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

Wrap exact mathematical strings in backticks. Use Unicode, never LaTeX.

• 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 Hom, Spec, Proj, $\Gamma$, Sym, det, Pic, End, Aut in upright Roman.

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

• Greek letters as plain Unicode in backticks: $\varphi$, $\psi$, $\rho$, $\sigma$, $\tau$, $\theta$, $\omega$, $\Delta$, $\Gamma$, $\Omega$, $\Sigma$.

• Display equations: a fenced ```text block, contents indented two spaces:

  ```text
      Δ_{X/Y} : X → X ×_Y X
  ```
  

  matching the SGA I model.

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

Proofs follow immediately as a $\ast \ast Proof.\ast \ast$ paragraph; no $\ast \ast Proof.\ast \ast$ 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)

• $SoientX,Y\cdots$ → "Let $X$, $Y$ …"
• On a → "We have"
• Posons → "Set"
• Démontrons → "We show"
• Montrons → "We show"
• Il suffit de → "It suffices to"
• $Celar\acute{e}sulteaussit\hat{o}tde\cdots$ → "This follows immediately from …"
• $Celar\acute{e}sultede\cdots$ → "This follows from …"
• $D^{\prime }apr\grave{e}s\cdots$ → "By …"
• $Envertude\cdots$ → "By virtue of …" or "By …" depending on register
• $Comptetenude\cdots$ → "Taking … into account"
• Réciproquement → "Conversely"
• On notera que → "Note that"
• On dit que → "We say that" (in definitions); "One says that" only when the source is being deliberately impersonal in a way that matters
• $Ilestimm\acute{e}diatque$ → "It is immediate that"
• $Toutrevient\grave{a}\cdots$ → "It comes down to …"
• D'autre part → "On the other hand"
• En particulier → "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 $>{}_{Translator}^{\prime }snote{.}_{\cdots }$ 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 → "it seems"
• $on{s}^{\prime }attend\grave{a}ceque$ → "one expects that"
• conjecturalement → "conjecturally"
• vraisemblablement → "presumably"
• manifestement → "manifestly"
• évidemment → "obviously"
• clairement → "clearly"

Do not collapse these into a single English register.