{
  "id": "projective-modules-and-locally-polynomial-algebras",
  "title": "Projective modules and locally polynomial algebras",
  "description": "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.",
  "units": [
    {
      "id": "patching-extended-modules",
      "title": "Patching extended modules",
      "reader": "patching-extended-modules.html",
      "source": "src/patching-extended-modules.md",
      "source_sha256": "8a8ee49eaefa461193711457fddb81a00aa8d94d1a204427051c8ccfcc2b22e1",
      "status": "author_self_checked"
    },
    {
      "id": "horrocks-and-quillen-suslin",
      "title": "Horrocks' theorem and the Quillen–Suslin theorem",
      "reader": "horrocks-and-quillen-suslin.html",
      "source": "src/horrocks-and-quillen-suslin.md",
      "source_sha256": "191b3b810425c770dde187a80e6e4a7c6b15b5540f1721156dd709636d517b52",
      "status": "author_self_checked"
    },
    {
      "id": "automorphisms-of-symmetric-algebras",
      "title": "Automorphisms of symmetric algebras",
      "reader": "automorphisms-of-symmetric-algebras.html",
      "source": "src/automorphisms-of-symmetric-algebras.md",
      "source_sha256": "76f4de5942feb139ab3991c03a939ac8487abc0c941e0ce8cc5755cb43628722",
      "status": "author_self_checked"
    },
    {
      "id": "locally-polynomial-algebras",
      "title": "Locally polynomial algebras",
      "reader": "locally-polynomial-algebras.html",
      "source": "src/locally-polynomial-algebras.md",
      "source_sha256": "517cd37c61cc4bcf1f36163812427f8f2a6f8e257c70d1e84c5ee5ea780b27ad",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "AG-CA lesson localization-local-properties-and-support (Theorems 2.1, 2.2, 4.2, 5.1: exactness of localization, localizing Hom out of a finitely presented module, Nakayama's lemma, local detection)",
    "AG-CA lesson projective-dimension-and-the-auslander-buchsbaum-formula (Lemma 1.1: Schanuel's lemma)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}