Translation conventions — EGA II

These conventions are locked. Changes after §1 require a deliberate update here, the translation ledger, and a sweep of already-translated sections.

1. Terminology

2. Mathematical glyphs

All mathematics is written in LaTeX and rendered with KaTeX: inline math in $...$, displayed math in $$...$$. Do not use raw Unicode mathematical symbols (script/Fraktur/Greek letters, arrows, relations, operators) in running text — write the LaTeX command instead.

• Structure sheaves and standard sheaves: ${\mathcal{O}}_X$, ${\mathcal{O}}_Y$, ${\mathcal{O}}_S$, ${\mathcal{O}}_X(1)$, $\mathcal{O}(n)$, $\mathcal{M}$, $\mathcal{F}$, $\mathcal{G}$, $\mathcal{S}$, $\mathcal{A}$, $\mathcal{B}$, $\mathcal{I}$, ${\mathcal{I}}_n$, $\mathcal{R}$, $\mathcal{J}$, $\mathcal{N}$.

• Ideals in rings: $\mathfrak{m}$, $\mathfrak{n}$, $\mathfrak{p}$, $\mathfrak{q}$, $\mathfrak{a}$, $\mathfrak{b}$, $\mathfrak{d}$, $\mathfrak{J}$ for ideal-sheaf locals.

• Maps: $\to$, $\mapsto$, $\hookrightarrow$, $\twoheadrightarrow$, $\cong$, $\simeq$, $\subset$, $\subseteq$, $\supset$, $\supseteq$, $\in$, $\ni$.

• Operators: $\otimes$, $\oplus$, $\prod$, $\coprod$, $\cap$, $\cup$, $\bigcap$, $\bigcup$, $\perp$.

• Functors: write $\mathrm{Hom}$, $\mathrm{Spec}$, $\mathrm{Proj}$, $\Gamma$, $\mathrm{Sym}$, $\det$, $\mathrm{Pic}$, $\mathrm{End}$, $\mathrm{Aut}$ — use \operatorname{...} so they render upright.

• Specs: $\mathrm{Spec}(A)$, $\mathrm{Proj}(S)$, $\mathrm{Spec}(\mathcal{S})$ for the relative Spec, $\mathrm{Proj}(\mathcal{S})$ similarly.

• Greek letters as LaTeX commands: $\varphi$, $\psi$, $\rho$, $\sigma$, $\tau$, $\theta$, $\omega$, $\Delta$, $\Gamma$, $\Omega$, $\Sigma$.

• Display equations: a $$...$$ block, e.g.

  $${\Delta }_{X/Y}:X\to X{\times }_{Y}X$$

3. Block labels

Use bold-label, blank line, HTML label comment, blank line, body:

**Proposition.**

<!-- label: II.1.2.4 -->

Every `S`-prescheme that is affine over `S` is separated over `S` (in other words, it is
an `S`-scheme).

Available labels: Theorem., Proposition., Lemma., Corollary., Definition., Example., Remark., Notation., Reminder.. The HTML label uses the form II.N.M.K (volume prefix II, then EGA’s decimal address).

Proofs follow immediately as a Proof. paragraph; no Proof. HTML label needed unless the proof is itself a numbered display.

4. Numbered displays and cross-references

EGA’s parenthesized identifiers (1.1.2.1) are kept exactly. When the original prose writes “Setting (8.1.1.4)”, we render

$$\mathcal{S}={\oplus }_{n\ge 0}{\mathcal{I}}_n.(\mathrm{8.1.1.4})$$

with the tag right-aligned in plain text (no special markup; the right-alignment is informational only). Inline citations stay literal: (I, 4.2.3), (0, 4.2.4), (III, 2.3.8), and within the chapter just (1.2.4) or (5.5.10).

5. Pagination

<!-- original page N --> at every page boundary in the IHÉS print (pages 5–222). The break should appear at the prose break, not mid-sentence; if a sentence straddles a page, place the comment after the period of the sentence containing the break, and note the page change in the comment if useful.

6. Proof idioms (French → English)

• Soient X, Y … $\to$ “Let $X$, $Y$ …”
• On a $\to$ “We have”
• Posons $\to$ “Set”
• Démontrons $\to$ “We show”
• Montrons $\to$ “We show”
• Il suffit de $\to$ “It suffices to”
• Cela résulte aussitôt de … $\to$ “This follows immediately from …”
• Cela résulte de … $\to$ “This follows from …”
• D'après … $\to$ “By …”
• En vertu de … $\to$ “By virtue of …” or “By …” depending on register
• Compte tenu de … $\to$ “Taking … into account”
• Réciproquement $\to$ “Conversely”
• On notera que $\to$ “Note that”
• On dit que $\to$ “We say that” (in definitions); “One says that” only when the source is being deliberately impersonal in a way that matters
• Il est immédiat que $\to$ “It is immediate that”
• Tout revient à … $\to$ “It comes down to …”
• D'autre part $\to$ “On the other hand”
• En particulier $\to$ “In particular”

Keep long Grothendieck sentences long when the chain of dependencies is doing mathematical work. Split only when an English reader genuinely loses the antecedent.

7. Source-trace footer

Each translated section ends with:

8. Translator notes

A short > _Translator's note._ … quoted block, used only when:

• A 1961 term differs from current usage and a reader could misread the math.
• The OCR was unsalvageable and a small reconstruction was needed.
• The existing LaTeX cross-reference disagrees with the French and a choice had to be made.

Do not silently modernize. Do not interpolate exposition.

9. Modality

Preserve modal weight:

• il semble $\to$ “it seems”
• on s'attend à ce que $\to$ “one expects that”
• conjecturalement $\to$ “conjecturally”
• vraisemblablement $\to$ “presumably”
• manifestement $\to$ “manifestly”
• évidemment $\to$ “obviously”
• clairement $\to$ “clearly”

Do not collapse these into a single English register.