Translation conventions — EGA III

These conventions are locked after the §III.1 calibration pass. They inherit the EGA II conventions at ../ii/conventions.md 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 LaTeX 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 ../ii/conventions.md 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:

• $(0_{III},\mathrm{10.3.1})$ 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:

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 LaTeX rendering; display long expressions in $$...$$ blocks as in EGA II.

• Complexes: $K\bullet$ and $K_{\bullet }$ (lower-degree boundary) for chain complexes, $K\bullet$ and $K^{\bullet }$ (upper-degree boundary) for cochain complexes. EGA’s $K^{\bullet }$ and $K_{\bullet }$ are rendered with explicit ${}^{\bullet }$ / ${}_{\bullet }$ 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 $Tot(K)$ 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: $H^i(X,\mathcal{F})$, $H^i(\mathfrak{U},\mathcal{F})$ (open cover), $H^i(U,\mathcal{F})$ (open set).
• Derived functors: $R^{i}F$, R F (right-derived; covariant), $L_{i}F$, LF (left).
• Hyper-derived: $R^{\bullet }T(K^{\bullet })$, $L_{\bullet }T(K_{\bullet })$ for hypercohomology and hyperhomology of a complex; $R^{\bullet }T(K^{\bullet },K^{\prime \bullet })$, $L_{\bullet }T(K_{\bullet },K_{\bullet }^{\prime })$ for bifunctors.
• Higher direct images: $R^i{f}_{\ast }\mathcal{F}$, $R{f}_{\ast }\mathcal{F}$.
• Tor and Ext: $To{r}_i(A,B)$, $To{r}_i^A(M,N)$, $Ex{t}^i(A,B)$, $Ex{t}_A^i(M,N)$.
• Hypertor: $\mathcal{T}o{r}_i(K_{\bullet },K_{\bullet }^{\prime })$, $To{r}_i(M_{\bullet },N_{\bullet })$, with subscripts as in the source.
• Filtration: $F^{\bullet }$, $F_{\bullet }$, $g{r}^{\bullet }$, $g{r}_{\bullet }$. Graded object associated to a filtration: $gr(K)$.

4. Spectral sequences

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

$$E_2^{p,q}=H^p(Y,R^q{f}_{\ast }\mathcal{F})⟹H^{p+q}(X,\mathcal{F}).$$

• The page index is the subscript: E_r^{p,q} (cohomological) or E^r_{p,q} (homological).
• The differential is $d_r:E_r^{p,q}\to E_r^{p+r,q-r+1}$.
• 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

• $C^{\bullet }(\mathfrak{U},\mathcal{F})$ for the Čech cochain complex of $\mathcal{F}$ with respect to the cover $\mathfrak{U}$.
• ${\check{H}}^{\bullet }(X,\mathcal{F})$ (with caron) when EGA distinguishes Čech from derived cohomology; otherwise $H^{\bullet }(\mathfrak{U},\mathcal{F})$ for the Čech cohomology of the cover $\mathfrak{U}$.
• We never silently identify Čech and derived cohomology; preserve EGA’s bookkeeping.

6. Modules and sheaves

• ${\mathcal{O}}_X$-module: ${\mathcal{O}}_X$-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 $\dim \le n$; render as $\text{cohomological dimension}\le n$ when EGA does, and ${\dim }_X\mathcal{F}\le n$ for support otherwise.
• Filtered, graded, bigraded $\mathcal{S}$–C-module, $g{r}^{\bullet }(\mathcal{S})$-C-module: render S-C-module filtré, etc., in the form $\mathcal{S}$-$\mathcal{C}$-module filtered, $g{r}^{\bullet }(\mathcal{S})$-$\mathcal{C}$-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:

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 $\to$ formal prescheme.
• fini sur $\mathfrak{D}$, $\mathfrak{D}$-fini $\to$ finite over $\mathfrak{D}$, $\mathfrak{D}$-finite.
• propre sur $\mathfrak{D}$ $\to$ proper over $\mathfrak{D}$.
• Stein factorization, geometric fiber, geometric number of connected components: standard English terms.

9. Spectral-sequence-specific terminology

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.

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:

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