Human sources and rights

Freely accessible sources provide material for writing the programme proofs. Every used result requires its actual proof here or in an earlier programme lesson. Attributed Stacks adaptations retain their GNU FDL terms; original contributions retain their stated component terms.

Stacks project authors; AI Integrated Stacks Project edition

Stacks project authors; AI Integrated Stacks Project edition. 565b10e987aba5969b21145a0833f42d69f96790.

Licence: GFDL-1.2-only

Licence evidence: https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/565b10e987aba5969b21145a0833f42d69f96790/COPYING

Tag 0ABS: complete statement and proof, quasi-finite separated algebraic spaces over an affine scheme are quasi-affine
Tag 09N9: complete affine-open curve lemma and proof

Linear algebraic groups

Tom De Medts. Academic year 2025–2026, preface January 2023.

Licence: CC-BY-NC-SA-4.0 for the licensed contributions

Licence evidence: PDF p. 6, printed p. vi; https://creativecommons.org/licenses/by-nc-sa/4.0/

Theorem 10.2.5 and direct proof: PDF pp. 119–121
Theorem 10.2.4: PDF pp. 118–119, proof omitted
Theorem 10.3.2 and Proposition 10.3.4: PDF p. 124, projectivity proof omitted, conjugacy proof depends on it
Definitions 11.2.1–11.2.3 and diagonal-torus example: PDF pp. 131–132

18.745 Lie Groups and Lie Algebras I

Pavel Etingof; MIT OpenCourseWare. Fall 2020.

Licence: CC-BY-NC-SA-4.0

Licence evidence: https://ocw.mit.edu/pages/privacy-and-terms-of-use/

Lecture 21, Definitions 21.1–21.2, Proposition 21.3 and Example 21.4: PDF p. 1
Lecture 22, Theorem 22.5: PDF pp. 1–2; Theorem 22.14 and Proposition 22.15: PDF pp. 3–4

18.755 Lie Groups and Lie Algebras II

Pavel Etingof; MIT OpenCourseWare. Spring 2024.

Licence: CC-BY-NC-SA-4.0

Licence evidence: https://ocw.mit.edu/pages/privacy-and-terms-of-use/

Lecture 17, Definition 43.11 and Example 43.12: PDF p. 5, complex reductive Lie groups and GL_n
Lecture 18, Corollary 44.2 and Theorem 44.3: PDF p. 1, compact/complex torus conjugacy
Lecture 25, §§51.4–51.5 and Example 51.8: PDF pp. 3–4, complex parabolics and block matrices
Lecture 25, Theorems 51.17–51.18 and Exercise 51.19: PDF pp. 6–7; Bruhat proof is an exercise sketch, not a complete written proof

Algebraic Groups

J. S. Milne. Version 2.00, 20 December 2015, freely accessible author notes.

Licence: Copyright 2014–2015 J. S. Milne; PDF p. 2 permits single paper copies for noncommercial personal use, not general adaptation/translation.

Theorem 18.14, PDF p. 316: geometric maximal-torus conjugacy
Theorem 18.44 and proof, PDF p. 324: connected torus centralizers
Theorem 18.65, PDF p. 329: general field torus-existence proof deferred to the final version
Theorem 21.44, PDF p. 371: rank-one projective-line quotient
Theorems 22.67 and 22.72, PDF pp. 402–403: field Bruhat cells
Theorem 22.94, PDF pp. 410–411: field root-datum isomorphism; Proposition 22.103 on PDF p. 413 is unproved in this draft
Theorem 22.85, PDF p. 407: field standard-parabolic classification
Chapter 14, Theorem 14.17: character classification in this exact free version; semilinear Hopf descent is proved directly in AG-RG-S08 G.5.1–G.5.3.

Reductive group schemes

Brian Conrad. 2014, Panoramas et Synthèses 42–43.

Licence: Public author-posted PDF; no general copying/translation grant established in the checked version.

Sections read include §§2.1–2.3, 3.2, rank-one/root-data sections, pinned classification and automorphisms.

Invariantentheorie, Osnabrück 2012–2013

Holger Brenner and Wikiversity contributors. .

Licence: CC-BY-SA-4.0 stated in the Wikiversity footer; contributor history is required for an adaptation.

Lecture 29 definitions of regular actions and matrix invariants; lead says Lectures 29–32 include linear reductivity. The later proofs were not checked.

Brian Conrad

Brian Conrad. 2014 published text.

Checked scope: Exact statement comparison for the named course theorem; no whole-reference reading or adaptation claim.

Michel Demazure; SGA 3 editors P. Gille and P. Polo

Michel Demazure; SGA 3 editors P. Gille and P. Polo. 13 October 2024 re-edition.

Checked scope: Exact statement comparison for the named course theorem; no whole-reference reading or adaptation claim.

Michel Demazure; SGA 3 editors P. Gille and P. Polo

Michel Demazure; SGA 3 editors P. Gille and P. Polo. 13 October 2024 re-edition.

Checked scope: Exact statement comparison for the named course theorem; no whole-reference reading or adaptation claim.

Michel Demazure; SGA 3 editors P. Gille and P. Polo

Michel Demazure; SGA 3 editors P. Gille and P. Polo. 13 October 2024 re-edition.

Checked scope: Exact statement comparison for the named course theorem; no whole-reference reading or adaptation claim.

Michel Demazure; SGA 3 editors P. Gille and P. Polo

Michel Demazure; SGA 3 editors P. Gille and P. Polo. 13 October 2024 re-edition.

Checked scope: Exact statement comparison for the named course theorem; no whole-reference reading or adaptation claim.

Michel Demazure; SGA 3 editors P. Gille and P. Polo

Michel Demazure; SGA 3 editors P. Gille and P. Polo. 13 October 2024 re-edition.

Checked scope: Exact statement comparison for the named course theorem; no whole-reference reading or adaptation claim.

Brauer groups, separable splitting fields and Galois descent

GPT-6.1 Sol (OpenAI); AG-ET links and citations revised by Claude Opus 5.5 (Anthropic). Read 4 October 2026 (commit 88d1416).

Licence: CC0-1.0

Licence evidence: Original text dedication in the actual public source and supporting readers

The exact projective-matrix classification proof and the two supporting splitting/descent proofs were read. The full Brauer-group computation and unrelated Tsen theorem are not consumed.

Certains schémas de groupes semi-simples

Claude Chevalley. Séminaire Bourbaki, exposé 219, May 1961, pp. 219–234; Numdam archive.

Licence: Reference only; the consulted archive reserves rights. No source prose or illustration is reproduced.

Reading qualification: The direct normalized longest-cell argument in lesson four, Proposition 8.2, is independently written. This record credits the mathematical construction and does not confer a reuse licence on the reference.

Section 4, Proposition 1 and its exterior-power construction, printed pp. 226–229; checked by the internal proof reader.

Smoothing Ring Maps: Néron–Popescu desingularization and Artin approximation

The Stacks Project authors; AI Integrated Stacks fork maintainers. Mathematical work of Michael Artin, Dorin Popescu, Richard G. Swan, Tetsushi Ogoma and notes of Michel André.. 565b10e987aba5969b21145a0833f42d69f96790.

Licence: GFDL-1.2-only

Licence evidence: https://github.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/blob/565b10e987aba5969b21145a0833f42d69f96790/COPYING

Reading qualification: Freely accessible licensed writing material.

Complete polynomial approximation proof for local G-rings and pointed étale neighbourhoods
Complete critical desingularization chain, including inseparable residue fields; explicit elementary source corrections are supplied in lesson one
Permanence and ubiquity of G-rings in More on Algebra

Noetherian models with geometrically connected fibres and descent of smoothness

The Stacks Project authors. Tags 05FI and 0C0C, read 4 October 2026.

Licence: GNU Free Documentation License

Licence evidence: https://github.com/stacks/stacks-project/blob/master/COPYING

Reading qualification: The supporting lesson Projective cohomology and smooth affine models contains the reconstructed model and smoothness arguments.

Complete connected-fibre model proof, Tag 05FI
Descent of smoothness, Tag 0C0C; used with affine finite-presentation approximation

Noetherian universal cohomology complexes and formal functions

The Stacks Project authors. 29a05ca0ba1328874b7aae53301e09c9bbbbaf7b.

Licence: GFDL-1.2-only

Licence evidence: https://raw.githubusercontent.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/29a05ca0ba1328874b7aae53301e09c9bbbbaf7b/COPYING

Reading qualification: The supporting lessons Algebra and sheaf cohomology before reductive groups and Projective cohomology and smooth affine models contain the required cohomological arguments, including formal functions and curve duality.

Perfection and arbitrary base change, Tags 0DJQ and 0A1D
Formal functions, with degree-zero application to the projective quotient proof

Lie Algebras, Algebraic Groups, and Lie Groups

. 2.00, 5 May 2013 (title page).

Checked scope: Actual comparison for the newly written AG-RG-S06 proof route; source citations do not discharge programme proofs.

Lie Groups and Lie Algebras, full MIT course notes

. MIT OpenCourseWare, Fall 2020.

Checked scope: Actual comparison for the newly written AG-RG-S06 proof route; source citations do not discharge programme proofs.

General flat quotient: units, dimensions, regular cuts, descent, slicing and finite subgroupoids

Stacks Project authors. a04446e57ec1fbc252a871afcec7752fb2807b14.

Licence: GFDL-1.2-or-later

Licence evidence: https://github.com/stacks/stacks-project/blob/a04446e57ec1fbc252a871afcec7752fb2807b14/COPYING

04MK pointed field case,04L5 units,00P4 point dimension,00RE polynomial coordinates,00RH Cohen–Macaulay locus,046Y regular cuts,0489 slicing,04RI finite parts,04RU subgroupoid,04S6 general flat quotient

Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field

J. S. Milne. Corrected author text, 5 October 2021, published 2022.

Licence: Author copyright retained; free reading access is verified and does not infer adaptation or translation permission.

Relevant contents: Chapter 10 Lie algebras, 11c Cartier duality, Chapter 12 characters and multiplicative type. Specific mathematical proof uses remain separately checked.

Perfect trace pairings and flat-fibre smoothness

The Stacks Project authors. Tags 0BJF and 0BVH, read 6 October 2026; Tag 00TF, read 7 October 2026.

Licence: GNU Free Documentation License

Licence evidence: https://github.com/stacks/stacks-project/blob/master/COPYING

Reading qualification: Freely accessible comparison material; the lesson contains the complete proofs of the results it uses.

Full trace/discriminant construction and finite locally free etale criterion, Tags 0BVH and 0BJF
Full arbitrary-base flat-fibre smoothness statement and proof, Tag 00TF

Symmetric algebras and presentations of algebras

The Stacks Project authors. Tags 07M6 and 07CD, read 7 October 2026.

Licence: GNU Free Documentation License

Licence evidence: https://github.com/stacks/stacks-project/blob/master/COPYING

Reading qualification: Freely accessible comparison. The full argument used by the course is written in the lesson; this reference does not replace a proof.

Full arbitrary-ring symmetric-algebra smoothness statement and proof; the consumed symmetric-algebra identities in the presentation argument.

Local complete intersection maps: presentation independence

The Stacks Project authors. Tag 07CZ, read 7 October 2026.

Licence: GNU Free Documentation License

Licence evidence: https://github.com/stacks/stacks-project/blob/master/COPYING

Reading qualification: The freely accessible source is writing material. The lesson contains the complete used argument and does not replace it with this citation.

Full arbitrary-ring presentation-independence statement and proof

Full source and version record · Component licences and expression boundaries · Unaltered GNU FDL 1.2