Translation conventions — EGA III

These conventions are locked after the §III.1 calibration pass. They inherit the EGA II conventions at $d ocs / b oo k s / e g a / ii / co n v e n t i o n s . m d$ verbatim; only the EGA-III-specific additions for cohomology, derived functors, spectral sequences, and the running Chapter 0_III preliminaries are recorded here.

1. Inherited from EGA II

Terminology table (§1), math glyphs and Unicode policy (§2), block labels (§3), numbered displays and cross-references (§4), pagination comments (§5), proof idioms (§6), source-trace footer format (§7), translator-note guidance (§8), and modality preservation (§9) all transfer unchanged. Re-read those sections in $d ocs / b oo k s / e g a / ii / co n v e n t i o n s . m d$ before extending the ledger or making a stylistic choice here.

2. Cross-volume citations specific to EGA III

EGA III uses the running Chapter 0 (suite) — call it Chapter 0_III — and cites the preliminaries of EGA I as plain "Chapter 0". To keep the two distinguishable in print we write them differently:

$(0_{III}, 10.3.1)$ cites a paragraph in 03-ch0-10-complements-flat-modules.md (Chap 0_III §10).
(0, 4.2.4) cites Chapter 0 in EGA I.
(I, 4.2.3), (II, 5.5.4) cite EGA I, EGA II respectively.
(III, 2.3.8) cites EGA III itself; later volumes use this form.

EGA III also routinely cites four external classics, which we render literally:

Source key in EGA III	Work
$(M, \dots)$	H. Cartan and S. Eilenberg, Homological Algebra (Princeton, 1956).
$(G, \dots)$	R. Godement, Topologie algébrique et théorie des faisceaux (Hermann, 1958).
$(T, \dots)$	A. Grothendieck, Sur quelques points d'algèbre homologique (Tôhoku Math. J., 1957).
$(F A C, \dots)$	J.-P. Serre, Faisceaux algébriques cohérents (Annals of Math., 1955).

Where the source spells these out (e.g. "Cartan–Eilenberg, loc. cit."), we keep the spelling and add the bracketed key in the bibliography.

3. Complex and cohomology notation

EGA III routinely manipulates chain and cochain complexes, bicomplexes, total complexes, derived and hyper-derived functors. We fix the following Unicode rendering; display long expressions in fenced ```text blocks as in EGA II.

Complexes: $K ∙$ and $K_{∙}$ (lower-degree boundary) for chain complexes, $K ∙$ and $K^{∙}$ (upper-degree boundary) for cochain complexes. EGA's $K^{∙}$ and $K_{∙}$ are rendered with explicit $^{∙}$ / $_{∙}$ whenever the context could confuse them.
Bicomplexes: $K^{p, q}$ , $K_{p, q}$ .
Total complex: $s (K)$ (EGA's preferred notation; matches Cartan–Eilenberg); occasionally $T o t (K)$ when the source uses that.
Cocycles, coboundaries, cohomology: $Z^{i} (K)$ , $B^{i} (K)$ , $H^{i} (K)$ , with matching lower-indexed $Z_{i}$ , $B_{i}$ , $H_{i}$ for chain complexes.
Cohomology of a sheaf or module: $H^{i} (X, F)$ , $H^{i} (U, F)$ (open cover), $H^{i} (U, F)$ (open set).
Derived functors: $R^{i} F$ , R F (right-derived; covariant), $L_{i} F$ , LF (left).
Hyper-derived: $R^{∙} T (K^{∙})$ , $L_{∙} T (K_{∙})$ for hypercohomology and hyperhomology of a complex; $R^{∙} T (K^{∙}, K^{' ∙})$ , $L_{∙} T (K_{∙}, K_{∙}^{'})$ for bifunctors.
Higher direct images: $R^{i} f_{*} F$ , $R f_{*} F$ .
Tor and Ext: $T o r_{i} (A, B)$ , $T o r_{i}^{A} (M, N)$ , $E x t^{i} (A, B)$ , $E x t_{A}^{i} (M, N)$ .
Hypertor: $T o r_{i} (K_{∙}, K_{∙}^{'})$ , $T o r_{i} (M_{∙}, N_{∙})$ , with subscripts as in the source.
Filtration: $F^{∙}$ , $F_{∙}$ , $g r^{∙}$ , $g r_{∙}$ . Graded object associated to a filtration: $g r (K)$ .

4. Spectral sequences

EGA III's central objects. We render every spectral sequence in the canonical "page, indices, abutment" form:

```text
  E_2^{p,q} = H^p(Y, R^q f_* ℱ) ⟹ H^{p+q}(X, ℱ).


- The page index is the subscript: `E_r^{p,q}` (cohomological) or `E^r_{p,q}` (homological).
- The differential is `d_r : E_r^{p,q} → E_r^{p+r, q-r+1}`.
- Abutment uses `⟹` ("abuts to" / "converges to" — both English forms occur in EGA; we render either by `⟹` when the
  source uses `⟹` or the equivalent French "aboutit à"; we keep "abuts to" in prose since it's the canonical English
  form).
- For filtered complexes we name the filtration: `F^p H^n` is the `p`th piece of the filtration on `H^n` induced by the
  filtration on the complex.
- Convergence properties: `weakly convergent`, `regular`, `coregular`, `biregular`, `degenerate` — match
  `(0_III, 11.1.3)` and `(0_III, 11.1.6)`.

## 5. Cech cohomology

- `C^•(𝔘, ℱ)` for the Čech cochain complex of `ℱ` with respect to the cover `𝔘`.
- `Ȟ^•(X, ℱ)` (with caron) when EGA distinguishes Čech from derived cohomology; otherwise `H^•(𝔘, ℱ)` for the Čech
  cohomology of the cover `𝔘`.
- We never silently identify Čech and derived cohomology; preserve EGA's bookkeeping.

## 6. Modules and sheaves

- `𝒪_X`-module: `𝒪_X`-module (lowercase per EGA II convention; the type comes from the prefix).
- Quasi-coherent / coherent: as in EGA II.
- A module on a topological space `X` whose support is at most `n`-dimensional: EGA writes `dim ≤ n`; render as
  `cohomological dimension ≤ n` when EGA does, and `dim_X ℱ ≤ n` for support otherwise.
- Filtered, graded, bigraded `𝒮`–C-module, `gr^•(𝒮)`-C-module: render `S`-`C`-module filtré, etc., in the form
  `𝒮`-`𝒞`-module filtered, `gr^•(𝒮)`-`𝒞`-module graded — following the EGA convention of attaching the module species
  to the ambient structure.

## 7. The Mittag–Leffler condition

EGA III §0_III.13 introduces condition `(ML)` for projective systems. We keep the EGA abbreviation. Related vocabulary:

| French                                     | English                                   |
| ------------------------------------------ | ----------------------------------------- |
| condition `(ML)`                           | condition (ML) / Mittag–Leffler condition |
| système projectif strict                   | strict projective system                  |
| système projectif essentiellement constant | essentially constant projective system    |
| objet des images universelles              | object of universal images                |

## 8. Formal preschemes (forward references)

EGA III §III.3 and §III.4 use formal preschemes (`préschémas formels`) and properness relative to a formal base,
anticipating EGA I, ch. I §10 (formal preschemes) and forthcoming material. Render:

- `préschéma formel` → formal prescheme.
- `fini sur 𝔇`, `𝔇-fini` → finite over `𝔇`, `𝔇`-finite.
- `propre sur 𝔇` → proper over `𝔇`.
- Stein factorization, geometric fiber, geometric number of connected components: standard English terms.

## 9. Spectral-sequence-specific terminology

| French                                 | English                             |
| -------------------------------------- | ----------------------------------- |
| aboutissement                          | abutment                            |
| bicomplexe                             | bicomplex                           |
| caractéristique d'Euler–Poincaré       | Euler–Poincaré characteristic       |
| co-séparé, co-discrète                 | co-separated, co-discrete           |
| cohomologiquement plat                 | cohomologically flat                |
| complexe défini par un bicomplexe      | complex defined by a bicomplex      |
| cup-produit                            | cup product                         |
| cochaîne bi-alternée                   | bi-alternating cochain              |
| filtration co-discrète                 | co-discrete filtration              |
| hypercohomologie                       | hypercohomology                     |
| hyperhomologie                         | hyperhomology                       |
| polynôme de Hilbert                    | Hilbert polynomial                  |
| résolution de Cartan–Eilenberg         | Cartan–Eilenberg resolution         |
| résolution cohomologique               | cohomological resolution            |
| résolution droite / gauche             | right resolution / left resolution  |
| résolution injective / projective      | injective / projective resolution   |
| résolution libre / plate               | free / flat resolution              |
| résolution homologique                 | homological resolution              |
| suite spectrale                        | spectral sequence                   |
| suite spectrale dégénérée              | degenerate spectral sequence        |
| suite spectrale faiblement convergente | weakly convergent spectral sequence |
| suite spectrale régulière              | regular spectral sequence           |
| système de coefficients                | system of coefficients              |

## 10. EGA-III-specific terminology

These extend the EGA II terminology table; they first appear in the §III.1 calibration and the Chap 0_III preliminaries.

| French                                             | English                                        |
| -------------------------------------------------- | ---------------------------------------------- |
| algébrisable (`𝒪_X`-Module)                        | algebraizable (`𝒪_X`-module)                   |
| algébrisable (schéma formel)                       | algebraizable (formal scheme)                  |
| analytiquement intègre                             | analytically integral                          |
| application quasi-compacte                         | quasi-compact map                              |
| augmentation d'une résolution                      | augmentation of a resolution                   |
| complexe de l'algèbre extérieure                   | exterior algebra complex                       |
| complexe de Koszul                                 | Koszul complex                                 |
| condition (TF), condition (TN)                     | condition (TF), condition (TN)                 |
| constructible (partie, ensemble)                   | constructible (subset, set)                    |
| constructible (fonction)                           | constructible (function)                       |
| dihomomorphisme                                    | di-homomorphism                                |
| exact (sous-ensemble) dans une catégorie abélienne | exact (subset) in an abelian category          |
| factorisation de Stein                             | Stein factorization                            |
| filtration                                         | filtration                                     |
| fini (morphisme de préschémas formels)             | finite (morphism of formal preschemes)         |
| foncteur représentable                             | representable functor                          |
| foncteur covariant canonique `C → Hom(C°, Ens)`    | canonical covariant functor `C → Hom(C°, Set)` |
| genre arithmétique                                 | arithmetic genus                               |
| géométriquement connexe                            | geometrically connected                        |
| localement constructible                           | locally constructible                          |
| loi de composition externe / interne               | external / internal composition law            |
| morphisme de suites spectrales                     | morphism of spectral sequences                 |
| nombre géométrique de composantes connexes         | geometric number of connected components       |
| `C`-objet en groupes                               | `C`-object in groups                           |
| `C`-groupe, `C`-anneau, `C`-module                 | `C`-group, `C`-ring, `C`-module                |
| objet final d'une catégorie                        | final object of a category                     |
| partie propre (sur 𝔇)                              | proper part (over `𝔇`)                         |
| pleine (sous-catégorie)                            | full (subcategory)                             |
| pleinement fidèle (foncteur)                       | fully faithful (functor)                       |
| polynôme de Hilbert                                | Hilbert polynomial                             |
| propre (morphisme de préschémas formels)           | proper (morphism of formal preschemes)         |
| représentable (foncteur)                           | representable (functor)                        |
| rétrocompact                                       | retrocompact                                   |
| unibranche (anneau, point)                         | unibranch (ring, point)                        |
| universellement ouvert                             | universally open                               |

## 11. Two-part packaging

The 1961 first part and the 1963 second part are translated in one repository tree. We keep:

- two front-matter files, one per part (`00-front-matter-part-1.md`, `13-front-matter-part-2.md`),
- one merged terminology index, alphabetized,
- one merged notation index, source-ordered with subheadings for Chap 0_III, Chap III Part 1, and Chap III Part 2,
- one merged bibliography.

The merged back matter is the reader's surface. Each translated file's `<!-- source: … -->` footer still points to its
original Part 1 or Part 2 OCR file.

## 12. Source-trace footer

Each translated section ends with:

```html

When no LaTeX cross-reference exists (e.g. for §III.6 and §III.7), the $cross - re f :$ line is omitted.

Éléments de Géométrie Algébrique — English

Translation conventions — EGA III

1. Inherited from EGA II

2. Cross-volume citations specific to EGA III

3. Complex and cohomology notation

4. Spectral sequences