Preface to Tome II

A note from the translator.

Tome II of the 2011 SMF re-edition of SGA 3 contains Exposés VIII through XVIII, with no separate front matter, errata sheet, or notation index of its own — the editors chose to consolidate the volume-level apparatus into Tomes I and III only. Reading-order navigation for the Tome II Exposés can be found in the README; references to the Avertissement, Introduction, and reference convention all live with Tome I's front matter; and the translation glossary applies uniformly across the three Tomes. The Tome I errata sheet flags Exposé II only (§ 1.2), and the Tome III errata sheet does not concern this Tome.

Topically, Tome II spans three threads. Exposés VIII–XII (Grothendieck) develop the theory of groups of multiplicative type — diagonalizable groups (VIII), homomorphisms between groups of multiplicative type (IX), characterization and classification (X), representability criteria (XI), and the foundational structure theorems on maximal tori, the Weyl group, Cartan subgroups, and the reductive center (XII), which together provide the language used throughout Tome III. Exposés XIII–XV (Grothendieck, with Serre's appendix in XIV, and Raynaud in XV) develop the theory of regular elements in algebraic groups and Lie algebras and apply it to the construction and parametrization of maximal tori, in particular in smooth group schemes over a non-perfect field. Exposés XVI–XVIII close the Tome with Raynaud's treatment of groups of zero unipotent rank and of unipotent algebraic groups and their extensions (XVI–XVII), and Artin's writing-up of Weil's theorem on the construction of a group from a rational law (XVIII).

Several of the Exposés in this Tome were inserted into the mimeographed Séminaire without a corresponding oral presentation, as Grothendieck notes in §5 of the Introduction: VII_A and VII_B from Tome I, and XV, XVI, XVII from this Tome, may be read as natural extensions of the techniques developed in the surrounding material rather than as steps in the main line; the reader heading for the reductive-group material of Tome III need not pause on them.

The Exposés in this Tome are listed below; for the full reading order across all three Tomes, see the README.

• Exposé VIII — Diagonalizable groups (Grothendieck)
• Exposé IX — Groups of multiplicative type: homomorphisms (Grothendieck)
• Exposé X — Characterization and classification of groups of multiplicative type (Grothendieck)
• Exposé XI — Representability criteria (Grothendieck)
• Exposé XII — Maximal tori, the Weyl group, Cartan subgroups, the reductive center (Grothendieck)
• Exposé XIII — Regular elements: algebraic groups and Lie algebras (Grothendieck)
• Exposé XIV — Regular elements: continuation, applications (Grothendieck, app. Serre)
• Exposé XV — Complements on sub-tori. Application to smooth groups (Raynaud)
• Exposé XVI — Groups of zero unipotent rank (Raynaud)
• Exposé XVII — Unipotent algebraic groups; extensions (Raynaud)
• Exposé XVIII — Weil's theorem on the construction of a group from a rational law (Artin)