Projective modules and locally polynomial algebras

Projective modules over polynomial rings and algebras that are locally polynomial. Quillen's patching theorem for extended modules and the method of Quillen induction; Horrocks' theorem, with an elementary proof over local rings; the Quillen–Suslin theorem that finitely generated projective modules over a polynomial ring over a field are free; automorphism groups of symmetric algebras, axiom Q and the factorization of an automorphism over two localizations; and the theorem of Bass, Connell and Wright that a finitely presented algebra which is locally a symmetric algebra is the symmetric algebra of a finitely presented module, so that locally polynomial algebras over k[y₁,…,y_r] are polynomial rings.

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. Patching extended modules
2. Horrocks' theorem and the Quillen–Suslin theorem
3. Automorphisms of symmetric algebras
4. Locally polynomial algebras

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Localization, local properties and support (Commutative algebra for geometry)
• Projective dimension and the Auslander–Buchsbaum formula (Commutative algebra for geometry)
• Localization: exactness, Hom out of a finitely presented module, Nakayama's lemma and local detection (AG-CA lesson Localization, local properties and support); Schanuel's lemma (AG-CA lesson Projective dimension and the Auslander–Buchsbaum formula).