Éléments de géométrie algébrique

V. Elementary procedures for the construction of schemes (unpublished — prenotes)

A. Grothendieck — prenotes for an unpublished fifth volume of EGA. Translated and edited by Piotr Blass and Joseph Blass; polished and formatted here in continuity with the EGA I-IV translations at $docs/books/ega/i,ii,iii,iv/$.

The intended fifth volume was announced in 1964 alongside EGA IV Part 1 under the title Procédés élémentaires de construction de schémas (elementary procedures for the construction of schemes), with descent techniques, the method of the generic fibre, and algebraic and formal schemes. It was never published. What survives are Grothendieck's handwritten prenotes — draft notes intended for later expansion — covering four fragments of the planned volume.

Source attribution

The Blass & Blass English translations sit at

• 01-section-1-and-2-supplements.md (237 lines) — §V.1 + §§V.2.15-2.16 prenotes.
• 02-section-5-hyperplane-sections.md (2,563 lines) — §V.5 prenotes.
• 03-section-6-invertible-sheaves-divisors.md (975 lines) — §V.6 prenotes.
• 04-vaiello-unified-translation.md (4,625 lines) — Vaiello unified edition; reconciliation reference only.

Source PDFs (digital-LaTeX, not scans):

• — §V.1 and §§V.2.15-2.16.
• — §V.5.
• — §V.6.
• — Vaiello unified edition.

Unlike EGA I-IV, the PDFs are digital LaTeX output rather than scans of typeset journal pages, so OCR pathology is limited to shifted Unicode subscripts/superscripts and the occasional ambiguous glyph. Math symbols are largely intact; prose is in English.

V↔IV mapping

Grothendieck's prenotes were drafted under the EGA IV numbering and renumbered to EGA V only after EGA IV's Part 1 appeared in 1964. The §V↔§IV correspondence is:

We lead with the V numbering and attach (formerly IV, M) parenthetically at the first occurrence in each translated file; after that first occurrence, the parenthetical is dropped.

Sommaire (the four published prenote fragments)

• §V.1. Singular and supersingular zeros of a function; differential criteria.
• §§V.2.15, V.2.16. Jacobian and regularity supplements.
• §V.5. Hyperplane sections and conic projections (Bertini-Zariski theorem, connectedness, generic sections, grassmannians, linear-system formulation).
• §V.6. Invertible sheaves and divisors relative to projective fibrations; linear systems.

The 1964 sommaire announced further sections — descent techniques, method of the generic fibre, algebraic and formal schemes — but no prenotes for those sections survive. The relevant published substitutes are Grothendieck's Séminaire Bourbaki exposés 232 (analytic construction techniques), 236 (descent techniques), and 261 (scheme construction), together with Mumford's Lectures on curves on an algebraic surface and the post-EGA-IV flatness/descent literature (Raynaud-Gruson 1971; SGA 4½). None of those is translated here.

House style

These prenotes are polished into idiomatic English mathematical prose in continuity with EGA I-IV. The Blass & Blass translation is the canonical source; we retire French residue, fix grammar, modernize archaic phrasings where the mathematics is unchanged, normalize OCR'd math glyphs to Unicode, and preserve Grothendieck's marginal author-to-self comments as italicized translator-marked inserts ($\ast Grothendiecknote:\cdots \ast$ or $>\ast Grothendiecknote.\ast \cdots$). Translator queries from Blass become numbered footnotes at the section foot. [illegible] markers are resolved against the source PDFs where possible; where the PDF is genuinely unreadable we keep the marker and add a translator footnote naming the surrounding context.

See conventions.md for the full locked house style and translation-ledger.md for the running Blass→idiomatic-English term map.

Citation form within this edition

• (V, N.M.K) cites EGA V itself (this edition).
• (formerly IV, M) parenthetical at first occurrence in each file, then dropped.
• $(IV,\cdots )$, $(III,\cdots )$, $(II,\cdots )$, $(I,\cdots )$, $(0_{IV},\cdots )$, $(0_{III},\cdots )$ cite the published earlier volumes.
• $(M,\cdots )$, $(G,\cdots )$, $(T,\cdots )$, $(FAC,\cdots )$, (Bourbaki, Alg. comm., …), $(Nagata,\cdots )$, $(Zariski-Samuel,\cdots )$ are inherited classical citations (see EGA IV conventions).
• $(Mumford,\cdots )$ cites D. Mumford, Lectures on curves on an algebraic surface (Princeton, 1966).
• $(BourbakiS\acute{e}m.232/236/261)$ cites Grothendieck's Séminaire Bourbaki exposés.