{
  "id": "riemann-existence-theorem",
  "title": "The Riemann existence theorem",
  "description": "Finite étale covers of a scheme of finite type over the complex numbers are the same as finite covering spaces of its complex points, with no normality hypothesis: covers of punctured polydiscs, the smooth case through compactification and GAGA, descent along proper surjective morphisms, the general theorem by descent along the resolution, its analytic form, and the comparison of étale and topological fundamental groups.",
  "units": [
    {
      "id": "finite-etale-covers-and-their-analytification",
      "title": "Finite étale covers and their analytification",
      "reader": "finite-etale-covers-and-their-analytification.html",
      "source": "src/finite-etale-covers-and-their-analytification.md",
      "source_sha256": "35e08b88f76942021fe4f2739f0817e30bc4c3ea8dfb32c17bd717beace9073a",
      "status": "author_self_checked"
    },
    {
      "id": "connectedness-and-full-faithfulness",
      "title": "Connectedness and full faithfulness",
      "reader": "connectedness-and-full-faithfulness.html",
      "source": "src/connectedness-and-full-faithfulness.md",
      "source_sha256": "7ff78ff204fd28afab1aece0ac3dd656ed0e0be4272da02b3d2e56db0ae08c78",
      "status": "author_self_checked"
    },
    {
      "id": "finite-covers-of-punctured-polydiscs",
      "title": "Finite covers of punctured polydiscs",
      "reader": "finite-covers-of-punctured-polydiscs.html",
      "source": "src/finite-covers-of-punctured-polydiscs.md",
      "source_sha256": "90e09b552f1629dea2e03062e8695d947e0497a24b732c46e18f72440cb71d83",
      "status": "author_self_checked"
    },
    {
      "id": "covers-of-smooth-varieties",
      "title": "Covers of smooth varieties",
      "reader": "covers-of-smooth-varieties.html",
      "source": "src/covers-of-smooth-varieties.md",
      "source_sha256": "291ac28f6fe75313a3b4633829805de88069b5ec3122f99f7b48ae4f324eadfe",
      "status": "author_self_checked"
    },
    {
      "id": "descent-along-proper-surjective-morphisms",
      "title": "Descent along proper surjective morphisms",
      "reader": "descent-along-proper-surjective-morphisms.html",
      "source": "src/descent-along-proper-surjective-morphisms.md",
      "source_sha256": "a65e3a09f6a80ac32c6a1970afe5be3cd97f5e2d1238c1f7058bf3de28cce960",
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    {
      "id": "the-riemann-existence-theorem",
      "title": "The Riemann existence theorem",
      "reader": "the-riemann-existence-theorem.html",
      "source": "src/the-riemann-existence-theorem.md",
      "source_sha256": "25e7aa1ddd6dd518c5ddd8e2b2f7246931cc9601e34968700964efd463aef784",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "AG-QC lessons on complex analytic spaces and analytification and on Serre's comparison theorems (GAGA)",
    "Complex analytic spaces and coherent sheaves, lesson 1 (Riemann extension theorem)",
    "Resolution of singularities in characteristic zero (Theorem 4.8, Corollary 5.2)",
    "Core course D60 Algebraic Topology (path lifting, monodromy, Theorem 10.1, Corollary 12.1, Theorem 17.1)",
    "AI Integrated Stacks Project: finite etale morphisms, henselian rings, fpqc descent, Galois categories",
    "Subanalytic triangulations on analytic manifolds: relative polyhedron theorem and its stated subanalytic prerequisites (Proposition 3.3)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}