Translation ledger — EGA III

Running French↔English term ledger for the EGA III translation. Seeded from zz-index-terminologique-part-1.md, zz-index-terminologique-part-2.md, zz-index-notations-part-1.md, and zz-index-notations-part-2.md. Extends the EGA II ledger at $docs/books/ega/ii/translation-ledger.md$.

Terms inherited from EGA II

The EGA II ledger transfers unchanged: préschéma → prescheme, schéma → scheme, morphisme structural → structure morphism, morphisme affine, morphisme propre, morphisme projectif, morphisme entier, morphisme fini, morphisme quasi-fini, $universellementferm\acute{e}$, séparé, $tr\grave{e}sample$, (TF) / (TN) conditions, éclatement → blow-up, $c\hat{o}neprojetant$ → projecting cone, fermeture projective → projective closure, homogénéisation → homogenization, etc.

EGA III additions

Chapter 0_III preliminaries

Chapter III, Part 1

Chapter III, Part 2

Translator's policy notes

• "Préschéma formel" → "formal prescheme" throughout. EGA III §III.3–§III.4 develop proper morphisms of formal preschemes anticipating the full Chapter I §10 treatment.
• "Système projectif satisfaisant (ML)" stays as "projective system satisfying (ML)" rather than "inverse system" — match EGA's vocabulary.
• "Koszul complex" is the modern name for what EGA calls $complexede{l}^{\prime }alg\grave{e}breext\acute{e}rieureK\bullet (f)$. We render the EGA name with a translator's note at first use in §III.1.1.1 and use "Koszul" elsewhere only when no ambiguity is risked.
• "Caractéristique d'Euler–Poincaré" stays as Euler–Poincaré characteristic (not "Euler characteristic" alone — EGA's nomenclature is settled).
• "Suites spectrales de changement de base" → base-change spectral sequences (EGA III §III.6.9). Distinct from the base-change spectral sequences for Tor and Ext in $(0_{III},12.5)$.
• "Suites spectrales d'associativité" → associativity spectral sequences (EGA III §III.6.8); the corresponding spectral functor is called "associativity spectral functor" (foncteur spectral d'associativité).
• "Foncteur spectral" rendered as "spectral functor" (not "spectral sequence functor"); EGA's term is the family of spectral sequences attached to a parameter (the ${\mathcal{P}}_{\bullet }^{(i)}$ here).
• For §III.6.7: EGA writes ${\mathcal{H}}^{-n}(f,{\mathcal{P}}_{\bullet })$ for hypercohomology in negative-degree convention (matching $R^n{f}_{\ast }$ via $n\mapsto -n$). We preserve this sign convention literally; the hypertor abutment $\mathcal{T}o{r}_n^S$ is indexed by $n$, and the (b)-type sequence relates it to ${\mathcal{H}}^{-p}(f_1{\times }_S{f}_2,-)$.
• "Edge-homomorphisme" → edge homomorphism (no hyphen in English); cf. ledger entry at $0_{III}.12.1.7$.

Part B of §III.6 (subsections 6.7–6.9)

Source file: 13-c3-s06-foncteurs-tor-formule-kunneth.md, lines 922–1613. Output file: 14-ch3-06-tor-functors-kunneth-formula.md (§§6.7–6.9 portion of the concatenated file). PDF pages: EGA-III-2 pp. 25–39.

Subsection coverage:

• §6.7 (lines 922–1257): global hypertor of complexes of quasi-coherent modules; Künneth formula (6.7.8); the six spectral sequences (a), (a'), (b), (b'), (c), (d) of (6.7.3); finite-index extension (6.7.11); functoriality under change of $S$-preschemes (6.7.10).
• §6.8 (lines 1258–1324): associativity spectral functor ${}^{(e)}\mathcal{E}$ of (6.8.2); affine corollary (6.8.3).
• §6.9 (lines 1325–1613): base-change spectral sequences ${}^{(e)}\mathcal{E},{}^{(f)}\mathcal{E},{}^{(f^{\prime })}\mathcal{E}$ of (6.9.3); flat-$g$ reduction (6.9.2); degenerate-case isomorphisms (6.9.6); uniform $N$-bound (6.9.7); $m=1$ restatement (6.9.8); fibrewise commutation (6.9.10).

Part C of §III.6 (subsection 6.10)

Source file: 13-c3-s06-foncteurs-tor-formule-kunneth.md, lines 1615–end. Output file: 14-ch3-06-tor-functors-kunneth-formula.md (§6.10 portion of the concatenated file). PDF pages: EGA-III-2 pp. 39–43.

Subsection coverage:

• §6.10 (lines 1615–end): local structure of certain cohomological functors. Proposition (6.10.1) establishes the existence of a quasi-coherent $S$-flat complex ${\mathcal{K}}_{\bullet }$ on $Y$ with $\mathcal{T}o{r}_{\bullet }^S\stackrel{\sim }{\to }{\mathcal{H}}_{\bullet }({\mathcal{K}}_{\bullet }{\otimes }_S{\mathcal{F}}_{\bullet }^{\prime })$ and the base-change-functorial diagram (6.10.1.2). Corollary (6.10.2) gives boundedness improvements. Remarks (6.10.3) record the spectral sequence (e), the homotopy non-uniqueness of ${\mathcal{K}}_{\bullet }$, and the free / finite-cover refinement. Scholium (6.10.4) explains why hypertor is the right tool when flatness fails. Theorem (6.10.5) is the noetherian proper-morphism specialization ($n=1$, $Y=S$, ${\mathcal{P}}_{\bullet }$ $Y$-flat coherent): the complex ${\mathcal{L}}_{\bullet }$ may be taken associated to free $A$-modules of finite type. Remark (6.10.6) records the single-coherent-sheaf case and asks the converse question about realizing a given complex of projectives.

Translator's policy notes — Part C

• ${\mathcal{K}}_{\bullet }$ and ${\mathcal{L}}_{\bullet }$: EGA uses cursive script-K and script-L for the two complexes appearing in (6.10.1) and (6.10.5). We render them as ${\mathcal{K}}_{\bullet }$ and ${\mathcal{L}}_{\bullet }$ respectively, matching the script-X conventions established in Parts A and B.
• ${\mathcal{H}}_{\bullet }({\mathcal{K}}_{\bullet }{\otimes }_S{\mathcal{F}}_{\bullet }^{\prime })$ denotes the homology sheaves of the chain complex of ${\mathcal{O}}_Y$-modules ${\mathcal{K}}_{\bullet }{\otimes }_S{\mathcal{F}}_{\bullet }^{\prime }$, taken termwise. This is not a hypercohomology of a functor; it is the homology of a complex of sheaves. The locked cohomology notation ${\mathcal{H}}^{\bullet }(f,\cdots )$ (hyperderived-functor of $f_{\ast }$) is reserved for the LHS of (6.10.5.1), matching Part B's sign-convention discussion at the bottom of the policy notes.
• (6.10.3.1) interprets the spectral sequence (e) of (6.9.3) in the present flat-${\mathcal{K}}_{\bullet }$ setting. EGA writes the E_2 term using a hyperhomology-of-${\mathcal{F}}_{\bullet }^{\prime }$ slot; when ${\mathcal{F}}_{\bullet }^{\prime }$ is concentrated in degree 0 this collapses to $\mathcal{T}o{r}_p^S({\mathcal{H}}_q({\mathcal{K}}_{\bullet }),{\mathcal{F}}^{\prime })$. We keep the general form ${\oplus }_{q^{\prime }+q^{\prime \prime }=q}\mathcal{T}o{r}_p^S({\mathcal{H}}_{q^{\prime }}({\mathcal{K}}_{\bullet }),{\mathcal{H}}_{q^{\prime \prime }}({\mathcal{F}}_{\bullet }^{\prime }))$.
• (0, 11.5.2.1) "dualized": EGA's literal "« dualisée »". We preserve the quoted hint as in the source.