{
  "id": "small-cohen-macaulay-modules",
  "title": "Small Cohen–Macaulay modules",
  "description": "Does every complete Noetherian local domain have a finitely generated module of maximal depth? The answer is no: the completed cone over a suitable surface has none. The proof: a small Cohen-Macaulay module is free over a power series ring and extends to a sheaf on a resolution of the cone; balancing its restriction to the exceptional surface and comparing Riemann-Roch on the surface and on the plane gives K² - 2c₂ ≤ 6H²; a cover of the plane branched along the complete quadrangle violates this. Proved in full from five named hypotheses on smooth projective surfaces, among them Bogomolov's inequality and Noether's formula. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "completed-cones-over-surfaces",
      "title": "Completed cones over surfaces",
      "reader": "completed-cones-over-surfaces.html",
      "source": "src/completed-cones-over-surfaces.md",
      "source_sha256": "2c7b0b7500aa768f2a7c09317092141f85eb8a61d849b25231cba791b1240b30",
      "status": "author_self_checked"
    },
    {
      "id": "elementary-modifications-and-a-variance-inequality",
      "title": "Elementary modifications and a variance inequality",
      "reader": "elementary-modifications-and-a-variance-inequality.html",
      "source": "src/elementary-modifications-and-a-variance-inequality.md",
      "source_sha256": "72604d219ccbb30ec24f79da526e3362951b4aa032fdb2601c203edce3e50154",
      "status": "author_self_checked"
    },
    {
      "id": "the-numerical-obstruction",
      "title": "The numerical obstruction",
      "reader": "the-numerical-obstruction.html",
      "source": "src/the-numerical-obstruction.md",
      "source_sha256": "fde7b6c5a9fa7cfb96bf31a151bdac7f019a312b6b16f2bde71dd336d8da84df",
      "status": "author_self_checked"
    },
    {
      "id": "a-hirzebruch-kummer-surface",
      "title": "A Hirzebruch–Kummer surface",
      "reader": "a-hirzebruch-kummer-surface.html",
      "source": "src/a-hirzebruch-kummer-surface.md",
      "source_sha256": "774a43988583af3247f46c9df57e5cc7324299b3d73fe01ae0e634ab222e5daf",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "course Commutative algebra for geometry: completion, faithful flatness, integral extensions, regular local rings, depth and Cohen-Macaulay modules, the Auslander-Buchsbaum formula",
    "courses Morphisms of schemes and Cohomology of quasi-coherent sheaves: finite and proper morphisms, Zariski's main theorem, ample invertible sheaves, normalization, Serre's theorems, flat base change, Euler characteristics and Hilbert polynomials",
    "course Flat, smooth and étale morphisms, lesson on étale morphisms (Theorem 5.1)",
    "course ℓ-adic cohomology and trace formulas, lesson on cycle classes: the intersection pairing on a surface, surface Riemann-Roch for line bundles, the Hodge index theorem",
    "the lessons prove every other fact they use, except the named hypotheses (CS) of the second lesson, (NF) and (BI) of the third, (BL) and (EU) of the fourth, and the related work quoted in the first"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}