Translation Glossary

Translation choices for the English EGA I. The first occurrence of a flagged term in each section file is rendered with a bracketed modern gloss (e.g., prescheme [modern: scheme]); subsequent occurrences in the same file use the historical term alone.

Core terminology

French	English	Notes
préschéma	prescheme	EGA's term for what modern texts call a `scheme`. First use: `prescheme [modern: scheme]`.
schéma	scheme	In EGA, a separated prescheme. First use: `scheme [in EGA: separated prescheme]`.
morphisme	morphism	Of preschemes, sheaves, modules — context-dependent.
immersion	immersion	EGA distinguishes open, closed, and local immersions; preserve.
faisceau	sheaf	Established.
application	map	Default `map`; `mapping` only when prose demands.
application rationnelle	rational map	EGA's term.
application continue	continuous map	Topology.
espace topologique	topological space	Standard.
espace annelé	ringed space	Standard.
anneau	ring	Commutative by default in EGA. Stated explicitly in Chapter 0 §1.
anneau local	local ring	Standard.
anneau intègre	integral ring	EGA's term for an integral domain.
corps	field	Never "body."
corps résiduel	residue field	Standard.
idéal premier	prime ideal	Standard.
idéal maximal	maximal ideal	Standard.
nilradical	nilradical	Standard.
radical	radical	`radical of a ring`: intersection of maximal ideals. `radical of an ideal`: see below.
racine d'un idéal	radical of an ideal	Use `radical`, not `root`; EGA writes $r (a)$ for what modern texts often write $a$ .
anneau de fractions	ring of fractions	$S^{- 1} A$ , $A_{f}$ , $A_{p}$ .
anneau adique	adic ring	EGA: separated and complete with respect to an ideal of definition.
anneau préadique	preadic ring	EGA: like adic but not necessarily separated or complete. First use: `preadic ring [historical EGA term]`.
anneau admissible	admissible ring	EGA term.
Module	Module (capital M)	EGA uses capital $M$ for sheaves of modules over $O_{X}$ ; preserve typography.
Algèbre	Algebra (capital A)	Same convention.
Idéal	Ideal (capital I)	Same convention, when referring to a sheaf of ideals.
sous-Module	sub-Module	Preserve.
quasi-cohérent	quasi-coherent	Standard.
cohérent	coherent	Standard.
plat	flat	Technical.
fidèlement plat	faithfully flat	Technical.
ouvert / fermé	open / closed	Topology.
voisinage	neighborhood	US spelling throughout.
dense	dense	Standard.
rare	nowhere dense	EGA's `rare` is the modern `nowhere dense`. First use: `nowhere dense [EGA: rare]`.
irréductible	irreducible	Standard.
composante irréductible	irreducible component	Standard.
générisation	generization	EGA coinage, now standard.
spécialisation	specialization	Standard.
point générique	generic point	Standard.
espace de Kolmogoroff	Kolmogorov space	Spelling adjusted to standard English.
espace noethérien	Noetherian space	Standard.
quasi-compact	quasi-compact	EGA's `quasi-compact` is the modern `compact without Hausdorff`; preserve EGA's term.
espace propre	proper space	When the algebraic-geometry sense is meant.
propre (morphisme)	proper (morphism)	Algebraic-geometry sense; never `own`.
espace fibré	fiber space	Period term; US spelling `fiber`.
limite projective	projective limit	First use: `projective limit [modern: inverse limit]`.
limite inductive	inductive limit	First use: `inductive limit [modern: direct limit]`.
système projectif	projective system	Goes with `projective limit`.
système inductif	inductive system	Goes with `inductive limit`.
produit tensoriel	tensor product	Standard.
produit fibré	fiber product	Standard.
canonique	canonical	Not `natural` unless naturality is the point.
fonctoriel	functorial	Standard.
foncteur	functor	Standard.
catégorie	category	Standard.
dévissage	dévissage	Keep as technical loanword.
étale	étale	Keep loanword with accent.
spectre premier	prime spectrum	Standard.
ouvert affine	affine open	EGA writes `ouvert affine`; English `affine open` is standard.
schéma affine	affine scheme	Standard.
schéma de Chevalley	Chevalley scheme	EGA's term, kept verbatim.
topologie de Zariski	Zariski topology	Standard.
topologie spectrale	spectral topology	EGA's name for the Zariski topology on $Spec (A)$ ; both names appear in EGA.
pseudo-discret	pseudo-discrete	EGA's term.
par abus de langage	by abuse of language	Idiom preserved.

Proof-movement vocabulary

French	English
Soit / Soient	Let / Let … and … be
Supposons que	Suppose that / Assume that
On a	We have
On suppose	We suppose / We assume
Posons	Set / Put
Montrons que	We show that
Il suffit de	It suffices to
Il reste à montrer	It remains to show
Il en résulte que	It follows that
Par suite	Consequently / hence
D'où	Whence / hence
Or	Now
En effet	Indeed
Cela démontre	This proves
C.Q.F.D.	(omitted; end of proof)
en vertu de	by
d'après	by (theorem); per (author)
il revient au même	this amounts to / equivalently
il en est de même	the same holds
à isomorphisme près	up to isomorphism
à une équivalence près	up to equivalence
autrement dit	in other words
en outre	moreover
en particulier	in particular
une et une seule manière	one and only one way
il existe	there exists
quel que soit	for every / for any
tout	every / any / all (per context)
quelconque	arbitrary
si et seulement si	if and only if
nécessaire et suffisant	necessary and sufficient

Quantifier and modality discipline

$i l es t imm \overset{e}{ˊ} d ia tq u e$ is rendered as it is immediate that, never strengthened to it is obvious that.
$o n v o i t a u ss i t \overset{o}{^} tq u e$ is rendered as one sees at once that.
il est facile de voir que is rendered as it is easy to see that, preserving the rhetorical force.
clairement / évidemment is rendered as clearly / evidently — never escalated to obviously unless EGA uses the stronger word.
il semble que is rendered as it seems that — never as a stronger claim.

Punctuation and notation

Decimal points are used in numerical references ((2.1.4), never (2,1,4)).
French guillemets « … » are rendered as English double quotes $" \dots "$ .
Italics, in EGA reserved for emphasis and for first-occurrence-defines-a-term, are rendered as Markdown italics $* \dots *$ for emphasis and as $* * \dots * *$ (bold) for definitional first occurrences inside an unbracketed paragraph. (This follows the SGA-I convention.)
Footnotes in the French source are converted to Markdown footnotes keyed by section and item: $[^{N} - M - K]$ .

Notation kept exactly

EGA's symbolic choices that should never be silently modernized:

EGA	Modern (informational)	Comment
$r (a)$	$a$	Radical of an ideal.
$J$ , $p$ , $q$ , $m$	as such	Fraktur letters for ideals; preserve glyphs.
$i_{M}^{s}$	as such	Canonical map $M \to S^{- 1} M$ ; preserve subscript-superscript form.
$S^{- 1} M$	$S^{- 1} M$ (modern same)	Module of fractions.
$Spec (A)$	same	Prime spectrum.
$Sp f (A)$	same	Formal spectrum.
$S u pp (M)$	same	Support of a module.
$M_{ϕ}$	$ϕ_{*} M$	EGA's notation for restriction of scalars; preserve.
$lim$	`colim` or $lim$	Direct/inductive limit; EGA writes `lim` with a rightward arrow above. We render as $lim$ .
$lim$	`lim` or $lim$	Inverse/projective limit; rendered as $lim$ .
$M \otimes_{A} N$	same	Tensor product.
$Hom_{A} (M, N)$	same	Module of homomorphisms.
$Ω_{X / Y}^{1}$	same	Sheaf of relative differentials.

Notes on Grothendieck's voice

EGA sentences are long because they are doing mathematical work: they accumulate hypotheses, comparisons, and viewpoint shifts in a single architectural unit. In translation we keep long sentences long when their structure carries the argument. We split a sentence only when English grammar or reader memory would otherwise hide the dependency structure of the proof. We do not flatten dense paragraphs into friendly textbook prose, and we do not modernize EGA's deliberate technical vocabulary (prescheme, radical, quasi-compact, rare, dévissage) without a flagged gloss.

Éléments de Géométrie Algébrique — English