Translation conventions — EGA IV

These conventions are locked after the §IV.1 calibration pass. They inherit the EGA III conventions at $docs/books/ega/iii/conventions.md$ verbatim (which in turn inherit EGA II); only the EGA-IV-specific additions for commutative algebra, formal smoothness, derivations and differentials, Jacobson preschemes, étale morphisms, regular immersions, meromorphic functions, and divisors are recorded here.

1. Inherited from EGA III

The full EGA III conventions transfer unchanged: the terminology table, math glyphs and Unicode policy, block label format (<!-- label: IV.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, Čech cohomology bookkeeping, condition (ML) terminology, formal-prescheme vocabulary, and the citation forms for $(I,\cdots )$, $(II,\cdots )$, $(III,\cdots )$, $(0,\cdots )$, $(0_{III},\cdots )$, $(M,\cdots )$, $(G,\cdots )$, $(T,\cdots )$, $(FAC,\cdots )$. Re-read those sections in $docs/books/ega/iii/conventions.md$ before extending the ledger or making a stylistic choice here.

Block labels. English: $\ast \ast Proposition(\mathrm{1.1.4}).\ast \ast$, $\ast \ast Definition(\mathrm{1.2.1}).\ast \ast$, $\ast \ast Corollary(\mathrm{1.2.3}).\ast \ast$, $\ast \ast Theorem(\mathrm{1.8.4})(Chevalley).\ast \ast$, $\ast \ast Lemma(\mathrm{3.2.1}).\ast \ast$, $\ast \ast Remark(\mathrm{1.6.4}).\ast \ast$, $\ast \ast Scholium(\mathrm{2.3.3}).\ast \ast$. Never French ("Définition", "Corollaire", etc.). Each labeled block is followed by a blank line, the <!-- label: IV.N.M.K --> comment, another blank line, then the italicized body.

Proofs. EGA IV follows the 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. (This is a deliberate divergence from EGA III's explicit-marker convention; it matches the typographical style of the printed EGA IV, where short proofs are not always marked "Démonstration." in the source.) 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. Cross-volume citations specific to EGA IV

EGA IV continues the running Chapter 0 from EGA III at §14 — call it Chapter 0_IV — and is cited by later volumes (and within itself) as Chapter IV. To keep the two distinguishable in print we write them differently:

• $(0_{IV},\mathrm{19.3.5})$ cites a paragraph in 06-ch0-19-formally-smooth-algebras.md (Chap 0_IV §19).
• (IV, 17.6.2) cites EGA IV itself (used by later EGA volumes and by cross-section references inside EGA IV).
• $(0_{III},\mathrm{13.4.2})$ cites Chap 0_III in EGA III.
• (III, 2.3.8), (II, 5.5.4), (I, 4.2.3), (0, 4.2.4) cite EGA III, EGA II, EGA I, and Chapter 0 of EGA I, respectively.

EGA IV cites several external classics. We extend the EGA III table:

Bourbaki citations are rendered in the EGA form (Bourbaki, Alg. comm., chap. II, §3, n° 4) or shorter (Bourbaki, Alg. comm., II, §3, n° 4) when that matches the source. Page numbers are kept when present; chapter/section/n° structure is preserved verbatim. Where EGA spells out the title (e.g. "Bourbaki, Algèbre commutative, chap. II, §3, n° 4"), we keep the spelling and add the bracketed key in the bibliography.

The Nagata text Local Rings (Interscience, 1962) is cited as $(Nagata,\cdots )$ with chapter and section number. Zariski- Samuel Commutative Algebra (Van Nostrand, 1958-1960) is cited as $(Zariski-Samuel,\cdots )$ with volume, chapter, section.

3. Differential and formal-smoothness notation

EGA IV §§0_IV.19-22 and §§IV.16-18 work systematically with derivations, differentials, and the family of formal- smoothness conditions. We fix the following Unicode rendering; display long expressions in fenced ```text blocks.

• Module of relative differentials (rings): ${\Omega }_{B/A}$. EGA writes ${\Omega }_{B/A}^1$; we render as ${\Omega }_{B/A}$ when no higher differentials are in play and as ${\Omega }_{B/A}^1$ when the source needs the index for disambiguation.
• Higher exterior powers: ${\Omega }_{B/A}^p={\Lambda }^p{\Omega }_{B/A}$.
• Module of relative differentials (schemes): ${\Omega }_{X/Y}^1$ (or ${\Omega }_{X/Y}$ when no index is needed); locally on $\mathrm{Spec}(B)\to \mathrm{Spec}(A)$ this is ${\Omega }_{B/A}^1$ glued.
• Universal derivation: $d:B\to {\Omega }_{B/A}$, $d_{B/A}$ when the source/target needs naming.
• Derivation modules: ${\mathrm{Der}}_A(B,M)$, $\mathrm{Der}(B,M)$ (when $A$ is understood); EGA's $T_{B/A}(M)$ notation is preserved when used.
• Frobenius: $F:A\to A$, $a\mapsto a^p$ (characteristic $p>0$).
• $p$-basis: a family $(b_i)$ whose images in $A/A^p$ form a basis (EGA's "$p$-base"). We render $p$-base → "$p$-basis".
• Imperfection module: ${{\rm Y}}_{B/A}$ (EGA's "module d'imperfection"; pronounce "upsilon" or render literally).
• Differential criteria for smoothness, étaleness, unramifiedness: EGA's (D_I), $(D_{II})$, $(D_{III})$ letter-pair labels are preserved literally.

4. Formal smoothness / étaleness / unramifiedness

EGA IV §0_IV.19 and §IV.17 introduce the family of "formally $P$" properties ($P\in smooth,\acute{e}tale,unramified$). We fix the terminology as follows:

Where EGA writes "formellement lisse pour la topologie $\mathcal{J}$-préadique", we render "formally smooth for the $\mathcal{J}$-preadic topology". The qualifier ("for the discrete topology", "for the $J$-adic topology", etc.) is kept; do not silently drop it. EGA's stylistic device of inserting the topology as a parenthetical adverb is preserved when it occurs.

5. Dimension, depth, regularity

EGA IV §§0_IV.14-17 fix the combinatorial-dimension and depth machinery. We fix:

• Combinatorial / Krull dimension of a topological space: $\dim (X)$.
• Local dimension at a point: ${\dim }_x(X)$.
• Codimension of $Y$ in $X$ at a point $y$: $codi{m}_y(Y,X)$.
• Depth (profondeur): $prof(M)$, $pro{f}_{\mathfrak{m}}(M)$. We render the EGA symbol prof rather than English depth because the symbol is the entry in the notation index; the term "depth" appears in English running prose.
• Regular system of parameters: $(t_1,\cdots ,t_n)$ with $(t_1,\cdots ,t_n)\cdot A=\mathfrak{m}$ and $n=\dim (A)$.
• Cohen-Macaulay: EGA's "anneau de Cohen-Macaulay" → "Cohen-Macaulay ring"; the abbreviation (CM) is preserved where used.
• Regular: EGA's "régulier" → "regular".
• $M$-regular sequence ($suiteM-r\acute{e}guli\grave{e}re$): a sequence $(f_1,\cdots ,f_n)$ such that each $f_i$ is a non-zero-divisor on $M/(f_1,\cdots ,f_{i-1})M$.
• $\mathcal{F}$-regular sequence ($suite\mathcal{F}-r\acute{e}guli\grave{e}re$): the sheaf-of-modules version; rendered with the script $\mathcal{F}$.

6. Jacobson preschemes and constructibility (§§IV.9-10)

• Jacobson prescheme: EGA's "préschéma de Jacobson"; we render literally and capitalize "Jacobson" throughout.
• Jacobson condition: (J), parenthesized as in EGA.
• Very dense subset: EGA's "partie très dense"; we render literally.
• Constructible function/set: inherited from $(0_{III},\mathrm{9.1.}x)$ rendering.

7. Étale morphisms, henselian rings (§§IV.17-18)

• Étale morphism: as above. The EGA family-of-properties is rendered exactly: étale, étale at a point, locally étale, …
• Henselian local ring: EGA's "anneau local hensélien" → "Henselian local ring" (capitalize Hensel).
• Strict Henselization: EGA's "hensélisation stricte" → "strict Henselization".
• Hensel's lemma: EGA's "lemme de Hensel" → "Hensel's lemma".
• Étale cover (revêtement étale): "étale cover" (EGA III ledger; reinforced here).

8. Regular immersions, divisors (§§IV.19, IV.21)

• Regular immersion of codimension $n$: EGA's "immersion régulière"; we render literally. Definition at $(0_{IV},15.x)$.
• Transversally regular immersion: EGA's "immersion transversalement régulière"; locked at §IV.19.1.
• Sheaf of meromorphic functions: ${\mathcal{M}}_X$, ${\mathcal{M}}_X^{\times }$. (Distinct from ${\mathcal{M}}_X$, which EGA uses for ideal sheaves; check context.)
• Cartier divisor: EGA's "diviseur de Cartier"; locked at §IV.21.1.
• Weil divisor: EGA's "diviseur de Weil"; locked at §IV.21.6.
• Picard group: $\mathrm{Pic}(X)$.
• Locally principal divisor / locally factorial scheme: render literally.

9. Excellent rings (§IV.7)

EGA IV §IV.7 introduces "excellent rings" with a long list of formal properties. The term is preserved verbatim: "excellent ring", "anneau excellent". Sub-properties:

• Japanese ring / Nagata ring: EGA's "anneau japonais"; we render literally (matches §0_IV.23). EGA's vocabulary is older than Nagata's "pseudo-geometric ring" / Matsumura's "Nagata ring"; we keep "Japanese ring" with a one-line translator's note at first appearance.
• Universally Japanese: EGA's "universellement japonais"; locked at §0_IV.23.1.
• (G)-ring, (G)-property: EGA's (G) for "Grothendieck"; preserved literally in (IV, 7.x).

10. EGA-IV-specific terminology

These extend the EGA III terminology tables; they first appear in the §IV.1 calibration and the Chap 0_IV preliminaries. Locked piecewise as each section lands. The current table is in translation-ledger.md.

11. Four-part packaging

The 1964 first part, 1965 second part, 1966 third part, and 1967 fourth part are translated in one repository tree. We keep:

• four front-matter files, one per part (00-front-matter-part-1.md, 13-front-matter-part-2.md, 20-front-matter-part-3.md, 28-front-matter-part-4.md),
• one merged terminology index, alphabetized, with chapter prefix where ambiguous (0_IV vs IV),
• one merged notation index, source-ordered with subheadings for Chap 0 (continued) and Chap IV Parts 1-4,
• one merged bibliography.

The merged back matter is the reader's surface. Each translated file's $<!--source:\cdots -->$ footer still points to its original Part 1, 2, 3, or 4 OCR file.

12. §IV.11

EGA IV Part 3 (1966) prints §§8, 9, 10, 11, 12, 13, 14, 15 — including §11. The 1964 sommaire announced §11 as "Topological properties of flat morphisms of finite presentation; local criteria of flatness" with the footnote that the order and content of §§11-21 are tentative; the published §11 followed that announcement. (Older secondary literature occasionally treats §11 as unpublished; this is incorrect.) The §11 file lives at 23a-ch4-11-flatness-loci-and-descent.md in the translation tree, ordered between §10 and §12. Subsequent flatness-descent developments that build on §11 include Raynaud-Gruson, Critères de platitude et de projectivité (Inventiones math. 13, 1971), and the étale-cohomology projects of SGA 4½.

13. Source-trace footer

Each translated section ends with:

When no LaTeX cross-reference exists, the $cross-ref:$ line is omitted. The PDF entry is narrowed to the relevant part (e.g. EGA-IV-1.pdf for Part 1 sections).