v0.5 Supporting Units — Comprehensive Scaffold
The v0.5 apex units (Strands A, C, D shipped by claude; B, E by gpt-codex) name many concepts they don't formally define — "divisor", "Picard group", "root system", "Weyl group", "Hodge decomposition", "theta function", "Verma module", and many more. This document scaffolds the supporting units beneath those apexes, anchoring each on the best modern textbook and citing the original paper(s) where the concept was conceived. The pedagogical goal is to convey not only the modern formalism but the intelligence of the geniuses who first made these ideas precise.
Read alongside docs/plans/CURRICULUM_V0_5_PLAN.md (apex strand plan), docs/catalogs/FASTTRACK_BOOKLIST.md (master booklist), docs/catalogs/CONCEPT_CATALOG.md (registered concepts), docs/specs/QUALITY_RUBRIC.md (the validator gate). Every unit follows the Phase 3.2 LM-prose template (three-tier, 27/27 validator).
1. Source-pairing principle
Each supporting unit cites two reference layers:
- Original-author tier — the paper(s) that introduced the concept. Where readable in 2026 (Riemann's dissertations, Cartan's 1894 thesis, Hodge's 1941 monograph, Serre's FAC), Master-tier prose mines the original prose for conceptual phrasing. Where the original is opaque (Killing's 1888-90, Grothendieck's EGA), modern translations do the work.
- Modern-textbook tier — the canonical contemporary reference, ideally drawn from the Fast Track booklist (Hartshorne, Forster, Donaldson, Griffiths-Harris, Voisin, Fulton-Harris, Humphreys, Bourbaki, Knapp, Mumford). Intermediate-tier prose synthesises these.
Where the Fast Track lists the original-author text directly (e.g., Serre Complex Semisimple Lie Algebras, Mumford Curves and their Jacobians, Donaldson Riemann Surfaces), it serves both layers.
Genius-prose principle. Master-tier sections of supporting units quote or paraphrase the original conception. For Riemann's genus, the prose channels Riemann's 1857 Theorie der Abelschen Functionen. For Cartan's classification, the prose sits in Cartan's 1894 dissertation and Killing's earlier (often-criticised) papers. The reader leaves understanding why these ideas seemed inevitable to the people who first saw them.
2. Strand A — Algebraic Geometry (22 supporting units)
Apex shipped (4): 04.01.01 Sheaf, 04.02.01 Scheme, 04.03.01 Sheaf cohomology, 04.04.01 Riemann-Roch theorem for curves.
Chapter expansion: existing 04.01–04.04 plus new 04.05 (Divisors and line bundles), 04.06 (Coherent sheaves), 04.07 (Projective geometry), 04.08 (Differentials and duality), 04.09 (Hodge theory), 04.10 (Moduli).
04.01 Sheaves (3 supporting)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 04.01.02 | Stalk of a sheaf | Leray 1946 (in prison camp); Cartan seminar 1948–54 | Hartshorne II.1; Vakil §2.4; Bredon Ch I |
| 04.01.03 | Sheafification | Cartan-Serre 1948–55 | Hartshorne II.1; Vakil §2.4 |
| 04.01.04 | Direct and inverse image of sheaves | Grothendieck Tôhoku 1957 | Hartshorne II.1; Vakil §2.6; Iversen Cohomology of Sheaves |
04.02 Schemes (3 supporting)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 04.02.02 | Affine scheme Spec(R) | Grothendieck-Dieudonné EGA I, 1960 | Hartshorne II.2; Vakil §3–5 |
| 04.02.03 | Projective scheme Proj(S) | Grothendieck EGA II, 1961 | Hartshorne II.2, II.5; Vakil §4–5 |
| 04.02.04 | Morphism of schemes | Grothendieck EGA I–IV | Hartshorne II.3; Vakil §6–9 |
04.05 Divisors and line bundles (5 supporting; new chapter)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 04.05.01 | Weil divisor on a scheme | André Weil Foundations of Algebraic Geometry, 1946 | Hartshorne II.6; Vakil §14 |
| 04.05.02 | Picard group | Émile Picard 1895 (analytic); Grothendieck (algebraic, 1962) | Hartshorne II.6; Vakil §14; Mumford Abelian Varieties |
| 04.05.03 | Line bundle / invertible sheaf | Cartan-Serre FAC 1955 | Hartshorne II.6; Vakil §14 |
| 04.05.04 | Cartier divisor | Pierre Cartier 1957–58 | Hartshorne II.6; Vakil §14 |
| 04.05.05 | Ample and very ample line bundle | Serre 1955; Grothendieck EGA II 1961 | Hartshorne II.7; Vakil §15–16; Lazarsfeld Positivity |
04.06 Coherent sheaves (2 supporting; new chapter)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 04.06.01 | Quasi-coherent sheaf | Serre FAC 1955 | Hartshorne II.5; Vakil §13 |
| 04.06.02 | Coherent sheaf | Cartan-Serre coherence theorems 1953; Serre FAC 1955 | Hartshorne II.5; Vakil §13; Cartan seminar exposés |
04.07 Projective geometry (2 supporting; new chapter)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 04.07.01 | Projective space ℙⁿ | Grassmann (1844 nineteenth-century synthetic), Grothendieck (modern functorial) | Hartshorne II.4–5; Vakil §4; Eisenbud-Harris 3264 and All That |
| 04.07.02 | Blowup | Zariski 1944; Hironaka 1964 (resolution of singularities) | Hartshorne II.7; Vakil §22; Kollár Lectures on Resolution of Singularities |
04.08 Differentials and duality (3 supporting; new chapter)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 04.08.01 | Sheaf of differentials Ω¹_{X/Y} | Grothendieck EGA IV, 1967 | Hartshorne II.8; Vakil §21 |
| 04.08.02 | Canonical sheaf ω_X | Implicit in Riemann (1857); formalised by Grothendieck | Hartshorne II.8, III.7; Vakil §21 |
| 04.08.03 | Serre duality | Serre 1955 Un théorème de dualité | Hartshorne III.7; Vakil §30; Hartshorne Residues and Duality |
04.09 Hodge theory (2 supporting; new chapter)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 04.09.01 | Hodge decomposition (Kähler manifolds) | W. V. D. Hodge 1941 Theory and Applications of Harmonic Integrals | Voisin Vol I Ch 6 (Fast Track 3.27); Griffiths-Harris Ch 0; Wells Differential Analysis on Complex Manifolds |
| 04.09.02 | Kodaira vanishing theorem | Kunihiko Kodaira 1953 | Voisin Vol I; Griffiths-Harris Ch 1; Lazarsfeld Positivity |
04.10 Moduli (2 supporting; new chapter)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 04.10.01 | Moduli space of curves M_g | Riemann 1857 (idea, dim = 3g−3); Mumford 1965 (rigorous via GIT) | Harris-Morrison Moduli of Curves (Fast Track 3.30); Mumford-Fogarty-Kirwan |
| 04.10.02 | Geometric invariant theory (GIT) | David Mumford GIT, 1965 | Mumford-Fogarty-Kirwan GIT (Fast Track 3.31); Newstead Lectures on Introduction to Moduli Problems and Orbit Spaces; Thomas Notes on GIT |
3. Strand C — Riemann Surfaces & Complex Analysis (18 supporting units)
Apex shipped (3): 06.01.01 Holomorphic function, 06.03.01 Riemann surface, 06.04.01 Riemann-Roch for compact Riemann surfaces.
Chapter expansion: existing 06.01, 06.03, 06.04 plus new 06.02 (Branch points), 06.05 (Divisors and bundles on RS), 06.06 (Jacobians and abelian differentials), 06.07 (Several complex variables).
06.01 Complex analysis (5 supporting)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 06.01.02 | Cauchy integral formula | Cauchy 1825 Mémoire sur les intégrales définies prises entre des limites imaginaires | Ahlfors Complex Analysis §4 (Fast Track 1.04); Stein-Shakarchi Vol II §2 |
| 06.01.03 | Residue theorem | Cauchy 1826 Mémoire sur les Intégrales définies | Ahlfors §4–5; Stein-Shakarchi Vol II §3 |
| 06.01.04 | Analytic continuation | Weierstrass 1841– (lectures published 1894) | Ahlfors §8; Conway Ch IX |
| 06.01.05 | Meromorphic function | Riemann 1851 Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse | Ahlfors §4; Forster §1 |
| 06.01.06 | Riemann mapping theorem | Riemann 1851 (with Dirichlet principle); Koebe 1907 (rigorous proof) | Ahlfors §6; Stein-Shakarchi Vol II §8; Conway Ch VII |
06.02 Branch points (1 supporting; new chapter)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 06.02.01 | Branch point and ramification | Riemann 1851 dissertation | Forster §1; Donaldson Ch 1 (Fast Track 1.07); Miranda Ch II |
06.03 Riemann surfaces (2 supporting)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 06.03.02 | Genus of a Riemann surface | Riemann 1857 Theorie der Abelschen Functionen | Forster §16; Donaldson Ch 3; Farkas-Kra Ch I |
| 06.03.03 | Uniformization theorem | Koebe 1907; Poincaré 1907 | Forster §IV–V; Donaldson Ch 6 (Fast Track 1.07); Ahlfors-Sario Riemann Surfaces (Fast Track 1.06) |
06.05 Divisors and bundles on Riemann surfaces (3 supporting; new chapter)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 06.05.01 | Divisor on a Riemann surface | Riemann 1857 (implicit); Klein 1882 (explicit) | Forster §16; Miranda Ch IV; Farkas-Kra Ch III |
| 06.05.02 | Holomorphic line bundle on a Riemann surface | Cartan-Serre 1953–55 | Forster §29; Donaldson Ch 8; Griffiths-Harris Ch 2 |
| 06.05.03 | Riemann-Hurwitz formula | Hurwitz 1891 Über Riemann'sche Flächen mit gegebenen Verzweigungspunkten | Forster §17; Miranda Ch II; Donaldson Ch 4 |
06.06 Jacobians and abelian differentials (5 supporting; new chapter)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 06.06.01 | Holomorphic 1-form / abelian differential | Riemann 1857; Klein 1882 Über Riemann's Theorie der algebraischen Functionen | Forster §17; Farkas-Kra Ch II; Springer Riemann Surfaces |
| 06.06.02 | Period matrix | Riemann 1857 (introduces period relations) | Farkas-Kra Ch III; Griffiths-Harris Ch 2; Mumford Curves and their Jacobians |
| 06.06.03 | Jacobian variety | Jacobi 1829 (elliptic case); Riemann 1857 (general); Weierstrass | Mumford Curves and their Jacobians; Farkas-Kra Ch VI; Birkenhake-Lange Complex Abelian Varieties |
| 06.06.04 | Abel-Jacobi map | Abel 1826 Mémoire sur une propriété générale d'une classe très-étendue de fonctions transcendantes; Jacobi 1832 | Mumford; Griffiths-Harris Ch 2; Forster §21 |
| 06.06.05 | Theta function | Jacobi 1828 Fundamenta Nova Theoriae Functionum Ellipticarum; Riemann 1866 (theta with characteristics) | Mumford Tata Lectures on Theta I–III; Farkas-Kra Ch VI; Igusa Theta Functions |
06.07 Several complex variables (2 supporting; new chapter)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 06.07.01 | Holomorphic functions of several variables | Hartogs 1906 Zur Theorie der analytischen Functionen mehrerer unabhängiger Veränderlicher | Krantz Function Theory of Several Complex Variables (Fast Track 3.23); Gunning-Rossi Analytic Functions of Several Complex Variables (Fast Track 3.24) |
| 06.07.02 | Hartogs phenomenon | Hartogs 1906 | Krantz; Hörmander Introduction to Complex Analysis in Several Variables |
4. Strand D — Representation Theory (22 supporting units)
Apex shipped (4): 07.01.01 Group representation, 07.01.02 Schur's lemma, 07.03.01 Highest weight representation, 07.04.01 Cartan-Weyl classification.
Chapter expansion: existing 07.01, 07.03, 07.04 plus new 07.02 (Character theory), 07.05 (Symmetric groups and Young tableaux), 07.06 (Lie-algebraic structure), 07.07 (Compact Lie groups).
07.01 Foundations (6 supporting)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 07.01.03 | Character of a representation | Frobenius 1896 Über die Charaktere endlicher Gruppen | Serre §2 (Fast Track 3.15); Fulton-Harris §2 (3.11); James-Liebeck |
| 07.01.04 | Character orthogonality | Frobenius 1896; Schur 1905 (refined) | Serre §2; Fulton-Harris §2 |
| 07.01.05 | Regular representation | Frobenius 1898 | Serre §1–2; Fulton-Harris §2 |
| 07.01.06 | Tensor product of representations | Schur 1901; Weyl 1925 | Fulton-Harris §1.1 |
| 07.01.07 | Induced representation | Frobenius 1898 Über Relationen zwischen den Charakteren einer Gruppe und denen ihrer Untergruppen | Serre §7; Fulton-Harris §3 |
| 07.01.08 | Frobenius reciprocity | Frobenius 1898 | Serre §7; Fulton-Harris §3 |
07.02 Character theory (1 supporting; new chapter)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 07.02.01 | Maschke's theorem | Heinrich Maschke 1899 | Serre §1 (Fast Track 3.15); Fulton-Harris §1.1; Lang Algebra (3.01) |
07.05 Symmetric groups and Young tableaux (3 supporting; new chapter)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 07.05.01 | Symmetric group representation | Frobenius 1900; Young 1900–1933 | Fulton-Harris §4–6 (Fast Track 3.11); Sagan The Symmetric Group |
| 07.05.02 | Young diagram and tableau | Alfred Young 1900–1933 On Quantitative Substitutional Analysis | Fulton Young Tableaux; Fulton-Harris §4; Sagan |
| 07.05.03 | Specht module | Wilhelm Specht 1935 | Sagan; James The Representation Theory of the Symmetric Groups |
07.06 Lie-algebraic structure (9 supporting; new chapter)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 07.06.01 | Lie algebra representation | Cartan 1894 | Humphreys §6 (Fast Track 3.12-equivalent); Fulton-Harris §9 (3.11) |
| 07.06.02 | Universal enveloping algebra | Poincaré 1900; Birkhoff 1937; Witt 1937 (PBW theorem) | Humphreys §17; Dixmier Enveloping Algebras; Bourbaki Ch I |
| 07.06.03 | Root system | Wilhelm Killing 1888–90 Die Zusammensetzung der stetigen endlichen Transformationsgruppen I–IV; Cartan 1894 | Bourbaki Ch VI (Fast Track 3.12); Humphreys §8–10; Serre Complex Semisimple Lie Algebras (3.12) |
| 07.06.04 | Weyl group | Hermann Weyl 1925–26 Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen | Bourbaki Ch VI; Humphreys §10; Hiller Geometry of Coxeter Groups |
| 07.06.05 | Dynkin diagram | Eugene Dynkin 1947 Classification of simple Lie algebras | Humphreys §11; Bourbaki Ch VI |
| 07.06.06 | Verma module | Daya-Nand Verma 1968 (PhD thesis) | Humphreys Representations of Semisimple Lie Algebras in the BGG Category O; Dixmier; Knapp Ch V |
| 07.06.07 | Weyl character formula | Weyl 1925–26 (the four landmark papers) | Fulton-Harris §24 (Fast Track 3.11); Humphreys §24; Knapp Ch V |
| 07.06.08 | Weyl dimension formula | Weyl 1925 | Fulton-Harris §24 |
| 07.06.09 | Borel-Weil theorem | Armand Borel 1954 (Bourbaki seminar); Bott 1957 (extended to higher cohomology) | Knapp Ch VI; Tu Differential Geometry; Wallach Real Reductive Groups I |
07.07 Compact Lie groups (3 supporting; new chapter)
| ID | Title | Originator | Modern anchor |
|---|---|---|---|
| 07.07.01 | Compact Lie group representation | Weyl 1925–26 (the unitarian trick) | Knapp Ch IV (Fast Track 3.11-anchor adjacent); Bröcker-tom Dieck Representations of Compact Lie Groups; Sepanski Compact Lie Groups |
| 07.07.02 | Peter-Weyl theorem | Peter & Weyl 1927 Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe | Knapp Ch IV; Bröcker-tom Dieck; Folland A Course in Abstract Harmonic Analysis |
| 07.07.03 | Haar measure | Alfréd Haar 1933 Der Massbegriff in der Theorie der kontinuierlichen Gruppen | Knapp Ch VIII; Folland Real Analysis; Bourbaki Intégration |
5. Production order (highest leverage first)
The 62 units divide into three priority bands. Each band is internally orderable, but units within a band can be produced in parallel without prerequisite collisions (one session per strand, as for v0.5 apex).
Band 1 — Foundational dependencies (28 units)
Cited explicitly by ≥2 apex units; should land before any further v0.6 work.
Strand A — divisors, line bundles, projective geometry (10): 04.07.01 Projective space; 04.05.01 Weil divisor; 04.05.03 Line bundle; 04.05.04 Cartier divisor; 04.05.02 Picard group; 04.06.01 Quasi-coherent sheaf; 04.06.02 Coherent sheaf; 04.02.02 Affine scheme Spec(R); 04.02.03 Projective scheme Proj(S); 04.05.05 Ample / very ample.
Strand C — divisors, bundles, key transforms (8): 06.01.05 Meromorphic function; 06.05.01 Divisor on a Riemann surface; 06.05.02 Holomorphic line bundle on RS; 06.06.01 Holomorphic 1-form; 06.01.02 Cauchy integral formula; 06.01.03 Residue theorem; 06.05.03 Riemann-Hurwitz; 06.02.01 Branch point and ramification.
Strand D — character theory and Lie-algebraic core (10): 07.01.03 Character; 07.02.01 Maschke's theorem; 07.01.04 Character orthogonality; 07.01.05 Regular representation; 07.01.06 Tensor product of reps; 07.06.01 Lie algebra rep; 07.06.03 Root system; 07.06.04 Weyl group; 07.06.05 Dynkin diagram; 07.06.07 Weyl character formula.
Band 2 — Synthesis dependencies (20 units)
Needed for the deeper Master-tier connections; not blocking apex traversability.
Strand A (7): 04.01.02 Stalk; 04.01.03 Sheafification; 04.01.04 Direct/inverse image; 04.02.04 Morphism of schemes; 04.07.02 Blowup; 04.08.01 Sheaf of differentials; 04.08.02 Canonical sheaf.
Strand C (5): 06.01.04 Analytic continuation; 06.01.06 Riemann mapping theorem; 06.03.02 Genus of a Riemann surface; 06.06.02 Period matrix; 06.03.03 Uniformization theorem.
Strand D (8): 07.01.07 Induced representation; 07.01.08 Frobenius reciprocity; 07.06.02 Universal enveloping algebra; 07.06.06 Verma module; 07.06.08 Weyl dimension formula; 07.07.01 Compact Lie group rep; 07.07.02 Peter-Weyl; 07.07.03 Haar measure.
Band 3 — Advanced topics (14 units)
Apex-adjacent; full breadth of Fast Track parity.
Strand A (5): 04.08.03 Serre duality; 04.09.01 Hodge decomposition; 04.09.02 Kodaira vanishing; 04.10.01 Moduli of curves; 04.10.02 GIT.
Strand C (5): 06.06.03 Jacobian variety; 06.06.04 Abel-Jacobi; 06.06.05 Theta function; 06.07.01 Several complex variables; 06.07.02 Hartogs phenomenon.
Strand D (4): 07.05.01 Symmetric group representation; 07.05.02 Young diagram; 07.05.03 Specht module; 07.06.09 Borel-Weil.
6. Cross-strand sharing notes
Several units sit at the boundary of two strands. We produce one canonical unit per concept and cross-reference rather than duplicating.
- Divisor.
04.05.01 Weil divisor on a scheme(algebraic) and06.05.01 Divisor on a Riemann surface(analytic) are separate units: the analytic version is what the Riemann-Roch apex uses, the algebraic version is what the schemes apex uses. GAGA is the bridge, and each unit's Master-tier section names it. - Line bundle.
04.05.03 Line bundle / invertible sheaf(algebraic, on a scheme) and06.05.02 Holomorphic line bundle on a Riemann surface(analytic) — same pattern. - Picard group. Single unit
04.05.02 Picard groupdefined in scheme-theoretic terms; Master-tier addresses the analytic case via GAGA. The Riemann-surface apex cites this rather than spawning a duplicate. - Period matrix / Hodge decomposition.
06.06.02 Period matrixfor Riemann surfaces is the genus-1+ rank-1 case of04.09.01 Hodge decomposition; both units cross-link, with Hodge being the general Kähler-manifold statement. - Jacobian.
06.06.03 Jacobian varietyis both an analytic complex torus and a smooth projective abelian variety; one unit, two perspectives in Master tier. - Lie algebra representation.
07.06.01is the Lie-algebra-side of07.01.01 Group representation; we shipped the group-side first (apex), and the Lie-algebra-side derives from it via differentiation. - Root system, Weyl group, Dynkin diagram. All three are buried inside the Cartan-Weyl apex (
07.04.01). Standalone units make them accessible to the symplectic strand (Lie-theoretic moment maps), the algebraic-geometry strand (toric varieties, flag varieties), and physics units.
7. Validation cadence
Same as v0.5 apex: each unit runs scripts/validate_unit.py to 27/27, then scripts/validate_all.py to confirm green at the manifest level. Lean stubs follow Mathlib status (partial if Mathlib has the concept but not the named lemma; none only if Mathlib has nothing related).
The lean_status: stub value is not validator-acceptable — use partial whenever there's any Mathlib infrastructure the unit can build on (which there always is for these units).
8. Final unit count after v0.5 supporting
- v0.x core: 66 units
- v0.5 apex (claude A/C/D + gpt-codex B): 32 units shipped, total 98 currently
- v0.5 apex (gpt-codex E): ~12 units pending → ~110
- v0.5 supporting (this scaffold): 62 units → ~172
- Adjacent gaps post-supporting (homological algebra, Morse theory, TQFT/cobordism — Fast Track 3.02–3.07): ~20 units pencilled for v0.6
Estimated total at v0.5 close: ~172 units, achieving roughly 65–70% Fast Track coverage at apex level and ≥80% within Strands A, C, D themselves.
Per Tyler's directive: "base this on the literal all time best literature and convey the intelligence of writing from the original geniuses who conceived the work" — the original-author column above is load-bearing, not decorative. Master-tier prose for each unit should quote and contextualise the original conception alongside the modern synthesis.