Reductive group schemes

Six main lessons, preceded by supporting lessons: tori, centralizers, root groups, Bruhat decomposition, integral pinned classification, and forms and flag schemes, with worked examples and solved exercises.

The relative Bruhat and Schubert arguments retain their scope over every base, including the integers.

Approximation, quotient, Lie and group foundations have written proofs. Self-checked by the writing AI. The prerequisite guide identifies the required statements and their proof positions.

Reading order

  1. Algebra and sheaf cohomology before reductive groups · PDF
  2. Projective cohomology and smooth affine models · PDF
  3. Completing a smooth affine curve and extending its group action · PDF
  4. Affine descent, Zariski Main and recognition of spaces · PDF
  5. Artin approximation for polynomial equations and its desingularization proof · PDF
  6. Lie proofs for the characteristic-zero group construction · PDF
  7. Flat quotient bootstrap over an arbitrary base · PDF
  8. Supporting proofs for the consumed group-scheme foundations · PDF
  9. Tori, maximal tori and their conjugacy · PDF
  10. Regular elements and centralizers · PDF
  11. Roots and reductive groups of rank one · PDF
  12. Root data, Weyl chambers and the Bruhat decomposition · PDF
  13. Pinnings and the classification of split reductive groups · PDF
  14. Automorphisms, forms and parabolic subgroups · PDF

Reading and reuse

Human sources and rights · Supporting statements and reading order

Original contributions retain CC0 1.0. Attributed Stacks adaptations retain their GNU Free Documentation License obligations; the cumulative edition includes the full version 1.2 licence.

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