfying the equivalent conditions of `(10.1.2)`; equivalently, the canonical injection is a quasi-homeomorphism.
quasi-homéomorphisme	quasi-homeomorphism	IV.10.2.2	Continuous $f : X_{0} \to X$ satisfying the equivalent conditions of `(10.2.1)`. Induces equivalences of sheaf categories.
quasi-isomorphisme (d'espaces annelés)	quasi-isomorphism (of ringed spaces)	IV.10.2.8	Morphism $(ψ, θ)$ with $ψ$ a quasi-homeomorphism and $θ ♯$ an isomorphism of sheaves of rings; determines $(X, O_{X})$ from $(Y, O_{Y})$ up to isomorphism.
espace de Jacobson	Jacobson space	IV.10.3.1	Topological space whose set of closed points is very dense; equivalently every closed subset is the closure of its closed points.
préschéma de Jacobson	Jacobson prescheme	IV.10.4.1	Prescheme whose underlying topological space is a Jacobson space. Capitalize "Jacobson" throughout.
anneau de Jacobson	Jacobson ring	IV.10.4.1	Ring $A$ such that $Spec (A)$ is a Jacobson prescheme; equivalent to Bourbaki's definition (every prime is an intersection of maximal ideals).
profondeur rectifiée	rectified depth	IV.10.8.1	`prof*_x(ℱ) = prof(ℱ_x) + dim(‾{x})`. Of local character only on Jacobson preschemes satisfying conditions 2°, 3° of `(10.6.1)`.
profondeur rectifiée le long de $Z$	rectified depth along $Z$	IV.10.8.1	`prof_Z(ℱ) = inf_{x ∈ Z} prof_x(ℱ)`; writes $p ro f * (F)$ when $Z = X$ .
spectre maximal	maximal spectrum	IV.10.9.3	$Sp m (A)$ : the ringed-space of maximal ideals of a Jacobson ring $A$ . Functor $S : C \to C^{'}$ on Jacobson preschemes.
partie ouverte ultra-affine	ultra-affine open subset	IV.10.9.5	Open subset of a ringed space whose induced ringed space is isomorphic to a $Sp m (A)$ for $A$ a Jacobson ring.
ultrapréschéma	ultra-prescheme	IV.10.9.5	Ringed space with a cover by ultra-affine open sets. The category $C_{0}^{''}$ of ultra-preschemes is equivalent to the category $C$ of Jacobson preschemes via $S$ `(10.9.6)`.
morphisme d'ultrapréschémas	morphism of ultra-preschemes	IV.10.9.5	Morphism of ringed spaces in local rings satisfying the local finite-type condition on ultra-affine charts.
espace préalgébrique sur $k$	pre-algebraic space over $k$	IV.10.10.2	$Sp m (k)$ -ultra-prescheme over an algebraically closed field; equivalently $S (X)$ for $X$ locally of finite type over $k$ .
espace algébrique de Serre	Serre algebraic space	IV.10.10.2	Pre-algebraic space $S (X)$ with $X$ a scheme; equivalently the image of the diagonal $Δ_{X^{'}}$ is closed in $X^{'} \times_{k} X^{'}$ .
espace $(k, K)$ -préalgébrique	$(k, K)$ -pre-algebraic space	IV.10.10.5	Variant for $k$ not algebraically closed, using a fixed algebraically closed extension $K$ of $k$ ; "signalled only to reject it".
faisceau $A (X^{'})$ (germes d'applications dans $k$ )	sheaf $A (X^{'})$ (germs of maps to $k$ )	IV.10.10.3	Sheaf of germs of maps $X^{'} \to k$ ; receives the canonical homomorphism $h : O_{X^{'}} \to A (X^{'})$ , injective iff $X^{'}$ is reduced.

$p a r t i e t r \overset{e}{ˋ} s d e n se$ → "very dense subset": render literally. The qualifier "very" carries content ((10.1.2) requires $Z \cap X_{0} \neq = \emptyset$ for every locally closed $Z$ , strictly stronger than density).
espace de Jacobson, anneau de Jacobson, $p r \overset{e}{ˊ} sc h \overset{e}{ˊ} ma d e J a co b so n$ : capitalize "Jacobson" throughout (proper noun after Nathan Jacobson).
$p ro f o n d e u rrec t i f i \overset{e}{ˊ} e$ → "rectified depth", written $p ro f *_{x} (F)$ in formulas (asterisk = "rectified"). The plain $p ro f (F_{x})$ from §0_IV.16 is the "depth"; the rectified version adds $dim (\overline{x})$ .
ultrapréschéma → "ultra-prescheme": hyphenated; matches the EGA-IV convention of hyphenating compounds with Greek prefixes ("pro-", "ultra-", etc.).
$es p a ce a l g \overset{e}{ˊ} b r i q u e d e S erre$ → "Serre algebraic space": render literally with capital S. EGA-IV defines these via Spm; they are distinct from algebraic spaces in the sense of M. Artin (later development).
$Sp m (A)$ notation preserved verbatim. The notation index entry is $Sp m (A)$ .
§§10.9 and 10.10 caveat: EGA flags both numbers as not used in the sequel ("Les résultats de ce numéro ne seront pas utilisés par la suite"). We italicize this caveat at the top of §10.9; for §10.10 the disclaimer is folded into the prose.

§11.4-11.10 — flatness descent, valuative criterion, separating and schematically dominant families

French	English	First appearance	Note
descente de la platitude	descent of flatness	IV.11.4	Section heading for $n^{\circ} 11.4 - 11.6$ ; "descent" is the standard term for the going-down direction of a flatness check.
cas d'un préschéma de base artinien	artinian base case	IV.11.4	Heading qualifier; render as adjective + "base case" in English headings.
cas d'un préschéma de base unibranche	case of a unibranch base prescheme	IV.11.6	Heading qualifier for `(11.6.1)`–`(11.6.2)`.
anneau idéalement séparé	ideally separated ring/module	IV.11.4.7	Inherits the $(0_{III}, 10.2.1)$ notion: $M$ is $J$ -ideally separated if $a \otimes_{A} \hat{M} \to \hat{M}$ is injective for every ideal $a$ .
puissance symbolique $n$ -ième	$n$ -th symbolic power	IV.11.4.7	$p^{(n)}$ : kernel of $A \to A_{p} / p^{n} A_{p}$ .
extension primaire (d'un corps)	primary extension (of a field)	IV.11.4.11	Extension $k (x) / k$ with $k$ separably closed in $k (x)$ ; cited from `(4.3.1)`.
anneau de Zariski	Zariski ring	IV.11.5.2	Inherited from Bourbaki/EGA III; ring complete for a topology defined by an ideal contained in the Jacobson radical.
critère valuatif de platitude	valuative criterion of flatness	IV.11.8	Section heading and theorem name; "valuative" is the standard English adjectival form of "valuation".
famille séparante (d'homomorphismes)	separating family (of homomorphisms)	IV.11.9.1	Family $(u_{λ} : F \to (f_{λ})_{*} (G_{λ}))$ with intersection of kernels null; section-local notion.
famille universellement séparante (relativement à $S$ )	universally separating family (relative to $S$ )	IV.11.9.14	Separating after every base change $S^{'} \to S$ .
homomorphisme universellement injectif	universally injective homomorphism	IV.11.9.14	Single-element case of a universally separating family.
famille schématiquement dominante	schematically dominant family	IV.11.10.2	Equivalent conditions in `(11.10.1)`; generalises "dominant morphism" to families.
sous-préschéma schématiquement dense	schematically dense subprescheme	IV.11.10.2	Special case when the $f_{λ}$ are canonical immersions.
famille universellement schématiquement dominante (relativement à $S$ )	universally schematically dominant family (relative to $S$ )	IV.11.10.8	Schematically dominant after every base change $S^{'} \to S$ .
sous-préschéma universellement schématiquement dense	universally schematically dense subprescheme	IV.11.10.8	The immersion-family special case.
image fermée (d'un morphisme)	closed image (of a morphism)	IV.11.10.3	Cited from `(I, 9.5.3)`; the schematic closed image of $f : Z \to X$ when $f_{*} (O_{Z})$ is quasi-coherent.

descente de la platitude → "descent of flatness": deliberately translated literally and not by the more idiomatic "flat descent" — the section name names the direction of the inference (from $F^{'}$ flat over $Y^{'}$ to $F$ flat over $Y$ ), not the descent-theory framework. We reserve "flat descent" for §IV.2 / SGA 1 vocabulary.
$ann e a u i d \overset{e}{ˊ} a l e m e n t s \overset{e}{ˊ} p a r \overset{e}{ˊ}$ → "ideally separated": inherited from $(0_{III}, 10.2.1)$ ; we do not render "ideal-adically separated" because the EGA notion is strictly stronger.
$f ami ll es \overset{e}{ˊ} p a r an t e$ vs intersection of kernels being null: the two coincide for finite families but diverge for infinite ones (cf. (11.9.4) and the discrete-valuation-ring counter-example). The word "separating" is preserved rather than translated as "faithful" or "jointly injective".
$sc h \overset{e}{ˊ} ma t i q u e m e n t d o minan t$ → "schematically dominant": EGA's term distinguishes the schematic from the topological notion ( $Z \to X$ with dense image vs $Z \to X$ with $I = 0$ cutting out the schematic image). We never abbreviate to "dominant" alone; the schematic qualifier carries content.
§11 packaging. EGA IV Part 3 prints §§8, 9, 10, 12, 13, 14, 15, skipping §11; this file collates the announced §11 material (11.4–11.10) from the surviving manuscript (OCR file 23-c4-s10-preschemas-jacobson.md). The numbering IV.11.N.M follows the 1964 sommaire; §11 Part A (11.1–11.3) lives in the companion file 23a-ch4-11-flatness-loci-and-descent.part-a.md. The conventions note §13 stating "there is no §IV.11 file" remains true for the 1966 printed part; the present file translates the unprinted manuscript.

§IV.19 additions (Part A, §§19.1-19.5)

French	English	First appearance	Note
immersion régulière	regular immersion	IV.19.1	Inherited from `(16.9.2)`; cross-section vocabulary for §19. Codimension is well-defined and equals the transversal codimension `(19.1.4)`.
immersion quasi-régulière	quasi-regular immersion	IV.19.1.5	Inherited from `(16.9.4)`; weaker than regular immersion in general but coincident under locally Noetherian or finite-presentation hypotheses.
codimension transversale ( $Y$ dans $X$ au point $y$ )	transversal codimension (of $Y$ in $X$ at the point $y$ )	IV.19.1.3	Rank of $(I / I^{2})_{y}$ as a free $(O_{Y, y})$ -module; written $co d i m_{y}^{t} (Y, X)$ . Equal to the codimension $co d i m_{y} (Y, X)$ for $Y$ regularly immersed `(19.1.4)`.
codimension (d'une immersion régulière)	codimension (of a regular immersion)	IV.19.1.4	For $Y$ regularly immersed in $X$ , "codimension" replaces "transversal codimension" by virtue of `(19.1.4)`, even when $X$ is not locally Noetherian.
suite transversalement $F$ -régulière (relativement à $S$ )	transversally $F$ -regular sequence (relatively to $S$ )	IV.19.2.1	Sequence $(f_{i})$ of sections of $O_{X}$ that is $F$ -regular and such that each $F / (\sum_{j ⩽ i} f_{j} F)$ is $S$ -flat. Relative analog of $M$ -regularity over a base.
Idéal transversalement régulier (relativement à $S$ )	transversally regular Ideal (relatively to $S$ )	IV.19.2.1	Locally generated by a transversally $O_{X}$ -regular sequence (relative to the structure morphism $X \to S$ ). Definition matches `(19.2.1)`.
immersion transversalement régulière (relativement à $S$ )	transversally regular immersion (relatively to $S$ )	IV.19.2.2	$S$ -immersion $Y \to X$ whose defining Ideal is locally transversally regular relative to $S$ . Equivalent characterizations in `(19.2.4)`.
immersion transversalement régulière au point $y$	transversally regular immersion at the point $y$	IV.19.2.2	Pointwise version; the locus of such points is open in $Y$ .
anneau d'intersection complète (absolue)	(absolute) complete intersection ring	IV.19.3.1	Noetherian local ring $A$ whose completion `Â` is the quotient of a complete regular Noetherian local ring by a regular ideal. Every regular local ring is a complete intersection.
préschéma intersection complète au point	prescheme that is a complete intersection at the point	IV.19.3.1	Locally Noetherian $X$ at $x$ with $O_{X, x}$ a complete intersection ring.
intersection complète relative à $S$ (au point)	relative complete intersection relative to $S$ (at the point)	IV.19.3.6	For $f : X \to S$ flat, locally of finite presentation: the fibre $f^{- 1} (f (x))$ is an absolute complete intersection at $x$ . Equivalent characterizations in `(19.3.7)`.
morphisme d'intersection complète	complete intersection morphism	IV.19.3.6	$f : X \to S$ flat, locally of finite presentation, such that $X$ is a relative complete intersection at every point. Open immersions are complete intersection morphisms.
homomorphisme régulier (au point $x$ )	regular homomorphism (at the point $x$ )	IV.19.4.11	EGA's terminology for $u : E \to O_{X}$ whose image-defined sub-prescheme $Y$ is regularly immersed with codimension equal to $r g_{O_{x}} (E_{x})$ . Forward reference: chap. V.
préschéma éclaté (le long de $Y$ )	blow-up prescheme (along $Y$ )	IV.19.4.1	Inherited from `(II, 8.1.3)`; $X^{'} = Proj (\oplus I^{n})$ . Used as test case in §19.4 for regularity / smoothness criteria.
critère de régularité (pour un préschéma éclaté)	regularity criterion (for a blow-up prescheme)	IV.19.4.4	Proposition `(19.4.4)`: under suitable hypotheses, $Y$ regular at $x$ iff $Y^{'}$ regular at $x^{'}$ ; in that case $X^{'}$ regular at $x^{'}$ .
critère de lissité (pour un préschéma éclaté)	smoothness criterion (for a blow-up prescheme)	IV.19.4.8	Proposition `(19.4.8)`: analogous statement for smoothness over a base $S$ .
morphisme de Cohen-Macaulay (au point)	Cohen-Macaulay morphism (at the point)	IV.19.2.9	Inherited from `(6.8.1)`. Used in `(19.2.9)` to construct flat quasi-sections through transversally regular immersions.
quasi-section (plate, étale)	(flat, étale) quasi-section	IV.19.2.9	Inherited from `(17.16.1)` and `(17.16.3)`; `(19.2.9)` strengthens the existence by adding transversal regularity.
$g r_{I}^{∙} (O_{X})$ -plate (Algèbre)	$g r_{I}^{∙} (O_{X})$ -flat (Algebra)	IV.19.4.6	Condition on the associated graded Algebra of $(O_{X}, I)$ ; equivalent to $S$ -flatness of every infinitesimal neighbourhood `(19.4.6, (i))`.
voisinage infinitésimal $n$ -ième de $Y$ dans $X$	$n$ -th infinitesimal neighbourhood of $Y$ in $X$	IV.19.4.6	Inherited from `(16.1.2)`; $Y_{n}$ is the sub-prescheme defined by $I^{n + 1}$ .
critère de $M$ -régularité	$M$ -regularity criterion	IV.19.5	Section heading; theorems `(19.5.1)`, `(19.5.3)`, `(19.5.5)` completing `(0, 15.1)`.
filtration $J$ -préadique sur un quotient / sous-module	$J$ -preadic filtration on a quotient / sub-module	IV.19.5.3	Separation hypothesis on the induced filtration enters as a sufficient condition for the equivalence of $M$ -regularity and homological vanishing $H_{1} (f, M) = 0$ .
seconde filtration $(M_{k}^{'})$ (sur $M$ filtré)	second filtration $(M_{k}^{'})$ (on filtered $M$ )	IV.19.5.4	$M_{k}^{'} = \sum_{i + j = k} J^{i} M_{j}$ . Source of the homomorphism $ψ_{M}$ of `(19.5.4)`.
suite $g r_{∙} (M)$ -régulière	$g r_{∙} (M)$ -regular sequence	IV.19.5.5	The $M$ -regularity criterion applied to the associated graded of a filtered module; theorem `(19.5.5)` due in part to Deligne.

$imm ers i o n r \overset{e}{ˊ} gu l i \overset{e}{ˋ} re$ / $imm ers i o n q u a s i - r \overset{e}{ˊ} gu l i \overset{e}{ˋ} re$ → "regular immersion" / "quasi-regular immersion": render literally. Under locally Noetherian or finite-presentation hypotheses the two notions coincide; §19.1 reinforces this. We never abbreviate "regular immersion of codimension $n$ " to "codimension- $n$ immersion": EGA's adjective placement is preserved.
codimension transversale vs codimension: EGA writes $co d i m^{t}$ for transversal codimension at a point and proves (19.1.4) that the two agree for regularly immersed sub-preschemes in the locally Noetherian case. After (19.1.4) we render the unqualified word "codimension".
$t r an s v ers a l e m e n t r \overset{e}{ˊ} gu l i \overset{e}{ˋ} re$ ( $re l a t i v e m e n t \overset{a}{ˋ} S$ ): the relativity to $S$ is load-bearing — passing to a flat base extension is automatic (19.2.3), but the choice of $S$ controls what flatness means. Always include the relative qualifier on first appearance in any paragraph.
$ann e a u d^{'} in t ersec t i o n co m pl \overset{e}{ˋ} t e$ (absolu) vs $in t ersec t i o n co m pl \overset{e}{ˋ} t ere l a t i v e$ : the absolute notion is local on $X$ and demands a regular-completion presentation (19.3.1); the relative notion is fibrewise + flatness (19.3.6). The "(absolue)" parenthetical is preserved on the first occurrence in each numbered block to keep the distinction visible.
$h o m o m or p hi s m er \overset{e}{ˊ} gu l i er$ ( $u : E \to O_{X}$ régulier au point $x$ ) → "regular homomorphism": EGA's terminology, with an explicit forward reference to chap. V. We preserve the EGA quotation marks ("régularité" → "regularity") on first appearance in (19.4.11, (iii)).
$p r \overset{e}{ˊ} sc h \overset{e}{ˊ} ma \overset{e}{ˊ} c l a t \overset{e}{ˊ}$ / $f ai s an t \overset{e}{ˊ} c l a t er Y$ → "blow-up prescheme" / "blowing up $Y$ ": inherited from EGA II vocabulary; reinforced here. We never render "éclater" as "explode".
Notation $g r_{I}^{∙} (O_{X})$ : the source uses $g r_{I}^{∙}$ (with $I$ either an ideal or an Ideal sheaf, by context). We keep the bullet $∙$ for the running graded index, matching the EGA III conventions on graded objects.
(II, 8.1.3), (II, 8.1.7), (II, 8.1.8): standard EGA II references to the blow-up construction; preserved verbatim in citations.
§19.5.5 footnote (Deligne): the EGA footnote credits P. Deligne with the proof of the implication b) ⟹ c) in (19.5.5) without separation hypothesis on the $g r_{k} (M)$ . We preserve the credit as a labeled footnote $[^{19} .5.5 - d e l i g n e]$ .
§19 packaging. §19 is split into two translated files. Part A (32-ch4-19-regular-immersions.part-a.md) covers §§19.1-19.5 (regular and transversally regular immersions, relative complete intersections in the flat case, blow-up regularity / smoothness criteria, $M$ -regularity criteria). Part B (forthcoming) covers §§19.6-19.10 (regular sequences relative to a filtered quotient module, normal flatness, applications to deformation).

§IV.20 additions (meromorphic functions; pseudo-morphisms)

French	English	First appearance	Note
faisceau d'anneaux de fractions de $O_{X}$ à dénominateurs dans $S$	sheaf of rings of fractions of $O_{X}$ with denominators in $S$	IV.20.1.1	$O_{X} [S^{- 1}]$ ; flat $O_{X}$ -Module. Stalk equal to $O_{x} [S_{x}^{- 1}]$ .
faisceau de modules de fractions de $F$ à dénominateurs dans $S$	sheaf of modules of fractions of $F$ with denominators in $S$	IV.20.1.2	$F [S^{- 1}] = F \otimes_{O_{X}} O_{X} [S^{- 1}]$ .
élément régulier (d'un anneau)	regular element (of a ring)	IV.20.1.3	Not a zero-divisor. $S (O_{X})$ denotes the subsheaf of germs of regular elements; the regularity is fibre-by-fibre.
faisceau des germes de fonctions méromorphes (sur $X$ )	sheaf of germs of meromorphic functions (on $X$ )	IV.20.1.3	$M_{X} = O_{X} [S (O_{X})^{- 1}]$ . Sections over $X$ are the meromorphic functions $M (X)$ .
fonction méromorphe	meromorphic function	IV.20.1.3	A section of $M_{X}$ over $X$ . Forms the ring $M (X)$ .
faisceau des germes de sections méromorphes (de $F$ )	sheaf of germs of meromorphic sections (of $F$ )	IV.20.1.3	$M_{X} (F) = F \otimes_{O_{X}} M_{X} = F [S^{- 1}]$ . Sections over $X$ form $M (X, F)$ .
section méromorphe (d'un $O_{X}$ -Module)	meromorphic section (of an $O_{X}$ -Module)	IV.20.1.3	Element of $M (X, F) = Γ (X, M_{X} (F))$ .
domaine de définition (d'une fonction méromorphe)	domain of definition (of a meromorphic function)	IV.20.1.4	$d o m (ϕ)$ : largest open on which $ϕ \in M (X)$ is a section of $O_{X}$ .
domaine de définition (d'une section méromorphe)	domain of definition (of a meromorphic section)	IV.20.1.7	$d o m (u)$ for $u \in Γ (X, M_{X} (F))$ .
strictement sans torsion ( $O_{X}$ -Module)	strictly torsion-free ( $O_{X}$ -Module)	IV.20.1.5	$1_{F} \otimes i : F \to M_{X} (F)$ is injective; equivalently, every regular section acts injectively on $F$ .
fonction méromorphe régulière	regular meromorphic function	IV.20.1.8	Section of $M_{X}^{\times}$ . We deviate from authors who reserve "regular" for sections of $O_{X}$ .
section méromorphe régulière (d'un Module inversible)	regular meromorphic section (of an invertible Module)	IV.20.1.8	Section of $(M_{X} (L))^{\times}$ . Local condition under any trivialisation $L ∥ U ≅ O_{U}$ .
image réciproque d'une fonction méromorphe par $f$	inverse image of a meromorphic function under $f$	IV.20.1.11	$ϕ \mapsto ϕ \circ f$ , defined on sections of $M_{f}$ ; $M_{f} = O_{X} [S_{f}^{- 1}]$ is a subsheaf of $M_{X}$ adapted to $f : X^{'} \to X$ .
pseudo-morphisme	pseudo-morphism	IV.20.2.1	Equivalence class of morphisms $U \to Y$ for $U$ schematically dense in $X$ ; equivalent if they coincide on a common schematically dense open.
application rationnelle stricte	strict rational map	IV.20.2.1	Synonym for pseudo-morphism; the "strict" qualifier distinguishes the schematic notion from `(I, 7.1.2)`.
pseudo- $S$ -morphisme	pseudo- $S$ -morphism	IV.20.2.1	Relative variant for $S$ -preschemes. $P s . h o m_{S} (X, Y)$ denotes the set; $P s . h o m_{S} (X, Y)$ the associated sheaf.
domaine de définition (d'un pseudo-morphisme)	domain of definition (of a pseudo-morphism)	IV.20.2.3	$d o m_{S} (ω)$ : open of $x$ where $ω$ is locally representable by an $S$ -morphism. Largest schematically dense representative when $Y / S$ is separated `(20.2.4)`.
pseudo-fonction (sur $X$ )	pseudo-function (on $X$ )	IV.20.2.8	Pseudo-morphism of $X$ into $Spec (Z [T])$ ; equivalently, class of sections of $O_{X}$ over schematically dense opens.
$M_{X}^{'}$ (faisceau des pseudo-fonctions)	$M_{X}^{'}$ (sheaf of pseudo-functions)	IV.20.2.8	Associated sheaf of $U \mapsto P s . h o m (U, Spec (Z [T]))$ ; an $O_{X}$ -Algebra. $M^{'} (X) = Γ (X, M_{X}^{'})$ .
Idéal des dénominateurs (d'une section méromorphe)	Ideal of denominators (of a meromorphic section)	IV.20.2.14	Annihilator $J$ of the image `ū` of $u$ in $M_{X} (F) / F$ ; quasi-coherent. Defines $X - d o m (u)$ as a closed sub-prescheme.
pseudo- $S$ -morphisme composé (de $ω$ et $f$ )	composed pseudo- $S$ -morphism (of $ω$ and $f$ )	IV.20.3.2	$ω \circ f$ : class of $u \circ (f ∥ V)$ when $V = f^{- 1} (d o m_{S} (ω))$ is schematically dense in $X^{'}$ .
graphe (d'un pseudo- $S$ -morphisme)	graph (of a pseudo- $S$ -morphism)	IV.20.4.1	$Γ_{ω}$ : closure of the graph of any representative in $X \times_{S} Y$ . Defined for $Y / S$ separated and $Γ_{u}$ admitting a closure.
fonction multiforme	multivalued function	IV.20.4.2	Synonym used by some authors for the set-valued map $ω : X \to P (Y)$ deduced from a pseudo- $S$ -morphism.
critère valuatif (pour qu'une application rationnelle soit définie en un point)	valuative criterion (for a rational map to be defined at a point)	IV.20.4.6	Condition (P): every morphism $Spec (V) \to X$ from a discrete valuation ring sending the generic point into $U$ and the closed point to $x$ extends compatibly with the given $h_{x}$ .
pseudo-morphisme de $X$ dans $Y$ relativement à $S$	pseudo-morphism of $X$ into $Y$ relative to $S$	IV.20.5.1	Equivalence class of $S$ -morphisms $U \to Y$ with $U$ universally schematically dense in $X$ relative to $S$ . Denoted $P s . h o m_{X / S} (X, Y)$ .
pseudo-fonction sur $X$ relative à $S$	pseudo-function on $X$ relative to $S$	IV.20.5.4	$ϕ \in M^{'} (X / S)$ ; $d o m (ϕ)$ universally schematically dense relative to $S$ . $M_{X / S}^{'}$ is the corresponding subsheaf of $M_{X}^{'}$ .
image réciproque (par changement de base) d'un pseudo-morphisme relatif	inverse image (under base change) of a relative pseudo-morphism	IV.20.5.8	$ω_{(S^{'})}$ : pseudo-morphism of $X_{(S^{'})}$ into $Y_{(S^{'})}$ relative to $S^{'}$ deduced from $ω : X ⇢ Y$ over $S$ by $S^{'} \to S$ .
fonction méromorphe sur $X$ relative à $S$	meromorphic function on $X$ relative to $S$	IV.20.6.1	Section of $M_{X / S} = O_{X} [S_{X / S}^{- 1}]$ , where $S_{X / S} (U)$ is the set of sections regular on every fibre. $M (X / S) = Γ (X, M_{X / S})$ .
sans torsion relativement à $S$ ( $O_{X}$ -Module)	torsion-free relative to $S$ ( $O_{X}$ -Module)	IV.20.6.2	$F \to M_{X / S} (F)$ injective; weaker than strictly torsion-free `(20.1.5)`.
fonction méromorphe régulière relative à $S$	regular meromorphic function relative to $S$	IV.20.6.5	Invertible in $M (X / S)$ ; equivalently, $ϕ_{s}$ regular on every fibre $X_{s}$ . Strictly stronger than regularity in $M (X)$ .

$M_{X}$ vs $M_{X}^{'}$ vs $R (X)$ . EGA-IV maintains three distinct sheaves in §20: $M_{X}$ (germs of meromorphic functions, via fractions with regular denominators); $M_{X}^{'}$ (germs of pseudo-functions, via classes of sections over schematically dense opens); $R (X)$ (germs of rational functions, via classes over topologically dense opens, defined in (I, 7.3.2)). The canonical maps $M_{X} \to M_{X}^{'} \to R (X)$ are injective in special cases only; (20.2.11) and (20.2.13, (ii)-(iii)) record when each collapses.
$M_{X}^{\times}$ notation. We render the multiplicative-group sheaf as $M_{X}^{\times}$ (matching $(0_{I}, 5.4.7)$ for $O_{X}^{\times}$ ). EGA's text uses $M_{X}^{*}$ typographically; we keep $^{\times}$ as the canonical Unicode rendering for the units sheaf.
Ps.hom(X, Y) and $P s . h o m (X, Y)$ . Roman-italic Ps.hom and $P s . h o m_{S}$ for the sets; calligraphic $P s . h o m$ and $P s . h o m_{S}$ for the sheaves. Same convention for $P s . h o m_{X / S}$ and $P s . h o m_{X / S}$ in §20.5.
pseudo-morphisme vs application rationnelle. §20.2 introduces "pseudo-morphism" precisely to handle non-reduced preschemes; on reduced preschemes the two notions coincide (20.2.7). We keep "pseudo-morphism" wherever EGA does; we do not collapse to "rational map" even when the prescheme is reduced.
$sc h \overset{e}{ˊ} ma t i q u e m e n t d e n se$ vs universellement schématiquement dense relativement à S. Inherited from §11.10. The relative notion is what makes §20.5 work; the absolute notion is what makes §20.2 work. Both are translated literally with the "schematically" qualifier preserved.
Main theorem (Zariski). EGA-IV (20.4.4) repeats the formulation already invoked in §IV.8.12. We italicize the English phrase ( $* M ain t h eore m *$ ) at first appearance in this section, matching the earlier convention.
§20.5-20.6 caveat. EGA flags §§20.5, 20.6 (and §21.15) as relative variants which "the reader will find it advantageous to omit on a first reading". We carry that note verbatim in the §20.0 introduction.

§IV.18 additions (Part C, §§18.10-18.12 — étale preschemes over geometrically unibranch / normal preschemes; complete Noetherian local algebras over a field; étale localization for quasi-finite morphisms)

French	English	First appearance	Note
préschéma géométriquement unibranche	geometrically unibranch prescheme	IV.18.10.1	Inherited from $(0_{III}, 23.2)$ ; criterion for étaleness `(18.10.1)` uses geometric unibranchness at the image point.
revêtement étale fini	finite étale cover	IV.18.10.9	An étale and finite morphism $X \to Y$ ; equivalently, for $Y$ normal integral, $X$ is the integral closure of $Y$ in a finite separable extension.
algèbre non ramifiée sur un préschéma normal intègre	algebra unramified over a normal integral prescheme	IV.18.10.10	Finite-rank $K$ -algebra $L$ (separable) whose integral closure $Y^{'}$ of $Y$ in $L$ is unramified (equivalently étale) over $Y$ .
algèbre non ramifiée sur un anneau intégralement clos	algebra unramified over an integrally closed ring	IV.18.10.10	Abuse of language for "unramified over $Spec (A)$ " when $A$ is integral and integrally closed; not to be confused with `(17.3.2, (ii))`.
transitivité de la non-ramification	transitivity of non-ramification	IV.18.10.13	If $L / K$ is unramified over $Y$ and $M / L$ unramified over the integral closure of $Y$ in $L$ , then $M / K$ is unramified over $Y$ .
translation de la non-ramification	translation property of non-ramification	IV.18.10.13	For a dominant $g : Y^{'} \to Y$ between normal integral preschemes, $L \otimes_{K} K^{'}$ is unramified over $Y^{'}$ whenever $L$ is unramified over $Y$ .
discriminant (de $Spec (C)$ sur $Spec (A)$ )	discriminant (of $Spec (C)$ over $Spec (A)$ )	IV.18.10.15	$d_{C / A}$ : invertible in $A$ iff $L = F r a c (C)$ is unramified over $A$ ( $A$ integrally closed, $C$ projective $A$ -module of finite type).
degré séparable (en un point fibre)	separable degree (at a fibre point)	IV.18.10.16	$n (y) = \sum_{x i so l a t e d in f^{- 1} (y)} [k (x) : k (y)]_{s}$ ; bounded above by the total separable degree $n_{s}$ of the generic fibre.
localisation étale	étale localization	IV.18.10.17	The method of replacing $O_{Y, y}$ by its strict Henselization `(18.8.7)` to remove Noetherian hypotheses from `(15.5.1)` and analogues.
morphisme essentiellement propre	essentially proper morphism	IV.18.10.20	Locally of finite type, separated, and the relative-valuation-ring criterion `(II, 7.3.2.2)` is bijective. Proper iff also of finite type.
$Y$ -section rationnelle (d'un morphisme)	rational $Y$ -section (of a morphism)	IV.18.10.19	Rational $Y$ -map $Y ⇢ X$ with domain of definition $U \subset Y$ . Extends to all of $Y$ when $Y$ is geometrically unibranch and $f$ is essentially proper.
algèbre locale noethérienne complète sur un corps	complete Noetherian local algebra over a field	IV.18.11.1	$k$ -algebra $A$ complete Noetherian local with residue field a finite extension of $k$ . Setting for the §18.11 generators/regularity criteria.
`Â'` (produit tensoriel complété par un corps)	`Â'` (completed tensor product by a field)	IV.18.11.4	$\hat{A}^{'} = A \hat{\otimes}_{k} k^{'}$ : complete semi-local, direct composite of complete local rings $A_{i}^{'}$ faithfully flat over $A$ .
nombre minimum de générateurs d'un module	minimum number of generators of a module	IV.18.11.5	For $(Ω_{A / k}^{1})_{p}$ ; controls existence of a local $k$ -homomorphism $B = k [[T_{1}, \dots, T_{m}]] \to A$ unramified at $p$ `(18.11.5)`.
exposant caractéristique (d'un corps)	characteristic exponent (of a field)	IV.18.11.3	$p \geq 1$ : equals the characteristic of $k$ if $c ha r k > 0$ , equals `1` if $c ha r k = 0$ . Used throughout §18.11.
algèbre géométriquement régulière	geometrically regular algebra	IV.18.11.3	Inherited from `(6.7.6)`; $A_{p}$ is geometrically regular over $k$ iff $A_{p} \otimes_{k} k^{'}$ is regular for every extension $k^{'} / k$ .
extension composée	composed extension	IV.18.10.14	Bourbaki, Alg., chap. VIII, §8, def. 1; an extension $L_{1} / K^{'}$ that is a composite of $L / K$ and $K^{'} / K$ .
morphisme birationnel (sans hypothèse de réduit)	birational morphism (without reducedness hypothesis)	IV.18.10.18	Footnote: EGA's `(6.15.4)` extended by dropping the assumption that $X$ and $Y$ are reduced. $f$ is a local isomorphism iff étale.
Henselized local ring	Henselized local ring	IV.18.12.1	$A^{h} =^{h} O_{Y, y}$ : limit of étale $O_{Y, y}$ -algebras with trivial residue extension. Step in proving `(18.12.1)`.
théorème de la double limite inductive	double inductive limit theorem	IV.18.12.1	Used to write $A^{h}$ (resp. $^{s h} A$ ) as a filtered inductive limit of étale (resp. strictly essentially étale) $O_{Y, y}$ -algebras.
immersion ouverte quasi-compacte	quasi-compact open immersion	IV.18.12.13	Open immersion with quasi-compact image, equivalently of finite presentation `(1.6.2)`. Output side of Zariski's "Main theorem".
fermeture intégrale d'une $O_{Y}$ -Algèbre dans une autre	integral closure of one $O_{Y}$ -Algebra inside another	IV.18.12.14	$B$ : largest sub- $O_{Y}$ -Algebra of $A$ integral over $O_{Y}$ `(II, 6.3.2)`. Featured in the alternative proof of `(18.12.13)`.
Main theorem (de Zariski)	Main theorem (Zariski)	IV.18.12.13	Italicized at first appearance; quasi-finite + separated $f : X \to Y$ over a quasi-compact quasi-separated $Y$ factors as open-immersion-then-finite.

(18.10.1) is the fundamental étaleness criterion over a geometrically unibranch base. Replaces the Noetherian version of the same statement; the proof passes through the strict Henselization $^{s h} A$ (18.8.7) to leverage the fact that strict henselizations of geometrically unibranch rings are integral (18.8.15). Remark (18.10.2, (ii)) records an alternative locally Noetherian proof via Chevalley's openness criterion (14.4.4) and (15.2.3).
(18.10.3) is the étaleness criterion for connected covers. A formally unramified, locally-of-finite-type morphism into an integral geometrically unibranch base with non-empty generic fibre is automatically étale; the target $X$ inherits integrality and geometric unibranchness. The quasi-compactness hypothesis in (18.10.3.1) cannot be dropped — the ℂ²-with-glued-affine-lines example after the remark shows.
(18.10.7)–(18.10.9) are the structural results for étale morphisms to a geometrically unibranch base. Every étale morphism $X \to Y$ (with $Y$ geometrically unibranch irreducible/integral) decomposes $X$ as a sum of irreducible / integral components indexed by $f^{- 1} (η)$ . When $Y$ is normal integral, étale covers correspond bijectively to finite separable $K$ -algebras unramified over $Y$ (18.10.12).
(18.10.16) is the étale-and-finite criterion via separable-degree count. Generalizes (18.5.13); the upper bound $n (y) \leq n_{s} \leq n$ always holds at a geometrically unibranch point, and equality $n (y) = n_{s}$ is both necessary and sufficient for étaleness-and-finiteness on a neighbourhood, when $X$ is reduced and $y$ is normal in $Y$ . The proof reduces in four steps to the strictly-local case and then uses Henselianness (18.5.11) plus normality.
(18.10.17.1) and (18.10.17.2) upgrade (15.5.1). Strict Henselization removes the Noetherian hypothesis from most of §§14-15; the method is recorded in EGA as the prototype of "étale localization", reused throughout SGA.
(18.10.18)–(18.10.19) package étale + birational and rational $Y$ -sections. (18.10.18): étale + birational = local isomorphism (and = open immersion when separated). (18.10.19): at a geometrically unibranch point of a reduced base, the image of a rational section is open-and-closed in $X$ . (18.10.20) records the "essentially proper" terminology used in chap. VI for Picard / Néron-Severi preschemes.
(18.10.21) extends (17.15.5) off the spectrum of a field. Smoothness at $x$ is characterized by the minimum-number-of-generators $r = r g (Ω_{X / Y}^{1} \otimes k (x))$ of the differential module being matched by the fibre dimension at a generization of $x$ over the generic point of the unique component containing $y$ .
(18.11.1)–(18.11.4) are the differential-module machinery for $k$ -algebras $A$ . (18.11.1): when $Ω_{K / k}^{1}$ is of finite rank, $\hat{Ω}_{A / k}^{1}$ is of finite type; smoothness gives a free-module rank formula. (18.11.2): in characteristic $p > 0$ with $[k : k^{p}] < \infty$ , the imperfection module $Υ_{L / k}$ is finite over $L$ and the $n = r g (Ω_{L / k}^{1}) - r g (Υ_{L / k})$ formula holds. (18.11.3): the differential module is free of rank $dim (A / q)$ at a geometrically regular prime — provided $[k : k^{p}] < \infty$ .
(18.11.5)–(18.11.9) are the generators-vs-finite- $B$ -algebra criterion. (18.11.5): $(Ω_{A / k}^{1})_{p}$ admits $m$ generators iff there is a local $k$ -homomorphism $k [[T_{1}, \dots, T_{m}]] \to A$ making $A$ finite and unramified at $p$ . (18.11.7)–(18.11.9) upgrade to "free of rank $dim (A)$ " and "étale at $p$ " under equidimensionality; consequently $A_{p}$ is geometrically regular (18.11.9, (iii)).
(18.11.10) is the four-way equivalence (regularity ⇔ free differentials ⇔ étale-over-power-series ⇔ geom. regular). Always: a), a'), b), c) $\Leftrightarrow$ (i.e., the differential module is locally free of the expected rank $n$ , the morphism $Spec (A) \to Spec (k [[T_{1}, \dots, T_{n}]])$ is étale at $p$ , and after extending scalars to a perfect (or any) field, the base-changed local ring is regular at every prime above $p$ ); these always imply d) (geometric regularity), and are equivalent to d) under $[k : k^{p}] < \infty$ . The remark (18.11.11, (i)) produces a counterexample to d) ⇒ b) when $[k : k^{p}] = \infty$ .
(18.11.12) is the radicial-extension cleanup of Cohen's structure theorem. For $[k : k^{p}] < \infty$ and $A$ complete Noetherian local integral, a finite radicial extension $k^{'} / k$ makes $\hat{A}_{re d}^{'}$ finite separable over a power-series sub-algebra $k^{'} [[T_{1}, \dots, T_{n}]]$ . Auxiliary (18.11.12.1): artinianness of $K \otimes_{k} k^{p^{-} \infty}$ $\Leftrightarrow$ a finite radicial extension trivializing the reduced tensor product.
§18.12 attributed to P. Deligne. The numbered results are systematically the non-Noetherian / étale-localized upgrades of classical Zariski-Main-Theorem-type statements. (18.12.1) and (18.12.3) build the étale-base change that makes a quasi-finite morphism finite near an isolated fibre point; (18.12.4) upgrades (8.11.1) (proper + quasi-finite = finite, in the locally-finite-type setting); (18.12.6)–(18.12.7) give the radicial-and-geometrically-reduced criterion for closed immersions; (18.12.8) upgrades the "integral = affine + universally closed" equivalence; (18.12.12) upgrades (8.11.2) (quasi-finite + separated $\Rightarrow$ quasi-affine); and (18.12.13) is the modern form of Zariski's Main Theorem over quasi-compact quasi-separated bases.
(18.12.15) is the non-Noetherian étale base change for integral closures. For $C \to C^{'}$ étale and $A$ any $C$ -algebra with integral closure $B \subset A$ , the base change $B \otimes_{C} C^{'}$ is the integral closure of $C^{'}$ in $A \otimes_{C} C^{'}$ . Reduces to the Noetherian case (6.14.4) via finite-type and noetherian-approximation arguments; the delicate (6.14.1) is not needed once $C$ is étale-changed.
Pagination. Part C covers pages 157-184 of EGA-IV-4.pdf; the file boundary aligns with the printed page break at the end of §18 (the next page starts §19 — handled in 32-ch4-19-regular-immersions.md). Part C is provisional; will be merged with Part A (18.1-18.5) and Part B (18.6-18.9) into the final §IV.18 file.

§IV.21 additions (Part A, §§21.1-21.7 — divisors, invertible fractional Ideals, linear equivalence, inverse/direct images, 1-codimensional cycles, subprescheme interpretation)

French	English	First appearance	Note
faisceau des diviseurs	sheaf of divisors	IV.21.1.2	$D i v_{X} = M_{X}^{\times} / O_{X}^{\times}$ ; sections over $X$ form the commutative group $Div (X)$ . Notation locked.
diviseur (sur $X$ )	divisor (on $X$ )	IV.21.1.2	Section of $D i v_{X}$ over $X$ . The group $Div (X)$ is always written additively `(21.1.3)`.
diviseur de $f$	divisor of $f$	IV.21.1.2	$d i v (f)$ (or $d i v_{X} (f)$ ), image of a regular meromorphic function $f \in Γ (X, M_{X}^{\times})$ in $Div (X)$ .
support d'un diviseur	support of a divisor	IV.21.1.2	$S u pp (D) = x \in X : D_{x} \neq = 0$ ; closed in $X$ .
diviseur de $s$ (section méromorphe régulière)	divisor of $s$ (regular meromorphic section)	IV.21.1.4	$d i v (s) \in Div (X)$ for $s$ a regular meromorphic section of an invertible $O_{X}$ -Module $L$ . Not necessarily principal `(21.2.9)`.
diviseur positif	positive divisor	IV.21.1.6	Section of $D i v_{X}^{+} \subset D i v_{X}$ , the canonical image of $S (O_{X})$ ; set denoted $Div^{+} (X)$ .
faisceau de groupes ordonnés	sheaf of ordered groups	IV.21.1.6	$D i v_{X}$ carries this structure via $D i v_{X}^{+}$ ; stalk-by-stalk and section-by-section comparison.
Idéal fractionnaire	fractional Ideal	IV.21.2.1	Sub- $O_{X}$ -Module of $M_{X}$ . Capital "Ideal" preserved in line with EGA's typographical convention for sheaves of ideals.
Idéal fractionnaire inversible	invertible fractional Ideal	IV.21.2.1	Fractional Ideal that is an invertible $O_{X}$ -Module; characterized by local generation $I ∥ U = O_{U} \cdot f$ for $f \in Γ (U, M_{X}^{\times})$ `(21.2.2)`.
Idéal entier	integral Ideal	IV.21.2.7	Ideal of $O_{X}$ that is an invertible $O_{X}$ -Module; synonym for the positive elements of `Id.inv(X)` (terminology used "sometimes" in EGA).
$Id . inv_{X}$	$Id . inv_{X}$	IV.21.2.4	Sheaf of commutative groups $U \mapsto I d . in v (U)$ of invertible fractional Ideals.
$J (f) = O_{X} \cdot f$	$J (f) = O_{X} \cdot f$	IV.21.2.5	Invertible fractional Ideal associated with a regular meromorphic function $f$ ; defines the homomorphism $M_{X}^{\times} \to Id . inv_{X}$ .
$O_{X} (D)$	$O_{X} (D)$	IV.21.2.8	$(I_{X} (D))^{- 1}$ ; invertible fractional Ideal locally equal to $O_{U} \cdot f^{- 1}$ when $D ∥ U = d i v_{U} (f)$ .
$s_{D}$ (section méromorphe régulière canonique)	$s_{D}$ (canonical regular meromorphic section)	IV.21.2.9	Section of $O_{X} (D)$ corresponding to `1` under the canonical isomorphism $M_{X} (O_{X} (D)) \sim M_{X}$ ; satisfies $d i v (s_{D}) = D$ .
classe d'équivalence des couples $(L, s)$	equivalence class of pairs $(L, s)$	IV.21.2.10	Set $D (X)$ ; commutative group via tensor product; isomorphic to $Div (X)$ via $D \mapsto c l (O_{X} (D), s_{D})$ `(21.2.11)`.
sous-préschéma fermé $Y (D)$	closed sub-prescheme $Y (D)$	IV.21.2.12	For $D$ positive, the closed sub-prescheme defined by $O_{X} (D) \subset O_{X}$ ; regularly immersed of codimension `1`.
diviseur principal	principal divisor	IV.21.3.1	Divisor of the form $d i v (f)$ for $f$ regular meromorphic on $X$ ; forms $Div . p r in c (X)$ , isomorphic to $Γ (X, M_{X}^{\times}) /Γ (X, O_{X}^{\times})$ .
diviseurs linéairement équivalents	linearly equivalent divisors	IV.21.3.1	$D - D^{'} \in Div . p r in c (X)$ ; principal divisors are those linearly equivalent to `0`.
groupe de Picard	Picard group	IV.21.3.2	$Pic (X)$ , group of isomorphism classes of invertible $O_{X}$ -Modules. Already in EGA III ledger; locked again here.
$l : Div (X) \to Pic (X)$	$l : Div (X) \to Pic (X)$	IV.21.3.2	Composite $Div (X) \sim D (X) \to Pic (X)$ sending $D \mapsto c l (O_{X} (D))$ . Also denoted $l_{X}$ . Kernel $= Div . p r in c (X)$ `(21.3.3)`.
image réciproque d'un diviseur	inverse image of a divisor	IV.21.4.2	$f^{} (D)$ : divisor on $X^{'}$ defined when $s_{D} \in Γ (X, M_{f} (O_{X} (D)))$ and $s_{- D} \in Γ (X, M_{f} (O_{X} (- D)))$ ; corresponds to $(f^{} (O_{X} (D)), s_{D} \circ f)$ .
$Div_{f} (X)$	$Div_{f} (X)$	IV.21.4.2	Subgroup of divisors on $X$ whose inverse image under $f$ is defined; $D \mapsto f^{*} (D)$ is an increasing homomorphism into $Div (X^{'})$ .
$M_{X}^{f \times}$	$M_{X}^{f \times}$	IV.21.4.3	Subsheaf of groups of $M_{X}^{\times}$ of germs of regular meromorphic functions whose inverse image under $f$ exists and is regular.
$D i v_{X}^{f}$	$D i v_{X}^{f}$	IV.21.4.3	$M_{X}^{f \times} / O_{X}^{\times}$ ; sections over $X$ are $Div_{f} (X)$ .
image directe (norme) d'un diviseur	direct image (norm) of a divisor	IV.21.5.5	$f_{*} (D^{'})$ (or $N_{X^{'} / X} (D^{'})$ ) for $f$ finite + (I) locally free or (II) locally Noetherian normal + section-wise norm condition.
$N_{X^{'} / X}$ (sur les sections méromorphes)	$N_{X^{'} / X}$ (on meromorphic sections)	IV.21.5.3	Norm extension $Γ (X^{'}, M_{X^{'}} (L^{'})) \to Γ (X, M_{X} (N_{X^{'} / X} (L^{'})))$ ; multiplicative; sends regular sections to regular sections.
$N x^{'} / x$ (homomorphisme de faisceaux ordonnés)	$N_{X^{'} / X}$ (homomorphism of ordered sheaves)	IV.21.5.5	$f_{*} (D i v_{X^{'}}) \to D i v_{X}$ ; arises from the section-wise norm.
morphisme fini localement libre	finite locally free morphism	IV.21.5.3	Finite morphism with $f_{*} (O_{X^{'}})$ locally free; case (I) in `(21.5.3)`.
cycle (sur $X$ )	cycle (on $X$ )	IV.21.6.1	Element of $z (X) \subset Z^{X}$ whose nonzero coordinates form a locally finite set. Coincides with $Z^{(X)}$ for $X$ Noetherian.
cycle premier	prime cycle	IV.21.6.1	Element of $J (X)$ , the set of irreducible closed parts of $X$ (identified with $X$ via $x \mapsto \overline{x}$ ).
multiplicité (d'un cycle en un point)	multiplicity (of a cycle at a point)	IV.21.6.1	$m u l t_{x} (Z) = n_{x}$ for $Z = \sum n_{x} \cdot \overline{x}$ . Integer, positive or negative.
support d'un cycle	support of a cycle	IV.21.6.1	$S u pp (Z) = ⋃ \overline{x} : m u l t_{x} (Z) \neq = 0$ ; closed in $X$ .
dimension / codimension d'un cycle	dimension / codimension of a cycle	IV.21.6.1	Of the support; written $dim (Z)$ , $co d im (Z)$ .
partie purement de codimension $p$	part purely of codimension $p$	IV.21.6.2	Every irreducible component of codimension $p$ in $X$ .
cycle $p$ -codimensionnel	$p$ -codimensional cycle	IV.21.6.2	Cycle whose support is purely of codimension $p$ ; forms $z^{p} (X)$ . $X^{(p)} = x : co d im (\overline{x}, X) = p$ .
$z_{X}$ , $z_{X}^{p}$ , $z_{X}^{+}$ , $z_{X}^{p +}$	$z_{X}$ , $z_{X}^{p}$ , $z_{X}^{+}$ , $z_{X}^{p +}$	IV.21.6.3	Sheaves of cycles (resp. $p$ -codimensional cycles, resp. their positive submonoids). Flasque sheaves.
$c : D i v_{X} \to z_{X}^{1}$	$c : D i v_{X} \to z_{X}^{1}$	IV.21.6.4	Canonical homomorphism of sheaves of commutative groups; restricts to the monoid map $c : D i v_{X}^{+} \to z_{X}^{1 +}$ .
$cyc (D)$ (cycle associé à un diviseur)	$cyc (D)$ (cycle associated with a divisor)	IV.21.6.5	$\sum_{x \in X^{(1)}} l o n g (O_{Y (D), x}) \cdot \overline{x}$ for $D$ positive; extended to all divisors as the homomorphism $Div (X) \to z^{1} (X)$ .
multiplicité d'un diviseur en un point	multiplicity of a divisor at a point	IV.21.6.7	`mult_x(D) = mult_x(cyc(D))` for $x \in X^{(1)}$ ; equals $l o n g (O_{Y (D), x})$ when $D \geq 0$ . Also written $m u l t_{\overline{x}} (D)$ .
ordre de $f$ au point $x$	order of $f$ at the point $x$	IV.21.6.7	$ω_{x} (f) = m u l t_{x} (d i v (f))$ for $f$ regular meromorphic, $x \in X^{(1)}$ ; equals $l o n g (O_{X, x} / f O_{X, x})$ when $f \in O_{X, x}$ .
cycle des zéros / cycle des pôles (cycle polaire)	cycle of zeros / cycle of poles (polar cycle)	IV.21.6.7	$Z^{+} (f)$ and $Z^{-} (f)$ ; positive `1`-codimensional cycles whose difference is $cyc (d i v (f))$ .
cycle principal / $z_{p r in c}^{1} (X)$	principal cycle / $z_{p r in c}^{1} (X)$	IV.21.6.7	Cycle of the form $cyc (d i v (f))$ ; subgroup of $z^{1} (X)$ . Also called linearly equivalent to `0`.
cycle localement principal	locally principal cycle	IV.21.6.7	Section of $c (D i v_{X}) \subset z_{X}^{1}$ ; characterized stalk-locally as principal on every $Spec (O_{X, y})$ .
principal au point $y$	principal at the point $y$	IV.21.6.7	$Z_{y}$ is principal on $Spec (O_{X, y})$ ; locus of such points is open.
$Cl (X)$ (groupe des classes de cycles `1`-codim.)	$Cl (X)$ (group of classes of `1`-codim. cycles)	IV.21.6.7	$z^{1} (X) / z_{p r in c}^{1} (X)$ . Receives the canonical homomorphism `Div(X) / Div.princ(X) → Cl(X)`.
diviseur (au sens de Bourbaki)	divisor (in Bourbaki's sense)	IV.21.6.8	For $A$ integrally closed Noetherian integral, $z^{1} (Spec (A))$ matches Bourbaki's group of divisors of the Krull ring $A$ .
préschéma localement factoriel	locally factorial prescheme	IV.21.6.9	$O_{X, x}$ factorial for every $x$ ; equivalent to $cyc : Div (X) \to z^{1} (X)$ bijective on normal locally Noetherian preschemes.
anneau parafactoriel	parafactorial ring	IV.21.6.14	Forward-referenced terminology (`(21.13)`): $Pic (U) = 0$ and $p ro f (A) \geq 2$ for $U = Spec (A) - m$ . Used to characterize factoriality in §21.6.14.
sous-préschéma fermé $Y (C)$ (image fermée)	closed sub-prescheme $Y (C)$ (closed image)	IV.21.7.1	For $C$ positive `1`-codimensional, closed image of $∐_{x \in X^{(1)}} Spec (O_{X, x} / m_{x}^{n_{x}})$ under the canonical morphism. Defined by $J_{X} (C)$ .
$J_{X} (C)$ (idéal définissant $Y (C)$ )	$J_{X} (C)$ (Ideal defining $Y (C)$ )	IV.21.7.1	Also written $J (C)$ . Quasi-coherent Ideal of $O_{X}$ .

Sheaf notation. $D i v_{X}$ for the sheaf of divisors, $Id . inv_{X}$ for the sheaf of invertible fractional Ideals, $M_{X}^{\times}$ for the multiplicative-group sheaf of regular meromorphic functions. The capital "I" in "Ideal" follows EGA's typographical convention for sheaves of ideals; we keep it throughout §21 for fractional Ideals as well.
Divisors are written additively. (21.1.3) locks this; we use +, $-$ , $\leq$ , $\geq$ throughout. Multiplicative composition is reserved for Id.inv(X) and $Pic (X)$ .
$s_{D}$ versus $s_{- D}$ . EGA distinguishes $s_{D}$ (the canonical section in $O_{X} (D)$ mapping to $1 \in M_{X}$ under (21.2.9.2)) from $s_{- D}$ (the canonical section in $O_{X} (- D) = O_{X} (D)^{- 1}$ mapping to 1). Both must be defined over $X^{'}$ for the inverse image $f^{*} (D)$ to exist (21.4.2).
cyc versus $c$ . $c : D i v_{X} \to z_{X}^{1}$ is the sheaf map; $cyc : Div (X) \to z^{1} (X)$ is the global-sections map. The terminology "locally principal cycle" refers to sections of the image $c (D i v_{X})$ , not to sections of $D i v_{X}$ itself; the difference matters in (21.6.9)–(21.7.4).
$X^{(p)}$ notation. EGA writes $X^{(p)}$ for the set of points whose closure has codimension $p$ (equivalently $dim (O_{X, x}) = p$ ). We preserve the parenthesized superscript throughout.
$z_{p r in c}^{1} (X)$ versus $Cl (X)$ . The principal subgroup of 1-codimensional cycles is $z_{p r in c}^{1} (X)$ , the quotient is $Cl (X)$ ; $Pic (X) \to Cl (X)$ is the canonical homomorphism (21.6.10.1) and is bijective exactly when $X$ is locally factorial (21.6.10, (ii)).
Parafactoriality forward reference. (21.6.14) introduces parafactoriality in passing, with the explicit promise (21.13) that the term will be defined later. We carry the parenthetical "(conditions which we shall later (21.13) express by saying that the ring $A$ is parafactorial)" verbatim, since it is the only definition the reader has at this point.
(21.7.1) "closed image". The sum prescheme $Y^{'} (C) = ∐ Spec (O_{X, x} / m_{x}^{n_{x}})$ is not a sub-prescheme of $X$ ; one takes the closed image in the sense of (I, 9.5.3 and 9.5.1). The corresponding Ideal $J_{X} (C)$ is quasi-coherent.
(21.7.2) versus (21.7.3). (21.7.2) characterizes closed sub-preschemes defined by positive 1-codimensional cycles via conditions (R_1) + purity + (S_1). (21.7.3) compares this with the Cartier sub-prescheme $Y (D)$ and identifies when the two agree. (21.7.3.1) records the upshot for normal $X$ : $Div^{+} (X)$ matches the set of regularly immersed sub-preschemes of codimension 1 satisfying conditions (i) and (ii).
(21.7.5) counter-example. The "plane meeting a line" example from (14.1.5) shows that surjectivity of cyc on prime cycles alone does not force normality; one needs additionally the injectivity of $c$ on the sheaf level, as recorded in condition a') of (21.7.4).
§21 packaging. §21 is split into three translated files. Part A (this file, 34-ch4-21-divisors.part-a.md) covers §§21.1-21.7 (divisors, invertible fractional Ideals, linear equivalence, inverse and direct images, 1-codimensional cycles, subprescheme interpretation). Part B (forthcoming) covers §§21.8-21.10 (divisors and normalization, dimension 1, etc.). Part C (forthcoming) covers §§21.11-21.15 (Auslander-Buchsbaum, Ramanujam-Samuel, parafactorial rings, the relative variant).

§IV.21 additions (Part B, §§21.8-21.11 — divisors and normalization, dimension-1 preschemes, image/preimage of 1-codimensional cycles, Auslander-Buchsbaum)

French	English	First appearance	Note
normalisation	normalization	IV.21.8	American spelling; the section heading.
normalisé (d'un anneau, d'un préschéma)	normalization (of a ring, of a prescheme)	IV.21.8.6	Integral closure of $A$ in its total ring of fractions, and the corresponding global object. Inherited from `(II, 6.3.8)`.
préschéma de dimension 1	prescheme of dimension 1	IV.21.9	Section heading. "of dimension $\leq 1$ " when EGA writes $dim \leq 1$ .
diviseur sur un préschéma de dimension 1	divisor on a prescheme of dimension 1	IV.21.9	Section heading.
valuation discrète (anneau de)	discrete valuation (ring of)	IV.21.9.8	"Discrete valuation ring" preserved as a fixed phrase; matches $(0_{I V}, 16)$ usage.
théorème de Krull-Akizuki	Krull-Akizuki theorem	IV.21.9.10	Cited as Bourbaki, Alg. comm., chap. VII, §2, n° 5, prop. 5.
faisceaux de germes de pseudo-fonctions	sheaves of germs of pseudo-functions	IV.21.8.5	$M_{X}$ / $M_{X^{'}}$ in the locally Noetherian setting; matches the §20.2 vocabulary.
$N = C o k er (θ^{*})$	$N = C o k er (θ^{*})$	IV.21.8.5	Cokernel of $O_{X}^{\times} \to f_{} (O_{X^{'}}^{\times})$ for $f : X^{'} \to X$ integral; isomorphic to $Ker (θ^{'' })$ for the divisor sheaves.
21.10. Images réciproques et images directes de cycles 1-codimensionnels	21.10. Inverse images and direct images of 1-codimensional cycles	IV.21.10	Section heading.
image réciproque (d'un cycle 1-codimensionnel)	inverse image (of a 1-codimensional cycle)	IV.21.10.3	$f^{*} (Z)$ . Matches the divisor terminology of §21.4.
image directe (d'un cycle 1-codimensionnel)	direct image (of a 1-codimensional cycle)	IV.21.10.14	$f_{*} (Z^{'})$ . Defined for $f$ finite sending maximal to maximal.
cycle 1-codimensionnel à coefficients rationnels	1-codimensional cycle with rational coefficients	IV.21.10.9	Sections of $J_{X}^{1} \otimes_{Z} Q$ . Absorbs non-integer ramification multiplicities at non-flat finite morphisms.
formule de projection	projection formula	IV.21.10.18	$f_{} (f^{} (Z)) = n \cdot Z$ under the finite-rank- $n$ hypothesis at maximal points.
sous-faisceau de support contenu dans $T$	subsheaf with support contained in $T$	IV.21.10.3	$Γ_{T} (J_{X}^{1})$ : largest subsheaf of $J_{X}^{1}$ supported in $T$ .
$χ (N, v)$ (caractéristique d'Euler-Poincaré)	$χ (N, v)$ (Euler-Poincaré characteristic)	IV.21.10.17	`long_R(Ker v) − long_R(Coker v)` for $v$ an endomorphism with finite-length kernel and cokernel.
puissance symbolique j-ème (d'un idéal premier)	$j$ -th symbolic power (of a prime ideal)	IV.21.10.17.7	$n_{j}$ : inverse image in $A$ of $(p A_{p})^{j}$ .
21.11. Factorialité des anneaux locaux réguliers	21.11. Factoriality of regular local rings	IV.21.11	Section heading.
Auslander-Buchsbaum (théorème de)	Auslander-Buchsbaum (theorem of)	IV.21.11.1	Hyphenated; Kaplansky-style proof retained verbatim.
courbe algébrique (réduite) sur un corps	(reduced) algebraic curve over a field	IV.21.8.6	Connects §21.8 normalization theory to the Riemann-Roch hypothesis of chap. V.
$Λ^{m a x} E$	$Λ^{m a x} E$	IV.21.11.1	Locally $Λ^{p} E$ where $p$ is the local rank of $E$ ; the rank may vary with connected component.

(21.8.1)–(21.8.2): integral morphisms trivialize locally free sheaves and kill $R^{1} f_{*} (O^{\times})$ . The semi-local reduction (Bourbaki, Alg. comm., V, §2, n° 1, prop. 3) gives stalkwise triviality; consequently $Pic (X^{'})$ is the direct-image cohomology $H^{1} (X, f_{*} (O_{X^{'}}^{\times}))$ . This is the engine for the divisor / Picard comparison between a Noetherian prescheme and its normalization.
(21.8.3)–(21.8.5) are the divisor/Picard diagrams for integral morphisms. (21.8.3.1) is the commutative 3-column diagram; under the integrality hypothesis the snake lemma (21.8.4) produces injective $O_{X}^{\times} \to f_{*} (O_{X^{'}}^{\times})$ and surjective $D i v_{X} \to f_{*} (D i v_{X^{'}})$ with $Ker = C o k er$ . (21.8.5.1) is the four-row diagram tying Div(X)/Div(X') ↔ Pic(X)/Pic(X') through the local $O_{X, s}^{' \times} / O_{X, s}^{\times}$ quotients.
(21.8.6, iii) records the $F_{2}$ -residue-field obstruction. Beyond the schematic conditions, $Div (X) \to Div (X^{'})$ can be bijective without $f$ being an isomorphism only via a residue field of size 2. The remark settles when one may freely substitute $Pic (X) \to Pic (X^{'})$ for $Div (X) \to Div (X^{'})$ .
(21.9.2)–(21.9.4): the structure sheaf $D i v_{X}$ on a dimension-1 prescheme. (21.9.2) is the topological "skyscraper-direct-sum" criterion (discrete support contained in the closed points). (21.9.4) localizes the divisor sheaf at the codimension-1 points $X^{(1)}$ and proves $D i v_{X}$ is flasque. This collapses Cartier divisors on a curve to a family $(D_{x})_{x \in X^{(1)}}$ .
(21.9.7) produces a positive divisor on a separated curve. A dense open in a Noetherian dimension-1 prescheme without isolated points carries a positive divisor $D$ meeting every irreducible component; this is the input to the Riemann-Roch ampleness proof of chap. V and to the quasi-projectivity of separated $k$ -curves.
(21.9.11)–(21.9.12): extending divisors and Picard classes from a closed subprescheme. Under the locally- closed- $Y$ condition $2^{\circ}$ , every $D_{0} \geq 0$ on X_0 of support disjoint from $Z_{0} = ⋃_{x \in A ss (O_{X})} \overline{x} \cap X_{0}$ extends to a divisor on $X$ . With an ample sheaf on X_0, the canonical $Pic (X) \to Pic (X_{0})$ is surjective. (21.9.12) packages this into the Henselian case: for $A$ Henselian local and $f : X \to S$ separated of finite presentation with $dim (X_{0}) \leq 1$ , $Pic (X) \to Pic (X_{0}^{'})$ is surjective; the proof goes via the Noetherian-Henselian approximation $A = lim A_{λ}$ (18.6.15) and quasi-finite splitting (18.5.11, c).
(21.9.13) is the projectivity remark. Under (21.9.12), a proper morphism with ample $O_{X_{0}^{'}}$ -Module is projective; the input is (9.6.4) applied with the observation that every neighbourhood of the closed point of a Henselian $S$ is the whole of $S$ .
(21.10.1)–(21.10.3) define $f^{*} (Z)$ for 1-codimensional cycles. The trichotomy $x \in / T$ / flat at $x^{'}$ / $O_{X, x}$ factorial with $m_{x} \in / A ss (O_{X^{'}, x^{'}})$ covers the three accepted cases; the cycle multiplicity $n_{x^{'}}$ is built from the length $λ_{x^{'}} = l o n g (O_{X^{'}, x^{'}} / m_{x} O_{X^{'}, x^{'}})$ or from (21.6.9) factoriality.
(21.10.4): $cyc \circ f^{*} = f^{*} \circ cyc$ on divisors. Local reduction to $X = Spec (A)$ and $D = d i v (t)$ ; the discrete-valuation-ring case checks both routes give $λ_{x^{'}} n_{y}$ via (4.7.1).
(21.10.6)–(21.10.8): flat morphisms accept all 1-cycles, and the chain rule. Flatness in codimension $\leq 1$ is enough to define $f^{*}$ on all $J^{1} (X)$ ; composition $(f \circ g)^{*} = g^{*} \circ f^{*}$ follows from the codimension-0 transitivity of flatness and the length-multiplication formula long(A''/𝔪 A'') = long(A'/𝔪 A') · long(A''/𝔪' A'').
(21.10.10)–(21.10.13): 1-cycles with rational coefficients absorb non-integer multiplicities. The fourth case (iv): $A^{'} = \hat{O}_{X^{'}, x^{'}}$ finite over $A = \hat{O}_{X, x}$ with $K^{'} = A^{'} \otimes_{A} K$ free over $K$ , with rational multiplicity $μ_{x^{'}} = r_{x^{'}} / q_{x^{'}}$ . The lemma (21.10.13) ( $d_{t}$ -additivity over a dimension-1 local ring) is the algebraic engine; (21.10.13.1) upgrades $cyc \circ f^{*} = f^{*} \circ cyc$ to this rational setting. (21.10.12) exhibits a non-integral example using (6.15.11, ii).
(21.10.14)–(21.10.17): direct images of cycles under finite morphisms and the divisor-norm formula. For $f : X^{'} \to X$ finite sending maximal to maximal, $f_{*} (Z^{'}) = \sum n_{x^{'}} [k (x^{'}) : k (x)] \cdot x$ . (21.10.17): for $f$ finite locally free, $f_{*} (cyc D^{'}) = cyc (f_{*} (D^{'}))$ , reducing to $l o n g_{A} (A^{'} / t^{'} A^{'}) = l o n g_{A} (A / t A)$ . The supporting (21.10.17.3) splits into four cases (discrete-valuation, complete integral, complete with $m \in / A ss (A)$ , general), with (21.10.17.7) the $χ$ -additivity over minimal primes.
(21.10.18)–(21.10.19): projection formula. $f_{*} (f^{*} (Z)) = n \cdot Z$ under the finite-rank- $n$ hypothesis at maximal points; in particular for $f$ finite locally free of rank $n$ . Proof routes through complete-local reduction and rejoins flat case (ii) in cases (iii), (iv).
(21.11.1) Auslander-Buchsbaum: regular Noetherian local $\Rightarrow$ factorial. Kaplansky's proof: induction on dimension via $Pic (U) = 0$ for $U = X - m$ , using (21.6.14) and the finite-projective resolution of a coherent extension. The lemma (21.11.1.2) ( $⨂ (Λ^{m a x} E_{i})^{(- 1)^{i}} ≅ O_{X}$ for a finite exact sequence of locally frees) is the abstract reduction; coincidence of the local glueings reduces to Bourbaki's "projection parallel to a sub-module".
Pagination. Part B covers pages 280-303 of EGA-IV-4.pdf (§§21.8-21.11). The file boundary at the start of §21.12 aligns with the printed page break at page 304. Part B is provisional; will be merged with Part A (§§21.1-21.7) and Part C (§§21.12-end) into the final §IV.21 file.

§IV.21 additions (Part C, §§21.12-21.15 — Van der Waerden purity for the ramification locus; parafactorial couples and parafactorial local rings; Ramanujam-Samuel; relative divisors)

French	English	First appearance	Note
enveloppe affine (du $X$ -préschéma $U$ )	affine envelope (of the $X$ -prescheme $U$ )	IV.21.12.1	$U^{\circ} = A ff (U / X) = Spec (f_{*} (O_{U}))$ ; represents the functor $V \mapsto Hom_{X} (U, V)$ on $X$ -preschemes affine over $X$ $(0_{III}, 8.1.11)$ .
défaut d'affinité (de l'ouvert $U$ relativement à $X$ )	affineness defect (of the open $U$ relative to $X$ )	IV.21.12.5	$D a f (U / X) = f^{\circ} (U^{\circ}) - U$ ; empty iff $U \to X$ is affine. Functorial under flat base change `(21.12.5.1)`.
théorème de pureté de Van der Waerden	Van der Waerden's purity theorem	IV.21.12.12	For locally Noetherian integral $X$ , $Y$ , birational locally-of-finite-type $g : X \to Y$ with $Y$ normal and each $O_{Y, y}$ satisfying (W) (e.g. $Y$ locally factorial), every irreducible component of the non-local-isomorphism locus $T = X - V$ has codimension `1` in $X$ .
ensemble de ramification (d'un morphisme birationnel)	ramification locus (of a birational morphism)	IV.21.12.14	The locus $T$ of `(21.12.12)` is also `{x ∈ X : g \text{ ramified at } x\}` by `(21.12.14, (ii))`; justifies the section title.
condition (W) (sur un anneau local noethérien)	condition (W) (on a Noetherian local ring)	IV.21.12.8	Every open $U \subset Y^{'} = Y - y$ containing no irreducible component of $Y^{'}$ and with $U \to Y^{'}$ affine is itself an affine open of $Y$ .
condition (W̃)	condition (W̃)	IV.21.12.8	Every closed $T \subset Y$ whose components are codimension-`1` and miss the closed point on every irreducible component of $Y$ gives an affine $Y - T$ . Implies (W).
condition (W̃')	condition (W̃')	IV.21.12.8	Irreducible- $Y$ simplification of (W̃): for every irreducible closed $T \subset Y$ of codimension `1`, $Y - T$ is affine.
théorème principal de Zariski / "Main theorem"	Zariski's Main theorem / Main theorem	IV.21.12.12	Italicized at first appearance, matching the §20.4 convention. Cited from `(8.12.10)` in `(21.12.12)` and from `(III, 4.4.9)` in `(21.12.14, (ii))`.
couple parafactoriel	parafactorial couple	IV.21.13.1	$(X, Y)$ with $Y$ closed in $X$ such that $L \mapsto L ∥ (U \cap V)$ is an equivalence of categories of invertible Modules for every open $V \subset X$ ( $U = X - Y$ ).
anneau local parafactoriel	parafactorial local ring	IV.21.13.7	Local ring $A$ such that $(Spec (A), m)$ is parafactorial. Equivalently $Pic (U) = 0$ and $p ro f (A) ⩾ 2$ ( $U$ punctured spectrum); in particular $dim (A) ⩾ 2$ .
$i$ -pur (couple, pour un faisceau de groupes)	$i$ -pure (couple, for a sheaf of groups)	IV.21.13.13	Categorical-faithfulness ladder on principal $G$ -sheaves: $i = 0$ faithful, $i = 1$ fully faithful, $i = 2$ equivalence. For $G = O_{X}^{\times}$ , `2`-pure ≡ parafactorial.
faisceaux de cohomologie locale $H_{Y}^{p} (G)$	local cohomology sheaves $H_{Y}^{p} (G)$	IV.21.13.13	Cited from chap. III, 3rd part; for commutative $G$ , $i$ -purity ≡ $H_{Y}^{p} (G) = 0$ for $p ⩽ i$ , generalizing the notion for every $i$ .
augmentation (anneau augmenté $D_{B} (B)$ )	augmentation (augmented ring $D_{B} (B)$ )	IV.21.13.9	Trivial type extension `(0, 18.2.3)`; example of non-reduced parafactorial ring $A = D_{B} (B) ≃ B [T] / (T^{2})$ in dim. $⩾ 3$ `(21.13.9, (iii))`.
anneau local d'intersection complète absolue	absolute complete intersection local ring	IV.21.13.9	`(19.3.1)`-version; in dimension $⩾ 4$ , parafactorial (cited `[41, XI, 3.13]`).
conducteur (de $A$ dans $A^{'}$ )	conductor (of $A$ in $A^{'}$ )	IV.21.13.9	Largest ideal of $A^{'}$ contained in $A$ ; figures in the dim-`2` non-factorial parafactorial classification `(21.13.9, (vi))`.
théorème de Ramanujam-Samuel	Ramanujam-Samuel theorem	IV.21.14.1	For $A$ Noetherian local with `Â` integral and integrally closed, $B$ Noetherian local formally smooth over $A$ with $dim (B) > dim (A)$ and $[k (B) : k (A)] < \infty$ : every `1`-codimensional cycle on $Spec (B)$ principal at $m B$ is principal.
algèbre formellement lisse pour les topologies préadiques	formally smooth algebra for the preadic topologies	IV.21.14.1	`(0, 19.3.1)`-style formal smoothness; the standing hypothesis for `(21.14.1)`, `(21.14.2)`, `(21.14.3)`.
diviseur sur $X$ relativement à $S$	divisor on $X$ relative to $S$	IV.21.15.2	Section of $D i v_{X / S} = M_{X / S}^{\times} / O_{X}^{\times}$ . They form $Div (X / S) \subset Div (X)$ .
diviseur sur $X$ transversal à $f$	divisor on $X$ transversal to $f$	IV.21.15.2	Synonym for "divisor relative to $S$ "; reflects the transversal-regularity interpretation `(21.15.3.3)`.
sous-préschéma fermé transversalement régulier de codimension `1`	closed sub-prescheme transversally regular of codimension `1`	IV.21.15.3.3	Identified canonically with $D \in Div^{+} (X / S)$ via $D \mapsto Y (D)$ using `(11.3.8)` and `(19.2.4)`.
famille de diviseurs sur $X$ relatifs à $S$ , paramétrée par $S^{'}$	family of divisors on $X$ relative to $S$ , parametrized by $S^{'}$	IV.21.15.9	Element of $Div (X^{'} / S^{'})$ with $X^{'} = X \times_{S} S^{'}$ . Functorial in $S^{'}$ via $D i v_{X / S} : S c h_{/ S}^{\circ} \to A b$ .
foncteurs $D i v_{X / S}$ , $D i v_{X / S}^{+}$	functors $D i v_{X / S}$ , $D i v_{X / S}^{+}$	IV.21.15.9	Contravariant functors $S c h_{/ S}^{\circ} \to A b$ (resp. $\to S e t$ ); representability discussed in chap. VI.

(21.12.1)–(21.12.5) build the affine envelope $A ff (U / X)$ . Universal affine $X$ -prescheme through which a quasi-compact quasi-separated $f : U \to X$ factors; the discrepancy $D a f (U / X) = f^{\circ} (U^{\circ}) - U$ measures how far $U$ is from being affine over $X$ . Flat base change preserves Aff and Daf (21.12.2), (21.12.5.1).
(21.12.6) is the codimension-2 constraint on the closure of Daf. For $X$ locally Noetherian and $U \to X$ quasi-compact, the closure $Z = X - U_{1}$ has codimension $⩾ 2$ ; conversely, if $co d im (T, X) ⩾ 2$ , $f^{\circ} : U^{\circ} \to X$ is surjective. The proof uses Cohen's structure theorem (0, 19.8.8) to reduce to a complete Noetherian local integral base and (II, 6.1.10) (proper + dominant $\Rightarrow$ surjective).
(21.12.8) introduces conditions (W), (W̃), (W̃'). Working hypotheses on local rings at a closed point; (W̃) $\Rightarrow$ (W) and (W̃') ≡ (W̃) in the irreducible case. Every dimension- $⩽ 1$ Noetherian local ring satisfies (W) trivially; every dimension-2 Noetherian local ring satisfies (W) by local duality (chap. III).
(21.12.10)–(21.12.11) are the technical core under condition (W). (21.12.10): for $g : X \to Y$ with $g ∣ g^{- 1} (Y^{'})$ an open immersion ( $Y^{'} = Y - y$ ) and $O_{Y, y}$ satisfying (W), every irreducible component of $X_{y}$ is of codimension $⩽ 1$ or has isolated generic point. (21.12.11): extends pointwise to the locus where $g$ is locally an isomorphism, on an irreducible $X$ .
(21.12.12) is Van der Waerden's purity theorem proper. For $g : X \to Y$ birational locally-of-finite-type with $Y$ normal satisfying (W) pointwise (e.g. locally factorial), the locus where $g$ fails to be a local isomorphism has irreducible components everywhere of codimension 1. The proof reduces to a fibre $T = X_{y}$ and applies Zariski's Main theorem (8.12.10).
(21.12.13) upgrades to "local isomorphism" / "open immersion". Quasi-finiteness at the points of X_1 plus the hypotheses of (21.12.12) force $T = \emptyset$ , hence $g$ is a local isomorphism; separated $\Rightarrow$ open immersion via (I, 8.2.8).
(21.12.14) (v) records the Zariski-Nagata purity conjecture. The étale-purity conjecture is stated but not proved here; the locally-quasi-finite case is attributed to Zariski-Nagata. The reformulation in terms of affinity of the unramified locus $Spec (B) \to Spec (A)$ for $B$ a finite $A$ -module with $A$ regular local is one of the threads picked up in SGA 1 / SGA 2.
(21.12.15)–(21.12.17) are the relative Main theorem applications. (21.12.15): a proper $S$ -morphism between a flat $X / S$ with geometrically irreducible fibres and a smooth $Y / S$ , which is a generic-fibre isomorphism, is an isomorphism. (21.12.16): under proper + smooth + flat with geom-irred fibres, the locus {s : f_s \text{ iso}\} is open-and-closed in $S$ . (21.12.17) gives counterexamples without the irreducibility or properness hypotheses, and conjectures an étale-morphism upgrade.
(21.13.1)–(21.13.6) set up parafactorial couples. $(X, Y)$ parafactorial means restriction $L \mapsto L ∥ (U \cap V)$ is an equivalence of categories of invertible Modules, for every open $V \subset X$ ( $U = X - Y$ ). Criterion (21.13.5): $O_{X} \to j_{*} (O_{U})$ bijective plus $j_{*} (L)$ invertible for every invertible $L$ on $U$ . Lemma (21.13.4) matches the first condition to $p ro f_{Y} (O_{X}) ⩾ 2$ . Faithfully-flat-quasi-compact base descent (21.13.6, (iii)) holds when $U = X - Y$ is retrocompact.
(21.13.7)–(21.13.9) parafactorial local rings. A Noetherian local ring is parafactorial iff $p ro f ⩾ 2$ and $Pic (U) = 0$ ( $U$ punctured spectrum). Examples: every factorial Noetherian local ring of dimension $⩾ 2$ ; the non-reduced $A = D_{B} (B) ≃ B [T] / (T^{2})$ over a regular local $B$ of dimension $⩾ 3$ ; an integrally-closed non-factorial example via B[[T]] for $B$ non-factorial complete; a complete-intersection example of dimension $⩾ 4$ ; and example (vi) classifies dim-2 non-factorial parafactorial rings (Cohen-Macaulay + factorial integral closure + conductor conditions).
(21.13.10)–(21.13.12) reduce global parafactoriality to local, plus a flat-descent of factoriality. (21.13.10): $(X, Y)$ is parafactorial iff every $O_{X, y}$ ( $y \in Y$ ) is. (21.13.11): parafactorial plus locally factorial off $Y$ $\Rightarrow$ locally factorial. (21.13.12): flat local $A \to B$ with $B$ factorial $\Rightarrow$ $A$ factorial. (21.13.12.1): in the parafactorial direction, descent requires $m B$ ideal-of-definition; in particular Â parafactorial $\Rightarrow$ $A$ parafactorial.
(21.13.13) situates parafactoriality inside local cohomology. $(X, Y)$ is $i$ -pure for $G$ if restriction on principal homogeneous $G$ -sheaves is faithful ( $i = 0$ ), fully faithful ( $i = 1$ ), or an equivalence ( $i = 2$ ). For $G = O_{X}^{\times}$ , 2-pure ≡ parafactorial. For commutative $G$ the notion is $H_{Y}^{p} (G) = 0$ for $p ⩽ i$ (chap. III, 3rd part), and generalizes for every integer $i$ .
(21.13.14)–(21.13.16) package the Pic-Cl comparison for normal schemes. (21.13.14): on a locally Noetherian normal $X$ with a filtering family $(U_{λ})$ , "every 1-codim. cycle locally principal on some $U_{λ}$ is locally principal" ≡ parafactoriality of all $O_{X, x}$ outside $⋂ U_{λ}$ . (21.13.15) is the punctual reformulation; (21.13.16) adds $lim Pic (U_{λ}) = 0$ for the cycle-support criterion.
(21.14.1)–(21.14.2) are Ramanujam-Samuel. Under formal smoothness $B / A$ with strictly larger dimension, finite residue extension, and Â integral-and-integrally-closed: every 1-codimensional cycle on $Spec (B)$ principal at $m B$ is principal. Proof reduces to $B = A [[T_{1}, \dots, T_{n}]]$ via Cohen-structure plus the $D_{0} = \hat{B} \otimes_{\hat{A}} C$ complete-formal-power-series construction, and then to $n = 1$ via a Weierstrass-preparation / Bourbaki series-reduction trick. (21.14.2): parafactorial reformulation — $B_{q}$ parafactorial for every $q \neq \subset m B$ of height $⩾ 2$ .
(21.14.3) is the parafactoriality of smooth morphisms over a normal base. For $f : X \to S$ smooth with $S$ normal: every $x \in X$ with $dim (O_{X, x}) ⩾ 2$ not maximal in its fibre has parafactorial local ring; every 1-codim. cycle on $X$ not supporting a fibre component is locally principal; and for $Y \subset X$ closed with fibrewise codimension $⩾ 2$ on the generic fibres, $(X, Y)$ is parafactorial. Proof uses Noetherian descent (write $A$ as an inductive limit of integrally-closed $Z$ -algebras of finite type, completing via excellence) plus a pointwise reduction to (21.14.2).
(21.14.4) records four remarks. (i): conjectural strengthening dropping the residue-field-finite hypothesis. (ii): finite-descent extension to reduced Â, valid for $dim B ⩾ dim A + 2$ ; analogous for (21.14.3). (iii): chap. III, 3rd part gives a codim-3 parafactoriality theorem for smooth morphisms; the dual-numbers example $A = k [T] / (T^{2})$ , $X = V_{S}^{2}$ is a codim-2 non-reduced counterexample. (iv): historical credit to C. Seshadri [44] for the $S \times_{k} T$ algebraic case; the footnote $[^{21} .14.4 - ses ha d r i]$ records the "semi-complete" hypothesis we drop by local-on- $S$ reasoning.
(21.15.1)–(21.15.9) are the relative-divisor module of §21.15. For $f : X \to S$ flat locally of finite presentation, the relative meromorphic sheaf $M_{X / S}$ (20.6.1) carries a units subsheaf $M_{X / S}^{\times}$ , and $D i v_{X / S} = M_{X / S}^{\times} / O_{X}^{\times}$ is the relative-divisor sheaf. Positive relative divisors $Div^{+} (X / S)$ correspond to transversally regular closed sub-preschemes of codimension 1 (21.15.3.3). Flat pullback (21.15.7), finite-flat pushforward via norm (21.15.8), and base change (21.15.9) are all functorial; the cofunctor $D i v_{X / S}$ on $S c h_{/ S}$ is representable in important cases (chap. VI).
§21.15 is "first reading optional". The §20.0 introduction (translated in 33-ch4-20-meromorphic-functions.md) flags §21.15 (with §§20.5, 20.6) as relative variants the reader may skip on a first reading. We preserve that caveat.
Pagination. Part C covers pages 304-332 of EGA-IV-4.pdf (§§21.12-21.15). The file boundary at the start of §21.12 aligns with the printed page break at page 304. Part C is provisional; will be merged with Part A (§§21.1-21.7) and Part B (§§21.8-21.11) into the final §IV.21 file 34-ch4-21-divisors.md referenced in README.md.

Éléments de Géométrie Algébrique — English