Glossary and Translation Ledger — SGA 3

This file records the translation choices made for the SGA 3 Markdown translation. It is the authoritative reference for the parallel translators of individual Exposés; every translator must consult it before drafting and must obey it where it applies.

Core terminology

French	English	Note
préschéma	prescheme	Historical SGA/EGA term. Used throughout SGA 3, matching SGA 1 and SGA 2 I–XIII.
schéma	scheme	Older sense: separated prescheme. The 2011 re-edition retains this.
$S$ -préschéma	$S$ -prescheme	A prescheme over $S$ . Sometimes typeset `S-préschéma`; preserve hyphen.
$S$ -schéma en groupes	$S$ -group scheme	The relative form. Title-level for the whole Séminaire.
faisceau	sheaf	Standard.
faisceau (quasi-)cohérent	(quasi-)coherent sheaf	Standard.
espace annelé	ringed space	Standard.
application	map	"Function" only in elementary set-theoretic contexts.
morphisme	morphism	Standard.
morphisme étale / lisse / plat / propre	étale / smooth / flat / proper morphism	Standard; matches SGA 1 glossary.
de présentation finie	of finite presentation	SGA 3 deliberately replaces de type fini by de présentation finie (cf. Intro §6); preserve.
de type fini	of finite type	Use only when the source uses de type fini.
revêtement étale	étale covering	Per SGA 1 glossary.
immersion / immersion fermée / ouverte	immersion / closed immersion / open immersion	Standard.
ouvert / fermé / localement fermé	open / closed / locally closed	Standard.
voisinage	neighborhood	American spelling.
recouvrement	covering	"Cover" only in casual prose.
corps	field	False friend: not "body".
anneau	ring	Commutative by SGA convention unless stated.
anneau local (noethérien)	(noetherian) local ring	Standard.
anneau local régulier	regular local ring	Standard.
corps résiduel	residue field	Denoted $κ (x)$ or $κ (A)$ (per SGA 1).
catégorie	category	Standard.
foncteur / foncteur dérivé	functor / derived functor	Standard.
foncteur exact / exact à gauche / à droite	exact / left exact / right exact functor	Standard.
sous-foncteur	subfunctor	Standard.
foncteur représentable	representable functor	Central to SGA 3.
représentation (d'un groupe par un foncteur)	representation	Be careful: SGA 3 uses représenter both for representable functors and for representations of groups; pick by context.
catégorie abélienne	abelian category	Standard.
limite projective / inductive	inverse / direct limit	Modern English.
fibre	fiber	American spelling.
fibre géométrique	geometric fiber	Standard. (Used heavily in Tome III for reductive groups.)
spécialisation	specialization	American spelling.
canonique	canonical	Do not translate as "natural" unless the source specifically appeals to naturality.
fonctoriel	functorial	Standard.
naturel	natural	Reserve for genuine naturality.
à isomorphisme près	up to isomorphism	Standard.
nécessaire et suffisant	necessary and sufficient	Sometimes "if and only if" in theorem statements.
si et seulement si	if and only if	"Iff" only if local idiom permits.

SGA 3 topics: foundations (Tome I)

French	English	Note
schéma en groupes	group scheme	Title-level term.
$S$ -groupe	$S$ -group	Shorthand for $S$ -schéma en groupes. Preserve.
schéma en monoïdes	monoid scheme	Standard.
schéma en anneaux	ring scheme	Standard.
structure algébrique	algebraic structure	Per Exposé I title.
cohomologie des groupes	group cohomology	Per Exposé I title.
univers (Univers)	universe	Set-theoretic universe (Grothendieck universe).
catégorie `(Ens)`	category of sets `(Ens)`	Keep the source notation `(Ens)`. Gloss only at first use.
catégorie `(Sch)`	category of schemes `(Sch)`	Keep notation. The source writes $(S c h) / S$ for $S$ -schemes.
catégorie `(Ann)`	category of rings `(Ann)`	Keep notation.
catégorie `(Gr)`	category of groups `(Gr)`	Keep notation.
`Ĉ`	`Ĉ`	Notation for $Hom (C^{\circ}, (E n s))$ . OCR may produce `Cb`; restore the hat.
$C^{\circ}$	$C^{\circ}$ (opposite category)	Keep $^{\circ}$ Unicode; OCR may give $o$ or `0`.
foncteur contravariant	contravariant functor	Standard.
topologie fpqc	fpqc topology	"Faithfully flat quasi-compact." Acronym kept; spell out on first use per Exposé.
topologie fppf	fppf topology	"Faithfully flat of finite presentation." Same convention.
topologie étale	étale topology	Standard.
topologie de Zariski	Zariski topology	Standard.
descente fidèlement plate	faithfully flat descent	Matches SGA 1 glossary.
catégorie fibrée	fibered category	American spelling.
faisceau (au sens d'une topologie)	sheaf (in the sense of a topology)	Standard.
faisceau associé	associated sheaf	Standard.
sous-foncteur représentable	representable subfunctor	Standard.
objet en groupes	group object	When the source uses objet en groupes rather than schéma en groupes.
algèbre de Hopf	Hopf algebra	Standard.
algèbre affine	affine algebra	Standard.
spectre	spectrum	Standard.
$Spec (A)$	$Spec (A)$	Keep as-is; the source sometimes writes $Spec A$ without parens.
extension infinitésimale	infinitesimal extension	Per Exposé III title.
déformation infinitésimale	infinitesimal deformation	Standard.
fibré tangent	tangent bundle	Per Exposé II title. Context may favor "tangent sheaf"; flag first occurrence.
fibré normal	normal bundle	Standard.
fibré principal	principal bundle	Standard.
algèbre de Lie	Lie algebra	Standard. $L i e (G / S)$ notation preserved.
$p$ -algèbre de Lie	$p$ -Lie algebra	Modern: restricted Lie algebra; the source's term is $p$ -algèbre de Lie. Keep it; flag with a one-time gloss.
schéma en groupes radiciel de hauteur 1	radicial group scheme of height 1	Per Intro §1. Radiciel = purely inseparable.
radiciel	radicial	Loanword (per EGA); not "radical".
opérateur différentiel	differential operator	Per Exposé VII_A title.
groupe formel	formal group	Per Exposé VII_B title.
$S$ -foncteur	$S$ -functor	A functor from $(S c h)^{\circ} / S$ to `(Ens)`. Preserve.
passage au quotient	passage to the quotient	Standard.
schéma quotient	quotient scheme	Per Exposé V title.
relation d'équivalence	equivalence relation	Standard.
groupoïde	groupoid	Standard.
sous-groupe distingué / normal	normal subgroup	Standard (English does not distinguish).
centralisateur / normalisateur	centralizer / normalizer	American spelling.
transporteur	transporter	Standard.
stabilisateur	stabilizer	American spelling.

SGA 3 topics: multiplicative-type and Lie-algebra material (Tome II)

French	English	Note
groupe diagonalisable	diagonalizable group	American spelling. Per Exposé VIII title.
groupe de type multiplicatif	group of multiplicative type	Per Exposé IX–X titles.
caractère / cocaractère	character / cocharacter	Standard.
$D (M)$	$D (M)$	The Cartier dual of the abstract group $M$ . Notation preserved.
groupe constant	constant group	Standard.
groupe tordu	twisted group	Standard.
forme tordue	twisted form	Standard.
critère de représentabilité	representability criterion	Per Exposé XI title.
tore	torus	Standard. Plural: tori.
tore maximal	maximal torus	Standard.
sous-tore	subtorus	Standard.
variété des tores maximaux	variety of maximal tori	Standard.
groupe de Weyl	Weyl group	Standard.
sous-groupe de Cartan	Cartan subgroup	Standard.
centre / centre réductif	center / reductive center	American spelling. $Z (G)$ , $r a d (G)$ , $r a d_{u} (G)$ notations preserved.
élément régulier	regular element	Per Exposés XIII–XIV titles.
élément semi-simple / unipotent	semisimple / unipotent element	"Semisimple" one word, no hyphen.
polynôme caractéristique	characteristic polynomial	Standard.
sous-groupe lisse	smooth subgroup	Standard.
groupe (algébrique) lisse	smooth (algebraic) group	Standard.
groupe de rang unipotent nul	group of zero unipotent rank	Per Exposé XVI title.
groupe algébrique unipotent	unipotent algebraic group	Per Exposé XVII title.
extension (de groupes algébriques)	extension (of algebraic groups)	Standard.
loi rationnelle	rational law	Per Exposé XVIII title (Weil's theorem on rational laws).
théorème de Weil	Weil's theorem	Standard.

SGA 3 topics: reductive groups (Tome III)

French	English	Note
groupe réductif	reductive group	Per Tome III title and Exposés XIX–XXVI.
groupe semi-simple	semisimple group	One word, no hyphen.
groupe résoluble	solvable group	American spelling.
groupe à rang semi-simple 1	group of semisimple rank 1	Per Exposé XX title.
déploiement / déployé	splitting / split	Standard.
groupe déployable	splittable group	Standard.
sous-groupe parabolique	parabolic subgroup	Per Exposé XXVI title.
sous-groupe de Borel	Borel subgroup	Standard.
sous-groupe radiciel	root subgroup	The $U_{α}$ of a reductive group.
donnée radicielle	root datum	Per Exposé XXI title. Plural: root data.
système de racines	root system	Standard.
racine	root	Standard. (Not "radical" — that is radical.)
coracine	coroot	Standard.
poids	weight	Standard.
groupe épinglé	pinned group	Per Exposé XXIII title. Épinglage = pinning; "framed group" only in passing.
épinglage	pinning	Standard (in this translation).
unicité des groupes épinglés	uniqueness of pinned groups	Per Exposé XXIII title.
automorphisme (d'un groupe réductif)	automorphism	Per Exposé XXIV title.
isogénie	isogeny	Standard.
groupe simplement connexe / adjoint	simply connected / adjoint group	Standard.
théorème d'existence	existence theorem	Per Exposé XXV title.
théorème d'unicité	uniqueness theorem	Standard.
groupes « de Tôhoku »	"Tôhoku" groups	Chevalley's groups over $Z$ . Italics on Tôhoku in journal references.
Bible (Séminaire Chevalley 1956/58)	the Bible (Chevalley Seminar 1956–58)	SGA 3 cites the Chevalley Seminar as "Bible". Render the same way.
Bourbaki	Bourbaki	Standard.
théorie de Borel-Chevalley	Borel–Chevalley theory	Use en-dash between author names.
théorie de Demazure	Demazure's theory	Standard.

Proof movement

These follow $re f ere n ces / f re n c h - ma t h - i d i o m . m d$ of the translation skill, with SGA-specific tweaks (inherited from SGA 1/SGA 2).

French	English	Note
Soit / Soient X / X et Y	Let X / X and Y be	Match singular / plural.
Supposons (que)	Suppose (that) / Assume	"Assume" if the sentence is a hypothesis register.
Posons	Set / Put	"Put" for variable assignments; "Set" for ad-hoc definitions.
Notons	Denote / Write / Note	Choose by what follows; Notons que is "Note that".
On a	We have	Not "one has" except for register.
On en déduit	We deduce	"It follows" if the inference is non-trivial.
Montrons (que)	We show (that)	"It remains to show" if it closes a reduction.
Il suffit de	It suffices to	Standard.
Il reste à montrer	It remains to show	Standard.
Il en résulte (que) / Il résulte (de)	It follows (that) / It follows (from)	Choose by structure; "hence" in compact proof prose.
Par suite	Consequently / hence	Avoid "as a result of this".
D'où	Hence / whence	"Whence" only if register supports it.
En effet	Indeed	Standard introduces a justification.
Or	Now	Pivot conjunction; "but" only if the sentence really contrasts.
Donc	Thus / hence / so	Choose by rhythm.
D'une part … d'autre part	On the one hand … on the other hand	Keep both clauses.
Cela achève la démonstration	This completes the proof	Standard.
CQFD	QED	Or "This proves the claim."
à savoir	namely	Standard.
compte tenu de	taking into account / in view of	Choose by clause weight.
en vertu de	by virtue of / by	"By" usually suffices.
moyennant	by means of / using	Standard.
quitte à	(possibly) by … / replacing X by Y if necessary	A standard French move.
à condition que	provided that	Standard.
dès que	as soon as	Standard.
chaque fois que	whenever	Standard.

Modality and certainty

Preserve modality exactly; do not upgrade certainty.

French	English
il est clair que	it is clear that
il est évident que	it is evident that
manifestement / il est manifeste que	manifestly / it is manifest that
il semble que	it seems that
on s'attend à ce que	one expects that
il est probable que	it is probable that
sans doute	doubtless / probably (context-dependent)
conjecturalement	conjecturally
il y a tout lieu de penser que	there is every reason to think that
on peut conjecturer	one may conjecture
il n'est pas exclu que	it is not excluded that
il se peut que	it may be that
vraisemblablement	presumably

Cross-volume citations

Source	English citation form
EGA I, II, III, IV (with chapter and number)	`EGA I, 1.2.3` / `EGA IV, 1.4`, etc. Preserve the chapter Roman numeral.
SGA 1 (with Exposé and number)	`SGA 1, I 5.1` / `SGA 1, VIII 1.7`, etc. Match the source's spacing.
SGA n in general	`SGA n, X y.z` per the Introduction §7 convention.
Bible = Séminaire Chevalley 1956/58	Bible (italicised) or $S \overset{e}{ˊ} minai re C h e v a ll ey 1956/58$ . Keep the source's wording.
TDTE	Grothendieck's Techniques de descente et théorèmes d'existence (Bourbaki 1959–62).
Mumford GIT	Mumford, Geometric Invariant Theory. Italics.
Demazure–Gabriel	Groupes algébriques (Demazure–Gabriel). Italics, en-dash between author names.

Common OCR repairs

The 2011 SMF re-edition is modern LaTeX, so OCR is high quality. Most math symbols survive. These items still recur:

Dropped sub/superscripts on big operators. $Γ$ , $H^{i}$ , $E x t^{i}$ , $R^{i} f_{*}$ , O_X, Lie, $T_{e} G$ , $μ_{n}$ , $G_{m}$ , $G_{a}$ , $α_{p}$ , $D (M)$ , $w α (X)$ , root operators $U_{α}$ , $T_{α}$ may lose their indices. Recover from context.
Cb instead of Ĉ. Hat on category symbol drops. Restore.
$(*)$ / $(* *)$ / $(* * *)$ for original footnote markers. Render as $[^{<} e x p os \overset{e}{ˊ} > - < sec > - < n >]$ slugs ( $[^{I} - 1 - 1]$ , $[^{I} - 1 - 2]$ , …).
(N) for N.D.E. footnote markers (small Arabic digit in parentheses). Render as $[^{N} . D . E - < e x p os \overset{e}{ˊ} > - N]$ ( $[^{N} . D . E - I - 3]$ , etc.) and translate the body of the note.
$\overset{}{¨}$ or ı (dotless i with floating diacritic) — restore to ï.
Backslash escapes surviving from LaTeX: \& → &, \# → #.
Arrows. $- \to$ → $\to$ , $7 \to$ → $\mapsto$ , $\sim=$ (often broken across lines) → $≅$ , \sim\n\to (OCR artefact) → $\sim$ or $\to$ annotated ~.
Greek subscripts. $π 1$ , H0, G m etc. should be $π_{1}$ , $H^{0}$ , $G_{m}$ (with backticks).
◦ for opposite category drops to $o$ or 0. Restore $^{\circ}$ or $^{\circ}$ (degree sign), matching local convention.
VIB, VIIA, VIIB citations. Render as VI_B, $V I I_{A}$ , $V I I_{B}$ . Preserve subscript via underscore.
« » guillemets. Translate to English quotes ( $" \dots "$ ) unless the source uses them to mark a Grothendieckian coinage, in which case keep them as quotation marks in the translation too.
Hat across line break for formal completions: $Et (b / X)$ → $\hat{E} t (\hat{X})$ . Same as SGA 2.
Math run-ons. Restore displayed equations as fenced ```text blocks.
Embedded N.D.E. footnotes. Source convention is $(N) N . D . E . :< t e x t >$ at page bottom. Recover the whole footnote (which may span lines) and translate as $[^{N} . D . E - < e x p os \overset{e}{ˊ} > - N]$ .

Style anchors (non-negotiable for cross-volume consistency)

Inherited from SGA 1 / SGA 2. Deviate only if the SGA 1/2 corpus is itself inconsistent.

File naming. 00-i-errata.md, 00-i-avertissement-introduction.md, 00-ii-preface.md, 00-iii-errata.md, NN-<slug>.md for each Exposé (using 06A, 06B, 07A, 07B for the split Exposés), glossary.md, README.md, zz-i-index-notations.md, zz-iii-index-notations.md, zz-iii-index-terminologique.md.
Heading hierarchy. # Exposé N. <English Title> (with _A / _B suffix where applicable), then numbered ## 1. <Section Title>, then sub-sections ### 1.1. <Subsection>. The Exposé heading carries a  comment on the next line.
Statement blocks. **Proposition.**, **Lemma.**, **Theorem.**, **Corollary.**, **Definition.**, **Remark.**, **Example.**, **Scholie.** on their own line. Statement numbers go inline: **Proposition 3.1.2.** *Statement…*. The label  follows on the next line.
Math. Unicode math wrapped in backticks for inline. Displayed math in fenced ```text blocks. Numbered displayed equations get .
Footnotes. Markdown $[^{s l ug}]$ syntax. Original footnotes: $[^{<} e x p os \overset{e}{ˊ} > - < sec > - < n >]$ (e.g. $[^{I} - 1 - 1]$ ). Editor footnotes: $[^{N} . D . E - < e x p os \overset{e}{ˊ} > - < n >]$ (e.g. $[^{N} . D . E - I - 3]$ ).
Page marks.  at page boundaries (page numbers from the 2011 re-edition).
Cross-references. Source's own convention: $(V I_{B}, 5.3)$ , (EGA III, 4.1.5), (SGA 1, IV 3.8), (Bible, XIV).
Bibliographies. Each Exposé's own bibliography section is preserved as ## Bibliography, entries as a numbered (or [Key]-keyed) list, journal titles in italics.

Cross-Exposé additions (discovered during translation)

Each translated Exposé file ends with a fenced HTML-comment block titled LEDGER DELTA — Exposé <N> recording new terminology choices that the translator encountered beyond what this glossary already covers. Those deltas are kept in place at the foot of each Exposé rather than copied here, both to preserve the audit trail and because consolidating the deltas (which span ~500–1,500 terms across the 26 Exposés) into a single table would make the glossary harder to scan than the per-Exposé tables. A future cleanup pass may de-duplicate the most cross-cutting entries (e.g. dévissage, the prescheme/scheme split, Demazure's épinglage → pinning, Bertin's "anti-affine" envelope) into the tables above; until then, consult the deltas in the Exposé files for the authoritative per-Exposé terminology choices.

Séminaire de Géométrie Algébrique du Bois-Marie — English