{
  "id": "jacobian-conjecture-and-affine-spaces",
  "title": "The Jacobian conjecture and affine spaces",
  "description": "Polynomial maps and affine spaces: Keller maps and the reduction of the Jacobian conjecture to injectivity, with a complete hand proof that an explicit map of three-space with Jacobian determinant -2 identifies three points; coordinates and a degree-17 polynomial in four variables whose zero set is affine three-space but which is not a coordinate; locally nilpotent derivations, additive actions and their lifts to line bundles; a fourfold whose cylinder is affine five-space but which is not affine four-space, so that Zariski cancellation fails in dimension four in characteristic zero; a polynomial in five variables that becomes a coordinate after adjoining one variable, with explicit complementary coordinates, but is not a coordinate; a degree-five stable coordinate in four variables that is not a coordinate; and smooth affine-space fibrations over the line, the plane and a discrete valuation ring, all of whose fibres are affine three-space, which are not locally trivial.",
  "units": [
    {
      "id": "keller-maps-and-the-jacobian-conjecture",
      "title": "Keller maps and the Jacobian conjecture",
      "reader": "keller-maps-and-the-jacobian-conjecture.html",
      "source": "src/keller-maps-and-the-jacobian-conjecture.md",
      "source_sha256": "33a8627d19589e7a02727eed4c4d34c8da026dc23d4820f18fba72710d77b0dd",
      "status": "author_self_checked"
    },
    {
      "id": "a-counterexample-in-dimension-three",
      "title": "A counterexample in dimension three",
      "reader": "a-counterexample-in-dimension-three.html",
      "source": "src/a-counterexample-in-dimension-three.md",
      "source_sha256": "5c604278be40463ed653434705e1ab502a7d448a6fd7f99cd615663950d2de91",
      "status": "author_self_checked"
    },
    {
      "id": "coordinates-and-the-abhyankar-sathaye-problem",
      "title": "Coordinates and the Abhyankar–Sathaye problem",
      "reader": "coordinates-and-the-abhyankar-sathaye-problem.html",
      "source": "src/coordinates-and-the-abhyankar-sathaye-problem.md",
      "source_sha256": "b4c7294f4194c8592765bac05b64d12e10be0d5e4471b7541b7814781f01a7b6",
      "status": "author_self_checked"
    },
    {
      "id": "locally-nilpotent-derivations",
      "title": "Locally nilpotent derivations",
      "reader": "locally-nilpotent-derivations.html",
      "source": "src/locally-nilpotent-derivations.md",
      "source_sha256": "42d05c4eea34716548db66e6fefd86e2641a1201162a56a000ca9179e08c11c6",
      "status": "author_self_checked"
    },
    {
      "id": "additive-actions-on-line-bundles",
      "title": "Additive actions on line bundles",
      "reader": "additive-actions-on-line-bundles.html",
      "source": "src/additive-actions-on-line-bundles.md",
      "source_sha256": "0587cd6a0e4ed17d40a2af6fbce413f4235505c0baa4c75c33f6292286731eb7",
      "status": "author_self_checked"
    },
    {
      "id": "a-fourfold-whose-cylinder-is-affine-space",
      "title": "A fourfold whose cylinder is affine space",
      "reader": "a-fourfold-whose-cylinder-is-affine-space.html",
      "source": "src/a-fourfold-whose-cylinder-is-affine-space.md",
      "source_sha256": "5d2e56fdea1873101d3a1b96ad643195fb4f7424c559bc8ca4e8779eedce1e9b",
      "status": "author_self_checked"
    },
    {
      "id": "cancellation-fails-in-dimension-four",
      "title": "Cancellation fails in dimension four",
      "reader": "cancellation-fails-in-dimension-four.html",
      "source": "src/cancellation-fails-in-dimension-four.md",
      "source_sha256": "088895223b0f2e5a93c0d75c66bb8ac4f5054d5f5c83a9cf873cf07ea44c6bdc",
      "status": "author_self_checked"
    },
    {
      "id": "a-stable-coordinate-in-five-variables",
      "title": "A stable coordinate in five variables",
      "reader": "a-stable-coordinate-in-five-variables.html",
      "source": "src/a-stable-coordinate-in-five-variables.md",
      "source_sha256": "c37e2c14eb41c74a9a0c9136df3e1f7012d7d4135f36289c1e6b1f160691d372",
      "status": "author_self_checked"
    },
    {
      "id": "a-stable-coordinate-that-is-not-a-coordinate",
      "title": "A stable coordinate that is not a coordinate",
      "reader": "a-stable-coordinate-that-is-not-a-coordinate.html",
      "source": "src/a-stable-coordinate-that-is-not-a-coordinate.md",
      "source_sha256": "e64078f957bd19540599ac69069444e75ef9eef50cedaa2fbfd023db6ca45543",
      "status": "author_self_checked"
    },
    {
      "id": "affine-fibrations-that-are-not-locally-trivial",
      "title": "Affine-space fibrations that are not locally trivial",
      "reader": "affine-fibrations-that-are-not-locally-trivial.html",
      "source": "src/affine-fibrations-that-are-not-locally-trivial.md",
      "source_sha256": "2dc7da28d0645e2e6113bdb793f75489bc437f448762bd635d56449fc7d8554f",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "AG-CA (commutative algebra: Nullstellensatz, Krull dimension, dimension theory of local rings, integral extensions)",
    "AG-MO lesson on the dimension of fibres",
    "Course projective-modules-and-locally-polynomial-algebras (lesson locally-polynomial-algebras: Lemmas 1.2, 1.3, Corollary 2.2; lesson horrocks-and-quillen-suslin: Lemma 1.1)",
    "AG-CA lessons tor-and-flat-modules (Propositions 3.1, 3.2, Theorem 3.3), faithful-flatness-and-the-local-criterion-for-flatness (Theorem 5.2, slicing lemma) and formally-smooth-unramified-and-etale-ring-maps (definitions of formally smooth and smooth ring maps)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}