Translation ledger — EGA V

Running Blass→idiomatic-English term ledger for the EGA V polish. Unlike EGA I-IV (which are translated from French OCR), the Blass & Blass source is already in English; this ledger records the few terms whose Blass rendering needed modernization, French residue retired, or notational standardization.

Extends $d ocs / b oo k s / e g a / i v / t r an s l a t i o n - l e d g er . m d$ verbatim. Most vocabulary inherits unchanged from EGA IV.

Inherited from EGA IV

The EGA IV ledger transfers without modification. In particular: préschéma → prescheme, schéma → scheme, the differential and formal-smoothness vocabulary $(Ω_{B / A}, Der, F, p - ba s i s, Υ, \dots)$ , the depth and regularity machinery (prof, coprof, dim, codim, Cohen-Macaulay, …), the regular-immersion and divisor vocabulary (immersion régulière, immersion transversalement régulière, 𝓜_X, 𝓜_X^×, Pic(X), …), the étale-morphisms vocabulary, the cohomology and spectral-sequence vocabulary, and the citation forms for $(M, \dots)$ , $(G, \dots)$ , $(T, \dots)$ , $(F A C, \dots)$ , (Bourbaki, Alg. comm., …), $(N a g a t a, \dots)$ , $(Z a r i s ki - S am u e l, \dots)$ .

The Bourbaki, Mumford, and Seidenberg references introduced in §V.5 and §V.6 are recorded below.

§V.1 — Singular and supersingular zeros (Blass prenote, formerly EGA IV §16)

Blass / OCR	Idiomatic English	Reference	Note
singular zero (or root) of $ϕ$	singular zero of $ϕ$	V.1.1	"root" dropped; "zero" alone is unambiguous in scheme theory.
supersingular zero	supersingular zero	V.1.1	Term preserved.
ordinary singular zero	ordinary singular zero	V.1.1	Used as the contrast to "supersingular".
non-singular zero	non-singular zero	V.1.2	Equivalent to "smooth zero" / "regular zero of $V (ϕ)$ ".
geometrically singular zero of $ϕ$ relative to $k$	geometrically singular zero (relative to $k$ )	V.1.3	The "relative to $k$ " qualifier preserved.
geometrically supersingular zero	geometrically supersingular zero	V.1.3
$V (ϕ) s in g$	$V (ϕ)_{s in g}$	V.1.5	The closed-set notation.
$V (ϕ) sup s in g$	$V (ϕ)_{s u p s in g}$	V.1.7	Or $V (ϕ)_{s u p s in g}$ if disambiguation is needed.
$d 0 ϕ$ , $d 1 ϕ$ , $d 2 ϕ$	$d^{0} ϕ$ , $d^{1} ϕ$ , $d^{2} ϕ$	V.1.5	Unicode superscripts; matches EGA IV §0_IV.20.
principal parts of order 1	principal parts of order 1	V.1	Inherited from EGA IV §0_IV.20.
$PX / Y 2$	$P_{X / Y}^{2}$	V.1.5	Script 𝒫 for principal-part bundle; matches EGA IV.
$g r 1 (PX / Y 1)$	$g r^{1} (P_{X / Y}^{1})$	V.1.5
$M (ϕ)$	$M (ϕ)$	V.1.5	Quadratic form on the cotangent space.
$D (ϕ)$	$D (ϕ)$	V.1.5	Determinant of $M (ϕ)$ .
sub-prescheme of zeros of $ψ$	subprescheme of zeros of $ψ$	V.1.5	Hyphenation modernized (no hyphen in "subprescheme"; matches EGA II).
$R am (V^{'} / Y)$	$R am (V^{'} / Y)$	V.1.7	Ramification subprescheme.
`en termes de papa` (Fr)	"in the old language"	V.1.7	First occurrence: translator footnote quotes the French verbatim.
`sous-entendu` (Fr)	"tacitly assumed", "understood"	V.1.5	Used to mark which auxiliary scheme is implicit.
`confondus` (Fr)	"coinciding"	V.1.7	Used in describing "infinitely near singular points" picture.

§V.2 supplements — §§V.2.15-2.16 (formerly EGA IV §§17.15-17.16)

Blass / OCR	Idiomatic English	Reference	Note
Jacobian criterion	Jacobian criterion	V.2.15	Reference to §0_IV.22.6 and §IV.17.
regular at a point	regular at a point	V.2.15	Inherited from §0_IV.17.
critical point (of $ϕ$ )	critical point (of $ϕ$ )	V.2.15	Synonym for "singular zero" in the smooth case.
$Ω_{X / k}^{1}$ ( $x$ )	$Ω_{X / k}^{1} (x)$	V.2.15	Stalk-at- $x$ notation.

§V.5 — Hyperplane sections and conic projections (formerly EGA IV §20)

Blass / OCR	Idiomatic English	Reference	Note
hyperplane section	hyperplane section	V.5.1
generic hyperplane section	generic hyperplane section	V.5.2
conic projection	conic projection	V.5.9	The projection from a sub-linear-space.
Bertini-Zariski theorem	Bertini-Zariski theorem	V.5.3	Inherited terminology; both names retained.
Seidenberg-style theorem	Seidenberg-style theorem	V.5.6	Connectedness/irreducibility of generic sections.
linear system	linear system	V.5.16	Subscheme-of-`Hilb`/`Pic` interpretation in §V.5.16.
projective bundle, projective fibration	projective bundle, projective fibration	V.5.1	Both terms used; "fibration" emphasises the morphism structure.
Grassmannian	Grassmannian	V.5.10	$G r a s s_{k} (V)$ or $G (k, n)$ ; preserve EGA-style.
$Λ$ , $Π$	$Λ$ , $Π$	V.5	Greek capitals for linear subspaces of $P$ .
$P_{X}^{n}$ , $P_{S}^{n}$	$P_{X}^{n}$ , $P_{S}^{n}$	V.5.1	Relative projective space.
$p \overset{e}{^} l e m \overset{e}{^} l e$ (Fr)	"pell-mell", or restructure	V.5.1	Used in Grothendieck's commentary; translator footnote at first occurrence.

§V.6 — Invertible sheaves and divisors; linear systems (formerly EGA IV §21)

Blass / OCR	Idiomatic English	Reference	Note
relative divisor	relative divisor	V.6.2
relative effective Cartier divisor	relative effective Cartier divisor	V.6.2	Term locked.
Picard group of a projective fibration	Picard group of a projective fibration	V.6.1
$Pic (X / S)$	$Pic (X / S)$	V.6.1	Relative Picard functor / scheme.
invertible sheaf	invertible sheaf	V.6.1	Inherited from EGA II/III/IV.
linear system of divisors	linear system of divisors	V.6.5	Relative version.
$O (1)$ , $O_{P} (1)$	$O (1)$ , $O_{P} (1)$	V.6.1	Standard EGA II form.
multiprojective fibration	multiprojective fibration	V.6.2

Standing abbreviations introduced in EGA V

Key	Work
$(M u m f or d, \dots)$	D. Mumford, Lectures on curves on an algebraic surface (Princeton Math. Notes, 1966).
$(Vai e ll o, \dots)$	Used in source-trace footers only; the Vaiello unified edition is a reconciliation reference, not a citable work in prose.
$(B o u r baki S \overset{e}{ˊ} m .232)$	A. Grothendieck, Techniques de construction en géométrie analytique (Sém. Bourbaki, exposé 232, 1961-62).
$(B o u r baki S \overset{e}{ˊ} m .236)$	A. Grothendieck, Techniques de descente et théorèmes d'existence en géométrie algébrique (Sém. Bourbaki, exposé 236, 1959-60+).
$(B o u r baki S \overset{e}{ˊ} m .261)$	A. Grothendieck, Techniques de construction de schémas (Sém. Bourbaki, exposé 261, 1960-61).

These extend the EGA IV standing-abbreviation table. The full bibliographic entries appear in bibliography.md.

Éléments de Géométrie Algébrique — English