Glossary and Translation Ledger — SGA 2

This file records the translation choices made for the SGA 2 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 through Exposés I–XIII. Not silently modernized.
schéma	scheme	In Exposés I–XIII, denotes a separated prescheme. In Exposé XIV, the modern meaning (= prescheme of Exp. I–XIII), per Raynaud's opening footnote.
schéma séparé (Exp. XIV)	separated scheme	Raynaud's modern terminology in Exposé XIV.
faisceau	sheaf	Standard.
faisceau (quasi-)cohérent	(quasi-)coherent sheaf	Standard.
espace annelé	ringed space	Standard.
espace topologique	topological 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.
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).
module de type fini	finitely generated module	"Of finite type" if matching adjacent SGA 1 prose.
de type fini (morphisme, présentation)	of finite type	Morphism-level terminology, EGA-standard.
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.
catégorie dérivée $D^{+} (X)$	derived category $D^{+} (X)$	Used principally in Exposé XIV.
catégorie abélienne	abelian category	Standard.
limite projective / inductive	inverse / direct limit	Modern English. Flag the SGA-era usage in a translator note on first occurrence per Exposé.
fibre	fiber	American spelling.
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 2 topics

French	English	Note
cohomologie locale	local cohomology	Title-level. Not "cohomology locally".
cohomologie à support dans Y	cohomology with supports in Y	Match the SGA 1 phrasing.
$H_{Y}^{i} (X, F)$ (sections with support)	$H_{Y}^{i} (X, F)$	Global, non-underlined.
$H_{Y}^{i} (F)$ (sheafified, underlined in source)	$H_{Y}^{i} (F)$	Use the script-H Unicode $H$ to mark the sheafified functor. Where OCR drops the underline, the surrounding French reveals which is meant.
$Γ_{Z}$ (global sections with support in $Z$ )	$Γ_{Z}$	Global, non-underlined functor.
$Γ Z$ underlined (sheafified)	$Γ_{Z}$ (sheafified) — render as $Γ Z$ or $Γ_{Z}$ with a one-line typographic note at first use per Exposé.	The translator should pin the convention at first use in each Exposé.
sorite	sorites	Loanword, kept (per SGA 1).
dévissage	dévissage	Loanword, kept.
Hartogs (phénomène de —)	Hartogs phenomenon	Standard.
profondeur	depth	Standard.
profondeur cohomologique	cohomological depth	Per source index.
profondeur étale	étale depth	Per source index.
profondeur homotopique	homotopical depth	Per source index.
profondeur homotopique rectifiée	rectified homotopical depth	Per source index.
profondeur géométrique	geometrical depth	Per source index. ("Geometrical" with -ical, as the index uses it.)
$M$ -régulier	$M$ -regular	For sequences regular on $M$ .
anneau de Cohen-Macaulay	Cohen-Macaulay ring	Standard.
théorème de comparaison	comparison theorem	Per source index.
théorème d'existence	existence theorem	Per source index.
théorème de finitude	finiteness theorem	Per source index.
théorème de dualité (locale / projective)	(local / projective) duality theorem	Per source index.
théorème d'excision	excision theorem	Per source index.
module dualisant / foncteur dualisant	dualizing module / dualizing functor	Per source index.
enveloppe injective	injective envelope	Per source index.
extension essentielle	essential extension	Per source index.
forme résidu	residue form	Per source index.
résolution injective / projective	injective / projective resolution	Standard.
résolution flasque	flasque (flabby) resolution	Use "flasque" (the SGA term) inline; "flabby" only if the local prose already uses it.
faisceau flasque	flasque sheaf	Per SGA usage.
faisceau mou / fin	soft / fine sheaf	Less common in SGA 2; if encountered, use these renderings.
suite spectrale	spectral sequence	Standard.
terme initial / aboutissement	initial term / abutment	Standard.
homomorphisme de Gysin	Gysin homomorphism	Per source index.
théorème de pureté de Zariski-Nagata	Zariski-Nagata purity theorem	Per source index.
théorème de pureté cohomologique	cohomological purity theorem	Per source index.
théorème de semi-pureté cohomologique	cohomological semi-purity theorem	Per source index.
couple parafactoriel de préschémas	parafactorial pair of preschemes	Per source index. ("Pair" preserves the couple sense.)
anneau local parafactoriel	parafactorial local ring	Per source index.
anneau local pur	pure local ring	Per source index.
couple pur de préschémas	pure pair of preschemes	Per source index.
géométriquement factoriel / parafactoriel	geometrically factorial / parafactorial	Per source index.
anneau strictement local	strictly local ring	Per source index.
condition de Lefschetz	Lefschetz condition	Per source index.
condition de Lefschetz effective	effective Lefschetz condition	Per source index.
théorème de Lefschetz affine	affine Lefschetz theorem	Per source index.
théorèmes de Lefschetz (du type —)	Lefschetz-type theorems / Lefschetz theorems	Either rendering, depending on rhythm.
groupes d'homotopie locale	local homotopy groups	Per source index.
platitude normale	normal flatness	Standard.
complétion formelle	formal completion	Standard.
complété formel (de $X$ le long de $Y$ )	formal completion (of $X$ along $Y$ )	Standard; the hat $\hat{X}$ notation is preserved.
section hyperplane	hyperplane section	Standard.
schéma algébrique projectif	projective algebraic scheme	Title-level for Exposé XII.
théorème de représentabilité de Brown	Brown's representability theorem	If cited in N.D.E. footnotes.
Tôhoku	Tôhoku	Cite as the journal/article; italicize.

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.

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; "replacing X by Y if necessary" reads best in English.
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

Common OCR repairs

These recur across the source files. Apply them locally as the translator reads each block; do not silently "correct" the mathematics — only repair what the OCR clearly mangled, and flag anything genuinely ambiguous with a  comment.

Dropped sub/superscripts on big operators. $Γ$ , $H^{i}$ , $E x t^{i}$ , $R^{i} f_{*}$ , O_X, $H^{∙}$ , $H_{∙}$ routinely lose their indices. The surrounding French sentence almost always names them ("la cohomologie locale de F le long de Y", "les $H^{i}$ de F", etc.); recover from context.
Underline loss on sheafified functors. SGA 2 distinguishes $Γ Z$ (the global section functor — non-underlined) from $Γ Z$ (the sheafified functor — underlined). The OCR routinely loses the underline. In the translation, render the sheafified functor with a script-H $H$ for the cohomology version ( $H_{Y}^{i} (F)$ ) and keep an explicit $Γ Z$ for the underlined section functor, pinning the convention at first use per Exposé with a small note.
Hat across line break. Formal-completion hats break across lines: $Et (b / X)$ is $\hat{E} t (\hat{X})$ . X b after a closing paren is $\hat{X}$ . Repair as needed.
OCR substitutions for arrows.
- $7 \to$ is $\mapsto$ .
- $- \to$ is $\to$ .
- $\sim=$ (often broken across lines) is $≅$ .
- \sim\n\to (OCR artefact, with literal newline) is $\sim$ or $\to$ annotated with ~; pick the rendering that matches what is being claimed.
OCR substitutions for accents.
- $Et (X)$ should be $\hat{E} t (X)$ (the étale site / category of étale objects).
- etale should be étale.
- Tohoku should be Tôhoku.
Greek subscripts. $π 1$ , $π 0$ , H1, H0 etc. should be $π_{1}$ , $π_{0}$ , $H^{1}$ , $H^{0}$ etc. (Unicode), inside backticks.
Math run-ons. When a displayed equation has been broken into prose by the OCR, restore it as a fenced ```text block. - Embedded N.D.E. footnotes. Source convention is $(N) N . D . E . :< t e x t >$ at the bottom of a page. Recover the whole footnote (which may span lines) and render it as $[^{N} . D . E - N]$ (or $[^{N} . D . E - < e x p ose > - < n >]$ if multiple Exposés might collide) with its English translation in the footnote body. - Dash lead-ins. Statement bodies preceded by — in the source (after Proposition X.Y., $T h \overset{e}{ˊ} or \overset{e}{ˋ} m e X . Y .$ , etc.) are rendered SGA 1-style: bold keyword on its own line, then a paragraph break, then the body.

Cross-Exposé additions (discovered during translation)

Consolidated from the per-Exposé ledger deltas appended at the foot of each translation file. Refer to that file for the original context. New cross-cutting terms encountered during translation:

French	English	Note
sous-faisceau	subsheaf	Standard.
extension par 0 (en dehors de Z)	extension by 0 (outside Z)	Standard for $j_{!}$ constructions.
famille de supports (au sens de Cartan)	family of supports (in Cartan's sense)	Kept verbatim from the source.
à support dans Z	with support in Z	Standard.
faisceautiser	"sheafify"	In quotes when the source treats it as a coinage.
par abus de langage	by abuse of language	Standard.
à valeurs dans	with values in	Standard.
aboutissement	abutment	Of a spectral sequence.
terme initial	initial term	Of a spectral sequence.
immersion canonique	canonical immersion	Standard.
partie localement fermée	locally closed subset	Standard.
suite exacte de cohomologie relative	exact sequence of relative cohomology	Standard.
$\partial$ -foncteur / $δ$ -foncteur	$\partial$ -functor / $δ$ -functor	Preserve Grothendieck's distinction.
foncteur cohomologique universel	universal cohomological functor	Standard.
système projectif essentiellement nul	essentially zero projective system	Standard SGA terminology.
opérateur bord	boundary operator	Standard.
cohomologie de Koszul / complexe de Koszul	Koszul cohomology / Koszul complex	Standard.
`Module`, `Idéal`, `Algèbre` (capitalised)	`Module`, `Ideal`, `Algebra` (capitalised)	Per SGA convention for sheaves of modules/ideals/algebras. Preserve case.
annulateur	annihilator	Standard.
assassin de M	"assassin of $M$ "	Bourbaki idiom; kept in quotes with explanatory gloss.
racine de $a$	radical of $a$	Symbol $r (a)$ preserved.
$M$ -régulier (élément, suite)	$M$ -regular (element, sequence)	Per glossary; matches the depth taxonomy.
homothétie de rapport $a$	multiplication by $a$	"Homothety of ratio" is unidiomatic in module-theoretic prose.
suite régulière	regular sequence	Standard.
anneau semi-local	semi-local ring	Standard.
codimension homologique ( $co d h_{A} M$ )	homological codimension ( $co d h_{A} M$ )	Older terminology for depth; symbol preserved.
platitude / fidèlement plat	flatness / faithfully flat	Standard.
antifiltre	antifilter	Order-theoretic loanword.
chaîne (condition des chaînes)	chain (chain condition)	Standard.
connexe en codimension $d - 1$	connected in codimension $d - 1$	Standard.
ensemblistement	set-theoretically	Standard.
intersection complète (absolue)	(absolute) complete intersection	Standard.
image directe supérieure	higher direct image	$R^{i} f_{*}$ notation preserved.
Module gradué	graded Module	Capital preserved per SGA convention.
modules gradués	graded modules	Lowercase when context is module-level, not sheaf-of-modules.
résolution injective	injective resolution	Standard.
augmentation canonique	canonical augmentation	Standard.
homotopes à zéro	homotopic to zero	Standard.
accouplement	pairing	Standard.
théorème des syzygies	syzygy theorem	Hilbert's syzygy theorem.
théorème de Cohen	Cohen's theorem	Structure theorem for complete local rings.
dimension homologique globale	global homological dimension	Standard.
système de paramètres	system of parameters	Standard.
composantes irréductibles	irreducible components	Standard.
point générique	generic point	Standard.
morphisme déduit de $f$ par passage aux complétés	morphism deduced from $f$ by passing to the completions	Standard SGA proof movement.
idéal de définition	ideal of definition	Standard.
$I^{'}$ -bonne (filtration)	$I^{'}$ -good (filtration)	EGA terminology for "good filtration".
morphisme adique	adic morphism	Standard EGA term.
condition de Mittag-Leffler uniforme	uniform Mittag-Leffler condition	Standard.
préschéma formel	formal prescheme	Standard.
anneau noethérien adique	noetherian adic ring	Standard.
séparé et complet pour la topologie $I$ -adique	separated and complete for the $I$ -adic topology	Standard.
anneau gradué associé $g r_{I}$	associated graded ring $g r_{I}$	Standard.
Module inversible	invertible Module	Capital preserved per source.
pleinement fidèle	fully faithful	Standard.
factoriel / factoriel en codimension $\geq k$	factorial / factorial in codimension $\geq k$	Standard (unique factorization).
critère de normalité de Serre	Serre's criterion of normality	Standard.
diviseur principal / localement principal	principal / locally principal divisor	Standard.
diviseur de Cartier	Cartier divisor	Standard.
anneau de valuation discrète	discrete valuation ring	Standard.
résolution libre finie	finite free resolution	Standard.
dimension cohomologique	cohomological dimension	Standard.
dimension projective (`dp`)	projective dimension (`dp`)	Auslander–Buchsbaum formula context.
base de filtre	filter base	"Decreasing filtered family" per N.D.E.
algébrisable	algebraizable	Standard formal-geometry term.
Hauptidealsatz	Hauptidealsatz	Krull's principal-ideal theorem; kept untranslated.
canularesque (Grothendieck slang)	farcical / a farce	From `canular` (hoax). Preserves the joking register.
Je dis que	I claim that	Preserves the first-person move.
"point-base"	"base-point"	In quotation marks.

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

These are inherited from SGA 1's translation; deviate only if SGA 1 itself is inconsistent.

File naming. 00-title-preface.md, 00-introduction.md, NN-<slug>.md for Exposés, glossary.md, zz-index-*.md for indexes.
Heading hierarchy. # Exposé N. <English Title>, then ## 1. <Section Title>, then ### <Subsection> if needed. The Exposé heading carries a  comment on the next line.
Statement blocks. **Proposition.**, **Lemma.**, **Theorem.**, **Corollary.**, **Definition.**, **Remark.**, **Example.** on their own line. The label  follows on the next line (after a blank line). The body begins after another blank line.
Math. Unicode math wrapped in backticks for inline. Displayed math in fenced ```text blocks. Numbered displayed equations get a  comment immediately after the closing fence.
Footnotes. Markdown [^slug] syntax. The footnote body goes at the end of the section (or end of file). Original Grothendieck-era footnotes: [^<exposenum>-<sec>-<n>] slugs (e.g. [^I-1-1]). Editor footnotes: $[^{N} . D . E - < e x p ose > - < n >]$ (e.g. $[^{N} . D . E - I V - 2]$ ).
Page marks.  on its own line at page boundaries.
Cross-references. Use the source's own convention: (VIII 2.3), (EGA III 4.1.5), (SGA 4 IV 3.8).
Bibliographies. Each Exposé's own bibliography section, at its end, is preserved as a ## Bibliography section with entries as a numbered list, journal titles in italics, with the original keys (e.g. [1], [2]) preserved as leading bracketed labels.

Séminaire de Géométrie Algébrique du Bois-Marie — English