{
  "id": "ag-etale-cohomology",
  "title": "Sites, topoi and étale cohomology",
  "language": "en",
  "lessons": [
    {
      "id": "sites-and-sheaves",
      "title": "Sites and sheaves",
      "source": "src/sites-and-sheaves.md",
      "source_sha256": "c28a5ab1c6f92a4f5c67135be04280858f6aff2315797109eafcc287402f647c",
      "reader": "sites-and-sheaves.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Public domain (CC0)."
    },
    {
      "id": "topoi-morphisms-and-points",
      "title": "Topoi, morphisms and points",
      "source": "src/topoi-morphisms-and-points.md",
      "source_sha256": "2fb1452d3062146970af2c1d1db0d96641ca0c1b29db5f430caad7a8cf1abe8e",
      "reader": "topoi-morphisms-and-points.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Public domain (CC0)."
    },
    {
      "id": "cohomology-on-sites",
      "title": "Cohomology on sites",
      "source": "src/cohomology-on-sites.md",
      "source_sha256": "384e96847423a3c67ec243453f0ce841daf0df464519fee1104b1d5efc40892e",
      "reader": "cohomology-on-sites.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Public domain (CC0)."
    },
    {
      "id": "hypercoverings",
      "title": "Hypercoverings",
      "source": "src/hypercoverings.md",
      "source_sha256": "b69bdbe9b0cbec6259541d4ad29a567f1a719921f2360a513c6212f0fc73b595",
      "reader": "hypercoverings.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Public domain (CC0)."
    },
    {
      "id": "topologies-on-schemes",
      "title": "Topologies on schemes",
      "source": "src/topologies-on-schemes.md",
      "source_sha256": "7c724450159c4c42dc51ba901dd6b7e1e062755f461c1a12978cafa41b9a768a",
      "reader": "topologies-on-schemes.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Public domain (CC0)."
    },
    {
      "id": "the-etale-site-and-its-points",
      "title": "The étale site and its points",
      "source": "src/the-etale-site-and-its-points.md",
      "source_sha256": "c92d591e2c7ec85c7b5ee6de26aa32c40275b38452a5458ceebfd5f9340f1d25",
      "reader": "the-etale-site-and-its-points.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. **Self-checked by the writing AI, GPT-6.1 Sol (OpenAI).** Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Public domain (CC0)."
    },
    {
      "id": "pushforward-pullback-and-finite-morphisms",
      "title": "Pushforward, pullback and finite morphisms",
      "source": "src/pushforward-pullback-and-finite-morphisms.md",
      "source_sha256": "8f23cccb927fe74b0c9d5a2e2a8562220b6a1b0c8662a8fa14442ed8100b81f3",
      "reader": "pushforward-pullback-and-finite-morphisms.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Public domain (CC0)."
    },
    {
      "id": "galois-cohomology-and-the-etale-cohomology-of-a-field",
      "title": "Galois cohomology and the étale cohomology of a field",
      "source": "src/galois-cohomology-and-the-etale-cohomology-of-a-field.md",
      "source_sha256": "1a42ed230d4e4263c27343b666091bb0434add0821d251cd0e7a24209f989802",
      "reader": "galois-cohomology-and-the-etale-cohomology-of-a-field.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised, and the valuation step in section 7 written, by Claude Opus 5.5 (Anthropic), October 2026. Public domain (CC0)."
    },
    {
      "id": "brauer-groups-and-tsen-s-theorem",
      "title": "Brauer groups and Tsen's theorem",
      "source": "src/brauer-groups-and-tsen-s-theorem.md",
      "source_sha256": "529a4e2dddd406bfc481c45239c236a19ec1878aee7165ccdb19f39f604f89b2",
      "reader": "brauer-groups-and-tsen-s-theorem.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Original text released under CC0."
    },
    {
      "id": "the-multiplicative-group-on-a-curve",
      "title": "The multiplicative group on a curve",
      "source": "src/the-multiplicative-group-on-a-curve.md",
      "source_sha256": "22420067f6c1a51298bc7dd8e66d0164f445ee82bc82e59dc72b44f986473539",
      "reader": "the-multiplicative-group-on-a-curve.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Original text released under CC0."
    },
    {
      "id": "constructible-sheaves-and-extension-by-zero",
      "title": "Constructible sheaves and extension by zero",
      "source": "src/constructible-sheaves-and-extension-by-zero.md",
      "source_sha256": "0ae2b42ab6f22917925f46b8c2f404721399a83c8536d88eac4b827f7bfe8a0a",
      "reader": "constructible-sheaves-and-extension-by-zero.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Original text released under CC0."
    },
    {
      "id": "torsion-sheaves-on-curves",
      "title": "Torsion sheaves on curves",
      "source": "src/torsion-sheaves-on-curves.md",
      "source_sha256": "14fa264db8612ad1a2ae6aaee7abbe1fa508199e3f28a4fbcb5dfb9c239bddc3",
      "reader": "torsion-sheaves-on-curves.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Original text released under CC0."
    },
    {
      "id": "the-proper-base-change-theorem",
      "title": "The proper base change theorem",
      "source": "src/the-proper-base-change-theorem.md",
      "source_sha256": "4df39c967783b868d5cb031b3732ea0044b11ff061c93f5454299af6315aa36a",
      "reader": "the-proper-base-change-theorem.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Original text released under CC0."
    },
    {
      "id": "cohomology-with-compact-support",
      "title": "Cohomology with compact support",
      "source": "src/cohomology-with-compact-support.md",
      "source_sha256": "d158e79e7b1ac85fa6d0a863b1b19998c1a15c1c2c58bd5a31f006a0ba8eb044",
      "reader": "cohomology-with-compact-support.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Original text released under CC0. Constructibility reduction and finiteness proof in Section 12 by GPT-6 Astra (OpenAI), October 2026; self-checked by the contributing AI."
    },
    {
      "id": "smooth-base-change-and-local-acyclicity",
      "title": "Smooth base change and local acyclicity",
      "source": "src/smooth-base-change-and-local-acyclicity.md",
      "source_sha256": "73cef01367538e8be557eb97312a3005d24541936fe4f98120cfc073b3ff5e9f",
      "reader": "smooth-base-change-and-local-acyclicity.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Original text released under CC0."
    },
    {
      "id": "cohomological-dimension-and-the-kunneth-formula",
      "title": "Cohomological dimension and the Künneth formula",
      "source": "src/cohomological-dimension-and-the-kunneth-formula.md",
      "source_sha256": "28bc2c57c9776999e671726873266c6c8ddd76bc78bf3c716df3d43d099e0a13",
      "reader": "cohomological-dimension-and-the-kunneth-formula.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Links and citations revised by Claude Opus 5.5 (Anthropic), October 2026. Original text released under CC0."
    },
    {
      "id": "poincare-duality-for-curves",
      "title": "Poincaré duality for curves",
      "source": "src/poincare-duality-for-curves.md",
      "source_sha256": "02f895a6b9fddf1a425e1acd42341caf87a842ae00805bbb29afdda967e19fe5",
      "reader": "poincare-duality-for-curves.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Links and citations revised by Claude Opus 5.5 (Anthropic). Connecting-map and trace-normalization arguments by GPT-6 Astra (OpenAI), Ultra, October 2026. Self-checked by the contributing AIs. Original text released under CC0."
    },
    {
      "id": "poincare-duality-for-smooth-varieties",
      "title": "Poincaré duality for smooth varieties",
      "source": "src/poincare-duality-for-smooth-varieties.md",
      "source_sha256": "f877d568aeca51fb2a010bb3066200e63ce7349d0b02b80afba61bb74265b54d",
      "reader": "poincare-duality-for-smooth-varieties.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026; the construction of the trace in Section 2 was written by GPT-6 Astra (OpenAI) in ChatGPT and checked by Claude Opus 5.5 (Anthropic) in a separate session. The supported Kummer and divisor-normalization argument is by GPT-6 Astra (OpenAI) in Codex, Ultra, October 2026. Self-checked by the contributing AIs. Original text released under CC0. Constructibility-hypothesis check by GPT-6 Astra (OpenAI), October 2026; self-checked by the contributing AI."
    },
    {
      "id": "comparison-with-singular-cohomology",
      "title": "Comparison with singular cohomology",
      "source": "src/comparison-with-singular-cohomology.md",
      "source_sha256": "87a623ca821563ed17d67433529220f4fb6570c3858b5de6afedf206ee8d768b",
      "reader": "comparison-with-singular-cohomology.html",
      "licence": "CC0-1.0",
      "authorship_and_checking": "Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI, GPT-6.1 Sol (OpenAI). Prerequisites linked, Sections 10, 13 and 14 revised, and the orientation theorem of Section 10 proved, by Claude Opus 5.5 (Anthropic), October 2026. Trace-normalization arguments and their examples reconciled by GPT-6 Astra (OpenAI) in Codex, Ultra, October 2026. Original text released under CC0."
    }
  ],
  "published": true,
  "reading_url": "https://kokunoyumeto.github.io/open-math-courses/courses/ag-etale-cohomology/",
  "licence": "CC0-1.0",
  "constructibility_proof_routes": {
    "source_edition": {
      "repository": "KokunoYumeto/unofficial-stacks-project-ai-drafts",
      "revision": "565b10e987aba5969b21145a0833f42d69f96790",
      "path": "more-etale.tex",
      "sha256": "13b8345b8f4d305bdb69afcae44e23deca80507354f3b65fd2cccdc3247c61c4",
      "url": "https://raw.githubusercontent.com/KokunoYumeto/unofficial-stacks-project-ai-drafts/565b10e987aba5969b21145a0833f42d69f96790/more-etale.tex"
    },
    "reader_edition": {
      "repository": "KokunoYumeto/stacks-zh-hans-cn",
      "revision": "f90daee5768b4d5491b30b0b782b9eeb586e7322",
      "path": "en/more-etale.html",
      "sha256": "9ad1099f6868c9f9af241713c710dd5d9dd16e0d9e7f61420f1dc5f1429a71d0"
    },
    "providers": [
      {
        "result": "General compact-support constructibility",
        "lesson": "cohomology-with-compact-support.html#proof-of-general-constructibility",
        "statement": "Theorem 12.3",
        "scope": "Separated finite presentation, qcqs schemes, Noetherian coefficient ring; bounded-below torsion constructible complexes, or all constructible complexes when the ring is torsion.",
        "geometric_input": "https://kokunoyumeto.github.io/stacks-zh-hans-cn/en/more-etale.html#more-etale-lemma-constructible-shriek-rel-dim-1",
        "native_label": "lemma-constructible-shriek-rel-dim-1",
        "native_lines": [
          4651,
          4705
        ]
      },
      {
        "result": "Compact-support finiteness",
        "lesson": "cohomology-with-compact-support.html#general-finiteness-over-an-algebraically-closed-field",
        "statement": "Corollary 12.4",
        "scope": "Separated finite-type scheme over an algebraically closed field; the same coefficient categories; finite generation in each degree, finite cardinality for finite coefficient rings."
      },
      {
        "consumer": "poincare-duality-for-smooth-varieties.html#checking-constructibility-in-the-relative-curve-argument",
        "uses": [
          "Section 5 strict-local relative curves",
          "Sections 7, 10 and 13 field finiteness"
        ],
        "hypothesis_check": "Affine strict-local bases are qcqs; the chosen affine smooth curve maps are separated and of finite presentation. Z/n is Noetherian and finite; mu_n is locally constant because n is invertible."
      }
    ],
    "boundary": "Exact reduction and consumer hypotheses checked. This record does not certify every lower prerequisite or independent review. Linked human-source expression retains its own terms and is not imported into these CC0 lessons."
  }
}