Small Cohen–Macaulay modules

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).

Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Completed cones over surfaces
2. Elementary modifications and a variance inequality
3. The numerical obstruction
4. A Hirzebruch–Kummer surface

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Completion (Commutative algebra for geometry)
• Faithful flatness and the local criterion for flatness (Commutative algebra for geometry)
• Integral extensions: lying over, going up and going down (Commutative algebra for geometry)
• Projective dimension and the Auslander–Buchsbaum formula (Commutative algebra for geometry)
• Regular local rings (Commutative algebra for geometry)
• Regular sequences, depth and Cohen–Macaulay modules (Commutative algebra for geometry)
• Étale morphisms and their local structure · Flat, smooth and étale morphisms
• Cycle classes and the Lefschetz fixed-point formula for curves (ℓ-adic cohomology and trace formulas)
• Affine morphisms, relative Spec, and finite morphisms (Morphisms of schemes)
• Ample invertible sheaves · Morphisms of schemes
• Normalization · Morphisms of schemes
• Zariski's Main Theorem (Morphisms of schemes)
• Base change and the Grothendieck complex · Cohomology of quasi-coherent sheaves
• Cohomology of projective space · Cohomology of quasi-coherent sheaves
• Euler characteristics and Hilbert polynomials · Cohomology of quasi-coherent sheaves
• Coherent sheaves on projective schemes: Serre's theorems · Cohomology of quasi-coherent sheaves
• Commutative algebra (course Commutative algebra for geometry); schemes and their cohomology (courses Morphisms of schemes and Cohomology of quasi-coherent sheaves); the Hodge index theorem (course ℓ-adic cohomology and trace formulas).