Translation conventions — EGA V

These conventions are locked after the §V.1 calibration pass. They inherit the EGA IV conventions at $docs/books/ega/iv/conventions.md$ verbatim (which in turn inherit EGA II and EGA III); only the EGA-V-specific additions for prenote character, V↔IV renumbering, and the differences in source pathology (digital-LaTeX OCR rather than scan OCR; English-source rather than French-source) are recorded here.

1. Inherited from EGA IV

The full EGA IV conventions transfer unchanged: the terminology table, math glyphs and Unicode policy, block label format (<!-- label: V.N.M.K -->), numbered display blocks in fenced ```text blocks, page-break comments (<!-- original page N -->), source-trace footer format, translator-note guidance, modality preservation, cohomology and spectral-sequence rendering, differential and formal-smoothness notation, regular-immersion and divisor vocabulary, and the citation forms for $(I,\cdots )$, $(II,\cdots )$, $(III,\cdots )$, $(IV,\cdots )$, $(0_{III},\cdots )$, $(0_{IV},\cdots )$, $(M,\cdots )$, $(G,\cdots )$, $(T,\cdots )$, $(FAC,\cdots )$, (Bourbaki, Alg. comm., …), $(Nagata,\cdots )$, $(Zariski-Samuel,\cdots )$. Re-read those sections in $docs/books/ega/iv/conventions.md$ before extending the ledger or making a stylistic choice here.

Block labels. English: $\ast \ast Proposition(1.1).\ast \ast$, $\ast \ast Definition(1.3).\ast \ast$, $\ast \ast Corollary(1.4).\ast \ast$, $\ast \ast Theorem(\mathrm{5.3.2}).\ast \ast$, $\ast \ast Lemma(\mathrm{5.3.4}).\ast \ast$, $\ast \ast Remark(1.2).\ast \ast$, $\ast \ast Scholium(5.8).\ast \ast$. Never French. Each labeled block is followed by a blank line, the <!-- label: V.N.M.K --> comment, another blank line, then the italicized body.

Proofs. EGA V inherits EGA IV's inline-prose convention: a proof begins immediately after the labeled italic body, in ordinary (non-italic) prose, without an explicit $\ast \ast Proof.\ast \ast$ marker. Long multi-step proofs may use sub-paragraph markers like "Step 1.", "(a)", or "(i)" inline; do not insert a $\ast \ast Proof.\ast \ast$ header retroactively.

2. EGA V citation form

EGA V is the never-published fifth volume; we cite it as Chapter V. The §V.N.M.K decimal numbering is used throughout.

• (V, 5.3.2) cites EGA V itself.
• (formerly IV, 20.3.2) is attached parenthetically only at the first occurrence in each section where a §V passage was originally drafted under the EGA IV numbering. After the first occurrence in a given file, the parenthetical is dropped; subsequent in-section cross-references use V numbering alone.
• (IV, 17.6.2), $(0_{IV},\mathrm{19.3.5})$, etc. cite the published EGA IV and earlier volumes as in the EGA IV conventions.
• $(Mumford,\cdots )$ cites D. Mumford, Lectures on curves on an algebraic surface (Princeton, 1966). Used in §V.5 and §V.6.
• $(BourbakiS\acute{e}m.N)$ cites Séminaire Bourbaki exposé number $N$. EGA V references Sém. 232, 236, and 261 in particular.

3. V↔IV renumbering map

Grothendieck's prenotes were drafted under the EGA IV numbering and renumbered into EGA V only after EGA IV's Part 1 appeared in 1964. The mapping from prenote section to EGA V section is:

The section opening of each translated file states this correspondence inline. Inside-the-section cross-references to "old §N" are normalized to (formerly IV, N) at first occurrence and to plain $(V,M)$ afterwards.

4. Prenote character

The Blass & Blass source preserves Grothendieck's prenote style: marginal author-to-self comments, occasional structural sketches, "would be better as Proposition" remarks, and untrimmed French residue (parentheticals tagged Fr). We preserve this character; we do not silently smooth it into clean EGA-style exposition.

Grothendieck marginal notes

Render Grothendieck's author-to-self comments as italicized translator-marked inserts. For short inline asides ("It would be a good thing to summarize the above construction into a Proposition 6"):

… defined by the vanishing of this section, whose underlying set is the right one.
*Grothendieck note: it would be a good thing to summarize the above construction into a Proposition 6.*

For longer asides (a sentence or more set off in the source), use an italicized blockquote:

> *Grothendieck note.* In the language of the fathers (which we should give as a remark) a point …

The marker $\ast Grothendiecknote\ast$ distinguishes these from the surrounding mathematical prose. Never merge them silently; that is the prenote character the reader needs.

[Tr.] translator queries

Blass occasionally inserts [Tr] or [Tr.] queries (typically uncertainties about a French phrase, an ambiguous notation, or a notational guess). Render these as numbered footnotes at the section foot, with key $[{}^v-N-k]$ where $N$ is the §V section number and $k$ is a per-section running index. The footnote body begins "Translator's note:".

… the elongated `G` is the tangent sheaf.[^v-1-1]

[^v-1-1]: Translator's note (Blass): is the elongated `G` the tangent sheaf?

When the [Tr] query is mathematically substantive (e.g. Blass asks whether a symbol is $G$ or $\mathcal{G}$), retain it in a footnote even if we settle the question in the body. When the query is purely typographic and has a clear answer from the PDF, resolve silently.

[illegible] markers

Each [illegible] instance: read the source PDF; if the symbol/word is recoverable, drop the marker silently in the body. If not, render as [illegible] inline and add a translator footnote describing the surrounding context. For example:

… homomorphism `M(φ)′` [illegible][^v-1-2] deduced from the section …

[^v-1-2]: Translator's note: the prenote source shows an indistinct mark following `M(φ)′`; possibly a prime, possibly
a tilde. The PDF does not resolve it. We follow the Vaiello edition in dropping the mark.

French residue

The Blass translation leaves French interjections at points where the prenotes' bilingual working notes surface: en termes de papa Fr, sous-entendu Fr, $p\hat{e}lem\hat{e}leFr$, confondus Fr. These are retired into idiomatic English:

• en termes de papa Fr → "in the old language" (translator footnote at first occurrence quoting the original).
• sous-entendu Fr → "tacitly assumed", "understood".
• $p\hat{e}lem\hat{e}leFr$ → "pell-mell", "jumbled together", or restructure the sentence.
• confondus Fr → "coinciding".

When the French carries information the English alone cannot (a Bourbaki-style aside, a deliberate stylistic choice), preserve the French in a translator footnote with the literal English gloss.

5. Math notation locked to EGA IV

All math notation inherits EGA IV. The Blass files render OCR fragments in non-standard forms (e.g. $\Omega 1X/k$, $mX/m2X$, $d2X/Y\varphi$, $PX/Y2$); these are normalized:

For the OCR's $\Lambda rR$ and similar Greek-Roman ambiguities, consult the PDF; in our experience the PDF resolves them as ${\Lambda }^{r}M$ (capital lambda with rank $r$ of the module $M$).

6. Source-trace footer

Each translated section ends with:

The reconciliation line is included whenever the Vaiello unified file was consulted to resolve [illegible] markers or ambiguities. The PDF entry is narrowed to the relevant fragment (e.g. EGA-V-1-2.pdf for §§V.1 and V.2.15-16, EGA-V-5.pdf for §V.5, EGA-V-6.pdf for §V.6).

7. Prescheme / scheme reconciliation

Inherit the EGA II-IV convention: préschéma → "prescheme", schéma → "scheme". The Blass translation is inconsistent ("prescheme" in some passages, "scheme" in others). We normalize: when the prenote semantically uses the 1961-1967 sense (separated/quasi-compact-versus-not distinction not yet collapsed), render "prescheme"; when the prenote uses "scheme" in a sense that survives into post-1970s terminology, render "scheme". The default for these prenotes — which were drafted in the 1962-1966 window — is "prescheme".

8. EGA-V-specific terminology

The Blass translation introduces no new French↔English terminology; the source is already English. The translation ledger at translation-ledger.md records the Blass→idiomatic-English term map (mostly disambiguations and modernizations), and a few additional terms specific to the §V.5 hyperplane-sections theory and the §V.6 invertible-sheaves theory.

9. Six-file layout

EGA V is presented as a single tree with five numbered section files plus six metadata files:

• 00-front-matter.md — title page, publication-history note, V↔IV mapping, sommaire.
• 01-ch5-01-singular-supersingular-sets.md, 02-ch5-02-jacobian-supplements.md, 03-ch5-05-hyperplane-sections-part-1.md, 04-ch5-05-hyperplane-sections-part-2.md, 05-ch5-06-invertible-sheaves-and-divisors.md — five numbered section files.
• One alphabetized terminology index, one source-ordered notation index, one bibliography.

Each translated file's <!-- source: ... --> footer points back to the Blass piecemeal source.