{
  "schema": "course-result-locations/v1",
  "course": "AG-QC",
  "results": [
    {
      "id": "AG-QC-01.Proposition-1.1",
      "unit": "AG-QC-01",
      "label": "Proposition 1.1",
      "statement_heading": "Proposition 1.1 (restriction). If \\(U\\subset X\\) is open, then",
      "source": "src/cohomology-of-sheaves-on-ringed-spaces.md",
      "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977",
      "source_line": 39,
      "reader": "cohomology-of-sheaves-on-ringed-spaces.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-01.Lemma-2.1",
      "unit": "AG-QC-01",
      "label": "Lemma 2.1",
      "statement_heading": "Lemma 2.1 (lifting with an extensible kernel). In a short exact sequence \\(0\\to\\mathcal A\\to\\mathcal B\\to\\mathcal C\\to0\\), if \\(\\mathcal A\\) is flasque, then",
      "source": "src/cohomology-of-sheaves-on-ringed-spaces.md",
      "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977",
      "source_line": 57,
      "reader": "cohomology-of-sheaves-on-ringed-spaces.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-01.Corollary-2.2",
      "unit": "AG-QC-01",
      "label": "Corollary 2.2",
      "statement_heading": "Corollary 2.2. If both \\(\\mathcal A\\) and \\(\\mathcal B\\) in Lemma 2.1 are flasque, then \\(\\mathcal C\\) is flasque.",
      "source": "src/cohomology-of-sheaves-on-ringed-spaces.md",
      "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977",
      "source_line": 62,
      "reader": "cohomology-of-sheaves-on-ringed-spaces.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-01.Theorem-2.3",
      "unit": "AG-QC-01",
      "label": "Theorem 2.3",
      "statement_heading": "Theorem 2.3 (flasque acyclicity). For a flasque sheaf of modules \\(\\mathcal F\\),",
      "source": "src/cohomology-of-sheaves-on-ringed-spaces.md",
      "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977",
      "source_line": 66,
      "reader": "cohomology-of-sheaves-on-ringed-spaces.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-01.Corollary-2.4",
      "unit": "AG-QC-01",
      "label": "Corollary 2.4",
      "statement_heading": "Corollary 2.4 (flasque resolutions). An exact resolution \\(0\\to\\mathcal F\\to\\mathcal L^0\\to\\mathcal L^1\\to\\cdots\\) with all \\(\\mathcal L^r\\) flasque computes cohomology on every open by taking sections.",
      "source": "src/cohomology-of-sheaves-on-ringed-spaces.md",
      "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977",
      "source_line": 81,
      "reader": "cohomology-of-sheaves-on-ringed-spaces.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-01.Theorem-3.1",
      "unit": "AG-QC-01",
      "label": "Theorem 3.1",
      "statement_heading": "Theorem 3.1 (independence of the coefficient category). The underlying groups of \\(H^q(U,\\mathcal F)\\), computed in modules over \\(\\mathcal O_X\\), are naturally the cohomology groups of the underlying abelian sheaf.",
      "source": "src/cohomology-of-sheaves-on-ringed-spaces.md",
      "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977",
      "source_line": 91,
      "reader": "cohomology-of-sheaves-on-ringed-spaces.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-01.Theorem-3.2",
      "unit": "AG-QC-01",
      "label": "Theorem 3.2",
      "statement_heading": "Theorem 3.2 (local description). The sheaf \\(R^qf_*\\mathcal F\\) is the sheaf associated to the presheaf",
      "source": "src/cohomology-of-sheaves-on-ringed-spaces.md",
      "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977",
      "source_line": 102,
      "reader": "cohomology-of-sheaves-on-ringed-spaces.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-01.Corollary-3.3",
      "unit": "AG-QC-01",
      "label": "Corollary 3.3",
      "statement_heading": "Corollary 3.3. Higher direct images commute with restricting the target to an open subset. They also agree, as underlying abelian sheaves, with higher direct images computed in abelian sheaves.",
      "source": "src/cohomology-of-sheaves-on-ringed-spaces.md",
      "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977",
      "source_line": 116,
      "reader": "cohomology-of-sheaves-on-ringed-spaces.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-01.Corollary-3.4",
      "unit": "AG-QC-01",
      "label": "Corollary 3.4",
      "statement_heading": "Corollary 3.4. A direct image of a flasque sheaf is flasque. A flasque sheaf is \\(f_*\\)-acyclic: \\(R^qf_*\\mathcal F=0\\) for \\(q>0\\).",
      "source": "src/cohomology-of-sheaves-on-ringed-spaces.md",
      "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977",
      "source_line": 120,
      "reader": "cohomology-of-sheaves-on-ringed-spaces.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-01.Theorem-4.1",
      "unit": "AG-QC-01",
      "label": "Theorem 4.1",
      "statement_heading": "Theorem 4.1 (Leray spectral sequence). For any morphism of ringed spaces and any sheaf of modules \\(\\mathcal F\\), there is a natural first-quadrant spectral sequence",
      "source": "src/cohomology-of-sheaves-on-ringed-spaces.md",
      "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977",
      "source_line": 130,
      "reader": "cohomology-of-sheaves-on-ringed-spaces.html#AG-QC-01-Leray",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-01.Theorem-5.1",
      "unit": "AG-QC-01",
      "label": "Theorem 5.1",
      "statement_heading": "Theorem 5.1 (Mayer–Vietoris). If \\(X=U\\cup V\\) is an open cover, there is a natural exact sequence",
      "source": "src/cohomology-of-sheaves-on-ringed-spaces.md",
      "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977",
      "source_line": 162,
      "reader": "cohomology-of-sheaves-on-ringed-spaces.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-02.Theorem-2.1",
      "unit": "AG-QC-02",
      "label": "Theorem 2.1",
      "statement_heading": "Theorem 2.1. The ordered and alternating complexes are isomorphic, and their inclusion into the full complex is a homotopy equivalence. No division by a factorial is involved.",
      "source": "src/cech-cohomology.md",
      "source_sha256": "0a82cf6d7b423f3d8c943b8e824f2b55d7c5639efd4d4320692fc5ca52f998c7",
      "source_line": 47,
      "reader": "cech-cohomology.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-02.Lemma-3.1",
      "unit": "AG-QC-02",
      "label": "Lemma 3.1",
      "statement_heading": "Lemma 3.1. For an injective sheaf of modules \\(\\mathcal I\\), the augmented Čech complex of any open cover is exact. Thus \\(\\check H^p(\\mathcal U,\\mathcal I)=0\\) for \\(p>0\\).",
      "source": "src/cech-cohomology.md",
      "source_sha256": "0a82cf6d7b423f3d8c943b8e824f2b55d7c5639efd4d4320692fc5ca52f998c7",
      "source_line": 83,
      "reader": "cech-cohomology.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-02.Theorem-3.2",
      "unit": "AG-QC-02",
      "label": "Theorem 3.2",
      "statement_heading": "Theorem 3.2 (acyclic covers). If \\(H^q(U_{i_0\\ldots i_p},\\mathcal F)=0\\) for all \\(q>0\\) and every nonempty finite intersection, then the natural comparison is an isomorphism",
      "source": "src/cech-cohomology.md",
      "source_sha256": "0a82cf6d7b423f3d8c943b8e824f2b55d7c5639efd4d4320692fc5ca52f998c7",
      "source_line": 108,
      "reader": "cech-cohomology.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-02.Theorem-4.1",
      "unit": "AG-QC-02",
      "label": "Theorem 4.1",
      "statement_heading": "Theorem 4.1 (basis criterion). Let \\(\\mathcal B\\) be a basis of opens, and let \\(\\mathrm{Cov}\\) be a collection of covers of members of \\(\\mathcal B\\). Assume:",
      "source": "src/cech-cohomology.md",
      "source_sha256": "0a82cf6d7b423f3d8c943b8e824f2b55d7c5639efd4d4320692fc5ca52f998c7",
      "source_line": 122,
      "reader": "cech-cohomology.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-02.Theorem-5.1",
      "unit": "AG-QC-02",
      "label": "Theorem 5.1",
      "statement_heading": "Theorem 5.1. Isomorphism classes of \\(\\mathcal F\\)-torsors are naturally classified by \\(H^1(X,\\mathcal F)\\). The comparison",
      "source": "src/cech-cohomology.md",
      "source_sha256": "0a82cf6d7b423f3d8c943b8e824f2b55d7c5639efd4d4320692fc5ca52f998c7",
      "source_line": 150,
      "reader": "cech-cohomology.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-02.Theorem-6.1",
      "unit": "AG-QC-02",
      "label": "Theorem 6.1",
      "statement_heading": "Theorem 6.1. There is a natural isomorphism of abelian groups",
      "source": "src/cech-cohomology.md",
      "source_sha256": "0a82cf6d7b423f3d8c943b8e824f2b55d7c5639efd4d4320692fc5ca52f998c7",
      "source_line": 170,
      "reader": "cech-cohomology.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-02.Proposition-7.1",
      "unit": "AG-QC-02",
      "label": "Proposition 7.1",
      "statement_heading": "Proposition 7.1 (constant coefficients on intervals). If \\(J\\) is an open interval and \\(A\\) an abelian group, then the constant sheaf \\(A_J\\) has zero positive cohomology on \\(J\\), and on every open subset of \\(J\\).",
      "source": "src/cech-cohomology.md",
      "source_sha256": "0a82cf6d7b423f3d8c943b8e824f2b55d7c5639efd4d4320692fc5ca52f998c7",
      "source_line": 192,
      "reader": "cech-cohomology.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-03.Lemma-2.1",
      "unit": "AG-QC-03",
      "label": "Lemma 2.1",
      "statement_heading": "Lemma 2.1 (exactness on a standard cover). Suppose the \\(f_i\\) generate the unit ideal of \\(R\\). For every \\(R\\)-module \\(M\\), the augmented usual Čech complex",
      "source": "src/affine-cohomology-and-serres-criterion.md",
      "source_sha256": "9113f6aa822a41c9f58e454bf0a86e0e440f71ccee048c2cf94cae34685af2db",
      "source_line": 23,
      "reader": "affine-cohomology-and-serres-criterion.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-03.Theorem-2.2",
      "unit": "AG-QC-03",
      "label": "Theorem 2.2",
      "statement_heading": "Theorem 2.2 (affine vanishing). For every ring \\(R\\), every \\(R\\)-module \\(M\\), and every \\(p>0\\),",
      "source": "src/affine-cohomology-and-serres-criterion.md",
      "source_sha256": "9113f6aa822a41c9f58e454bf0a86e0e440f71ccee048c2cf94cae34685af2db",
      "source_line": 49,
      "reader": "affine-cohomology-and-serres-criterion.html#AG-QC-03-affine-vanishing",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-03.Theorem-3.1",
      "unit": "AG-QC-03",
      "label": "Theorem 3.1",
      "statement_heading": "Theorem 3.1 (affine-cover computation). Let \\(\\mathcal F\\) be quasi-coherent on a scheme \\(X\\). If an affine open cover has affine finite intersections, its Čech complex computes \\(H^p(X,\\mathcal F)\\). For a finite such cover with \\(r\\) members,",
      "source": "src/affine-cohomology-and-serres-criterion.md",
      "source_sha256": "9113f6aa822a41c9f58e454bf0a86e0e440f71ccee048c2cf94cae34685af2db",
      "source_line": 61,
      "reader": "affine-cohomology-and-serres-criterion.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-03.Theorem-3.2",
      "unit": "AG-QC-03",
      "label": "Theorem 3.2",
      "statement_heading": "Theorem 3.2 (an affine morphism has no higher direct images). Let \\(f:X\\to S\\) be affine and \\(\\mathcal F\\) quasi-coherent. Then",
      "source": "src/affine-cohomology-and-serres-criterion.md",
      "source_sha256": "9113f6aa822a41c9f58e454bf0a86e0e440f71ccee048c2cf94cae34685af2db",
      "source_line": 70,
      "reader": "affine-cohomology-and-serres-criterion.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-03.Lemma-4.1",
      "unit": "AG-QC-03",
      "label": "Lemma 4.1",
      "statement_heading": "Lemma 4.1 (localization of cohomology). Let \\(X\\) be quasi-compact and quasi-separated, let \\(X\\to\\operatorname{Spec}A\\) be a morphism, and let \\(\\mathcal F\\) be quasi-coherent. For \\(a\\in A\\), put \\(X_a=f^{-1}D(a)\\). The natural map is an isomorphism in every degree:",
      "source": "src/affine-cohomology-and-serres-criterion.md",
      "source_sha256": "9113f6aa822a41c9f58e454bf0a86e0e440f71ccee048c2cf94cae34685af2db",
      "source_line": 85,
      "reader": "affine-cohomology-and-serres-criterion.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-03.Lemma-4.2",
      "unit": "AG-QC-03",
      "label": "Lemma 4.2",
      "statement_heading": "Lemma 4.2 (a uniform bound). Fix a finite affine cover of a nonempty quasi-compact quasi-separated scheme \\(X\\). For each nonempty intersection \\(W_I\\), choose a finite affine cover with \\(t_I\\) members. Set",
      "source": "src/affine-cohomology-and-serres-criterion.md",
      "source_sha256": "9113f6aa822a41c9f58e454bf0a86e0e440f71ccee048c2cf94cae34685af2db",
      "source_line": 104,
      "reader": "affine-cohomology-and-serres-criterion.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-03.Theorem-4.3",
      "unit": "AG-QC-03",
      "label": "Theorem 4.3",
      "statement_heading": "Theorem 4.3 (quasi-coherent higher direct images). Let \\(f:X\\to S\\) be quasi-compact and quasi-separated. For every quasi-coherent \\(\\mathcal F\\), all \\(R^qf_*\\mathcal F\\) are quasi-coherent. If \\(S\\) is quasi-compact, there is an integer \\(d\\), depending only on \\(f\\), such that these sheaves vanish for \\(q\\ge d\\). The same \\(d\\) works after any base change, for every quasi-coherent sheaf on the changed source.",
      "source": "src/affine-cohomology-and-serres-criterion.md",
      "source_sha256": "9113f6aa822a41c9f58e454bf0a86e0e440f71ccee048c2cf94cae34685af2db",
      "source_line": 114,
      "reader": "affine-cohomology-and-serres-criterion.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-03.Lemma-4.4",
      "unit": "AG-QC-03",
      "label": "Lemma 4.4",
      "statement_heading": "Lemma 4.4 (finite-type approximation). On a quasi-compact, quasi-separated scheme \\(X\\), every quasi-coherent sheaf \\(F\\) is the directed union of its quasi-coherent subsheaves of finite type.",
      "source": "src/affine-cohomology-and-serres-criterion.md",
      "source_sha256": "9113f6aa822a41c9f58e454bf0a86e0e440f71ccee048c2cf94cae34685af2db",
      "source_line": 132,
      "reader": "affine-cohomology-and-serres-criterion.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-03.Lemma-5.1",
      "unit": "AG-QC-03",
      "label": "Lemma 5.1",
      "statement_heading": "Lemma 5.1 (an affine principal cover detects affineness). Suppose finitely many global functions \\(f_i\\) generate the unit ideal in \\(R=\\Gamma(X,\\mathcal O_X)\\), and the opens \\(X_{f_i}\\) are affine and cover \\(X\\). Then \\(X\\) is affine.",
      "source": "src/affine-cohomology-and-serres-criterion.md",
      "source_sha256": "9113f6aa822a41c9f58e454bf0a86e0e440f71ccee048c2cf94cae34685af2db",
      "source_line": 148,
      "reader": "affine-cohomology-and-serres-criterion.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-03.Theorem-5.2",
      "unit": "AG-QC-03",
      "label": "Theorem 5.2",
      "statement_heading": "Theorem 5.2 (Serre's criterion). For a quasi-compact scheme \\(X\\), the following are equivalent:",
      "source": "src/affine-cohomology-and-serres-criterion.md",
      "source_sha256": "9113f6aa822a41c9f58e454bf0a86e0e440f71ccee048c2cf94cae34685af2db",
      "source_line": 156,
      "reader": "affine-cohomology-and-serres-criterion.html#AG-QC-03-Serre-affineness",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-03.Corollary-5.3",
      "unit": "AG-QC-03",
      "label": "Corollary 5.3",
      "statement_heading": "Corollary 5.3 (finite-type ideals suffice on a quasi-separated scheme). If \\(X\\) is quasi-compact and quasi-separated, it is enough in condition 3 to test quasi-coherent ideals of finite type.",
      "source": "src/affine-cohomology-and-serres-criterion.md",
      "source_sha256": "9113f6aa822a41c9f58e454bf0a86e0e440f71ccee048c2cf94cae34685af2db",
      "source_line": 190,
      "reader": "affine-cohomology-and-serres-criterion.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-04.Lemma-2.1",
      "unit": "AG-QC-04",
      "label": "Lemma 2.1",
      "statement_heading": "Lemma 2.1 (the three weight cases). For a weight \\(e\\), the cohomology of \\(C^\\bullet(e)\\) is:",
      "source": "src/cohomology-of-projective-space.md",
      "source_sha256": "b5a86a2f9b9071147238d8df41ab476f2a8734e0f8b77ee6a6218389e3c02367",
      "source_line": 41,
      "reader": "cohomology-of-projective-space.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-04.Theorem-2.2",
      "unit": "AG-QC-04",
      "label": "Theorem 2.2",
      "statement_heading": "Theorem 2.2 (cohomology of every twist). If \\(n\\ge1\\), then",
      "source": "src/cohomology-of-projective-space.md",
      "source_sha256": "b5a86a2f9b9071147238d8df41ab476f2a8734e0f8b77ee6a6218389e3c02367",
      "source_line": 76,
      "reader": "cohomology-of-projective-space.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-04.Theorem-3.1",
      "unit": "AG-QC-04",
      "label": "Theorem 3.1",
      "statement_heading": "Theorem 3.1 (perfect monomial pairing). This pairing is perfect: each module is the \\(A\\)-linear dual of the other.",
      "source": "src/cohomology-of-projective-space.md",
      "source_sha256": "b5a86a2f9b9071147238d8df41ab476f2a8734e0f8b77ee6a6218389e3c02367",
      "source_line": 110,
      "reader": "cohomology-of-projective-space.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-04.Proposition-3.2",
      "unit": "AG-QC-04",
      "label": "Proposition 3.2",
      "statement_heading": "Proposition 3.2 (base change and polynomial multiplication). The preceding identifications commute with every ring map \\(A\\to B\\), including nonflat ones. If \\(f\\in S_s\\), multiplication by \\(f\\) on top cohomology is the dual of",
      "source": "src/cohomology-of-projective-space.md",
      "source_sha256": "b5a86a2f9b9071147238d8df41ab476f2a8734e0f8b77ee6a6218389e3c02367",
      "source_line": 120,
      "reader": "cohomology-of-projective-space.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-04.Lemma-5.1",
      "unit": "AG-QC-04",
      "label": "Lemma 5.1",
      "statement_heading": "Lemma 5.1 (canonical top trace). For \\(r\\ge2\\), there is a canonical isomorphism",
      "source": "src/cohomology-of-projective-space.md",
      "source_sha256": "b5a86a2f9b9071147238d8df41ab476f2a8734e0f8b77ee6a6218389e3c02367",
      "source_line": 166,
      "reader": "cohomology-of-projective-space.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-04.Theorem-5.2",
      "unit": "AG-QC-04",
      "label": "Theorem 5.2",
      "statement_heading": "Theorem 5.2 (cohomology of a projective bundle). If \\(r\\ge2\\), then",
      "source": "src/cohomology-of-projective-space.md",
      "source_sha256": "b5a86a2f9b9071147238d8df41ab476f2a8734e0f8b77ee6a6218389e3c02367",
      "source_line": 206,
      "reader": "cohomology-of-projective-space.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-05.Lemma-1.1",
      "unit": "AG-QC-05",
      "label": "Lemma 1.1",
      "statement_heading": "Lemma 1.1 (extension after a power). If \\(X\\) is quasi-compact and quasi-separated, \\(F\\) is quasi-coherent, and \\(s\\) is as above, the natural map",
      "source": "src/serres-theorems-on-projective-schemes.md",
      "source_sha256": "ddac40fd2bf24adfb7184702da4b3adcf5596f2739bbd9fb2d81bf9941f3f116",
      "source_line": 13,
      "reader": "serres-theorems-on-projective-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-05.Proposition-1.2",
      "unit": "AG-QC-05",
      "label": "Proposition 1.2",
      "statement_heading": "Proposition 1.2 (finite generation by twists). For every coherent \\(F\\) on \\(P\\), there is a finite surjection",
      "source": "src/serres-theorems-on-projective-schemes.md",
      "source_sha256": "ddac40fd2bf24adfb7184702da4b3adcf5596f2739bbd9fb2d81bf9941f3f116",
      "source_line": 28,
      "reader": "serres-theorems-on-projective-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-05.Lemma-1.3",
      "unit": "AG-QC-05",
      "label": "Lemma 1.3",
      "statement_heading": "Lemma 1.3 (an ample power gives an immersion). Let \\(X\\) be quasi-compact and quasi-separated and of finite type over an affine scheme \\(\\operatorname{Spec}R\\). Let \\(L\\) be ample. There are \\(b>0\\) and an immersion \\(i:X\\to\\mathbf P_R^N\\) with \\(L^b\\cong i^*\\mathcal O(1)\\). If \\(X\\) is proper over \\(R\\), the immersion is closed.",
      "source": "src/serres-theorems-on-projective-schemes.md",
      "source_sha256": "ddac40fd2bf24adfb7184702da4b3adcf5596f2739bbd9fb2d81bf9941f3f116",
      "source_line": 38,
      "reader": "serres-theorems-on-projective-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-05.Theorem-2.1",
      "unit": "AG-QC-05",
      "label": "Theorem 2.1",
      "statement_heading": "Theorem 2.1 (finiteness and vanishing on projective space). For a coherent \\(F\\) on \\(\\mathbf P_R^N\\), with \\(R\\) Noetherian:",
      "source": "src/serres-theorems-on-projective-schemes.md",
      "source_sha256": "ddac40fd2bf24adfb7184702da4b3adcf5596f2739bbd9fb2d81bf9941f3f116",
      "source_line": 52,
      "reader": "serres-theorems-on-projective-schemes.html#AG-QC-05-Serre-projective",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-05.Theorem-2.2",
      "unit": "AG-QC-05",
      "label": "Theorem 2.2",
      "statement_heading": "Theorem 2.2 (proper schemes with an ample line bundle). Let \\(X\\to\\operatorname{Spec}R\\) be proper, with \\(R\\) Noetherian, and let \\(L\\) be ample. For coherent \\(F\\), all \\(H^q(X,F)\\) are finite over \\(R\\); the groups \\(H^q(X,F\\otimes L^d)\\), \\(q>0\\), vanish for all sufficiently large \\(d\\); and \\(F\\otimes L^d\\) is globally generated for all sufficiently large \\(d\\).",
      "source": "src/serres-theorems-on-projective-schemes.md",
      "source_sha256": "ddac40fd2bf24adfb7184702da4b3adcf5596f2739bbd9fb2d81bf9941f3f116",
      "source_line": 83,
      "reader": "serres-theorems-on-projective-schemes.html#AG-QC-05-Serre-ample",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-05.Corollary-2.3",
      "unit": "AG-QC-05",
      "label": "Corollary 2.3",
      "statement_heading": "Corollary 2.3 (relative finiteness and vanishing). If \\(S\\) is locally Noetherian, \\(f:X\\to S\\) is locally projective, and \\(F\\) is coherent, every \\(R^qf_*F\\) is coherent. If \\(S\\) is Noetherian, \\(f\\) is proper, and \\(L\\) is \\(f\\)-ample, then there is a single \\(d_0\\) such that",
      "source": "src/serres-theorems-on-projective-schemes.md",
      "source_sha256": "ddac40fd2bf24adfb7184702da4b3adcf5596f2739bbd9fb2d81bf9941f3f116",
      "source_line": 89,
      "reader": "serres-theorems-on-projective-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-05.Lemma-3.1",
      "unit": "AG-QC-05",
      "label": "Lemma 3.1",
      "statement_heading": "Lemma 3.1 (finite section module). The module \\(G(F)\\) is finitely generated over \\(B\\).",
      "source": "src/serres-theorems-on-projective-schemes.md",
      "source_sha256": "ddac40fd2bf24adfb7184702da4b3adcf5596f2739bbd9fb2d81bf9941f3f116",
      "source_line": 106,
      "reader": "serres-theorems-on-projective-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-05.Lemma-3.2",
      "unit": "AG-QC-05",
      "label": "Lemma 3.2",
      "statement_heading": "Lemma 3.2 (a sheaf sees the high-degree tail). For a finite graded \\(B\\)-module \\(M\\), the natural map",
      "source": "src/serres-theorems-on-projective-schemes.md",
      "source_sha256": "ddac40fd2bf24adfb7184702da4b3adcf5596f2739bbd9fb2d81bf9941f3f116",
      "source_line": 116,
      "reader": "serres-theorems-on-projective-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-05.Theorem-3.3",
      "unit": "AG-QC-05",
      "label": "Theorem 3.3",
      "statement_heading": "Theorem 3.3 (Serre's graded equivalence). Sheafification gives an equivalence",
      "source": "src/serres-theorems-on-projective-schemes.md",
      "source_sha256": "ddac40fd2bf24adfb7184702da4b3adcf5596f2739bbd9fb2d81bf9941f3f116",
      "source_line": 140,
      "reader": "serres-theorems-on-projective-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-05.Theorem-4.1",
      "unit": "AG-QC-05",
      "label": "Theorem 4.1",
      "statement_heading": "Theorem 4.1 (cohomological criterion). Let \\(X\\) be proper over a Noetherian ring \\(R\\), and let \\(L\\) be invertible. The following conditions are equivalent:",
      "source": "src/serres-theorems-on-projective-schemes.md",
      "source_sha256": "ddac40fd2bf24adfb7184702da4b3adcf5596f2739bbd9fb2d81bf9941f3f116",
      "source_line": 156,
      "reader": "serres-theorems-on-projective-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-06.Theorem-1.1",
      "unit": "AG-QC-06",
      "label": "Theorem 1.1",
      "statement_heading": "Theorem 1.1 (Artin–Rees for coherent sheaves). Let \\(F\\) be coherent, \\(G\\subset F\\) quasi-coherent, and \\(J\\subset\\mathcal O_X\\) a quasi-coherent ideal. There is an integer \\(c\\ge0\\) such that",
      "source": "src/proper-morphisms-and-coherent-direct-images.md",
      "source_sha256": "57e9b16f91b17b5a2bc6b4a812ab552ea2d9eed0d017e55dcbaac95adec75686",
      "source_line": 19,
      "reader": "proper-morphisms-and-coherent-direct-images.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-06.Lemma-1.2",
      "unit": "AG-QC-06",
      "label": "Lemma 1.2",
      "statement_heading": "Lemma 1.2 (extend a morphism across a closed complement). Let \\(U=X\\setminus V(J)\\), and let \\(F,G\\) be coherent. Any map \\(G|_U\\to F|_U\\) extends to a map \\(J^aG\\to F\\) for some \\(a\\ge0\\). Two such maps that agree on \\(U\\) agree after restricting to a sufficiently higher power. Equivalently,",
      "source": "src/proper-morphisms-and-coherent-direct-images.md",
      "source_sha256": "57e9b16f91b17b5a2bc6b4a812ab552ea2d9eed0d017e55dcbaac95adec75686",
      "source_line": 42,
      "reader": "proper-morphisms-and-coherent-direct-images.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-06.Lemma-2.1",
      "unit": "AG-QC-06",
      "label": "Lemma 2.1",
      "statement_heading": "Lemma 2.1 (a generic lattice). Let \\(i:Z\\hookrightarrow X\\) be integral, with generic point \\(\\eta\\). If \\(F\\) is coherent and \\(\\mathfrak m_\\eta F_\\eta=0\\), there is a coherent ideal \\(I\\subset\\mathcal O_Z\\) and an injection",
      "source": "src/proper-morphisms-and-coherent-direct-images.md",
      "source_sha256": "57e9b16f91b17b5a2bc6b4a812ab552ea2d9eed0d017e55dcbaac95adec75686",
      "source_line": 62,
      "reader": "proper-morphisms-and-coherent-direct-images.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-06.Theorem-2.2",
      "unit": "AG-QC-06",
      "label": "Theorem 2.2",
      "statement_heading": "Theorem 2.2 (filtration by ideals on integral subschemes). Every coherent \\(F\\) on \\(X\\) has a finite filtration whose nonzero successive quotients are",
      "source": "src/proper-morphisms-and-coherent-direct-images.md",
      "source_sha256": "57e9b16f91b17b5a2bc6b4a812ab552ea2d9eed0d017e55dcbaac95adec75686",
      "source_line": 75,
      "reader": "proper-morphisms-and-coherent-direct-images.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-06.Theorem-3.1",
      "unit": "AG-QC-06",
      "label": "Theorem 3.1",
      "statement_heading": "Theorem 3.1 (one generic-rank-one witness per integral support). Suppose a property \\(\\mathcal P\\) of coherent sheaves is invariant under isomorphism, holds for zero, and satisfies two out of three in short exact sequences. Assume that for every integral closed \\(Z\\subset X\\), with generic point \\(\\eta\\), there is a coherent sheaf \\(G_Z\\) such that",
      "source": "src/proper-morphisms-and-coherent-direct-images.md",
      "source_sha256": "57e9b16f91b17b5a2bc6b4a812ab552ea2d9eed0d017e55dcbaac95adec75686",
      "source_line": 91,
      "reader": "proper-morphisms-and-coherent-direct-images.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-06.Lemma-4.0",
      "unit": "AG-QC-06",
      "label": "Lemma 4.0",
      "statement_heading": "Lemma 4.0 (the integral Chow construction). Let \\(Z\\) be an integral separated scheme of finite type over a Noetherian ring \\(A\\). There is an integral quasi-projective \\(A\\)-scheme \\(Y\\) and a projective surjection \\(\\pi:Y\\to Z\\), an isomorphism over a nonempty open. If \\(Z\\) is proper over \\(A\\), then \\(Y\\) is projective over \\(A\\).",
      "source": "src/proper-morphisms-and-coherent-direct-images.md",
      "source_sha256": "57e9b16f91b17b5a2bc6b4a812ab552ea2d9eed0d017e55dcbaac95adec75686",
      "source_line": 113,
      "reader": "proper-morphisms-and-coherent-direct-images.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-06.Theorem-4.1",
      "unit": "AG-QC-06",
      "label": "Theorem 4.1",
      "statement_heading": "Theorem 4.1 (Grothendieck's coherence theorem). Let \\(S\\) be locally Noetherian, \\(f:X\\to S\\) proper, and \\(F\\) coherent on \\(X\\). Then every \\(R^qf_*F\\), \\(q\\ge0\\), is coherent on \\(S\\).",
      "source": "src/proper-morphisms-and-coherent-direct-images.md",
      "source_sha256": "57e9b16f91b17b5a2bc6b4a812ab552ea2d9eed0d017e55dcbaac95adec75686",
      "source_line": 136,
      "reader": "proper-morphisms-and-coherent-direct-images.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-06.Corollary-5.1",
      "unit": "AG-QC-06",
      "label": "Corollary 5.1",
      "statement_heading": "Corollary 5.1. If \\(X\\) is proper over a Noetherian ring \\(R\\) and \\(F\\) is coherent, every \\(H^q(X,F)\\) is a finite \\(R\\)-module.",
      "source": "src/proper-morphisms-and-coherent-direct-images.md",
      "source_sha256": "57e9b16f91b17b5a2bc6b4a812ab552ea2d9eed0d017e55dcbaac95adec75686",
      "source_line": 163,
      "reader": "proper-morphisms-and-coherent-direct-images.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-07.Proposition-1.1",
      "unit": "AG-QC-07",
      "label": "Proposition 1.1",
      "statement_heading": "Proposition 1.1 (additivity). For a short exact sequence of coherent sheaves on a proper \\(k\\)-scheme,",
      "source": "src/euler-characteristics-and-hilbert-polynomials.md",
      "source_sha256": "4a60fc9c4f97065c8bddd8c2ae1c0a4337c04eefa167d5bb8f3586ab33e8a263",
      "source_line": 17,
      "reader": "euler-characteristics-and-hilbert-polynomials.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-07.Proposition-1.2",
      "unit": "AG-QC-07",
      "label": "Proposition 1.2",
      "statement_heading": "Proposition 1.2 (proper pushforward). If \\(f:Y\\to X\\) is a \\(k\\)-morphism between proper \\(k\\)-schemes and \\(G\\) is coherent on \\(Y\\), then",
      "source": "src/euler-characteristics-and-hilbert-polynomials.md",
      "source_sha256": "4a60fc9c4f97065c8bddd8c2ae1c0a4337c04eefa167d5bb8f3586ab33e8a263",
      "source_line": 25,
      "reader": "euler-characteristics-and-hilbert-polynomials.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-07.Theorem-2.1",
      "unit": "AG-QC-07",
      "label": "Theorem 2.1",
      "statement_heading": "Theorem 2.1 (polynomial Euler characteristic). Let \\(X\\) be proper over \\(k\\), \\(L\\) invertible, and \\(F\\) coherent. There is a unique numerical polynomial \\(P_{F,L}\\) such that",
      "source": "src/euler-characteristics-and-hilbert-polynomials.md",
      "source_sha256": "4a60fc9c4f97065c8bddd8c2ae1c0a4337c04eefa167d5bb8f3586ab33e8a263",
      "source_line": 53,
      "reader": "euler-characteristics-and-hilbert-polynomials.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-07.Theorem-3.1",
      "unit": "AG-QC-07",
      "label": "Theorem 3.1",
      "statement_heading": "Theorem 3.1 (degree and positive leading coefficient). If \\(L\\) is ample and \\(F\\ne0\\), then",
      "source": "src/euler-characteristics-and-hilbert-polynomials.md",
      "source_sha256": "4a60fc9c4f97065c8bddd8c2ae1c0a4337c04eefa167d5bb8f3586ab33e8a263",
      "source_line": 83,
      "reader": "euler-characteristics-and-hilbert-polynomials.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-07.Lemma-4.1",
      "unit": "AG-QC-07",
      "label": "Lemma 4.1",
      "statement_heading": "Lemma 4.1 (the high-twist flat complex). Locally on \\(S\\), there is an integer \\(m_0\\) such that for every \\(m\\geq m_0\\), the module \\(H^0(X,F(m))\\) is finite locally free and commutes with every base change. All higher cohomology of these twists is zero after every base change.",
      "source": "src/euler-characteristics-and-hilbert-polynomials.md",
      "source_sha256": "4a60fc9c4f97065c8bddd8c2ae1c0a4337c04eefa167d5bb8f3586ab33e8a263",
      "source_line": 124,
      "reader": "euler-characteristics-and-hilbert-polynomials.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-07.Theorem-4.2",
      "unit": "AG-QC-07",
      "label": "Theorem 4.2",
      "statement_heading": "Theorem 4.2 (constancy in a flat projective family). The function \\(s\\mapsto P_{F_s}\\in\\mathbf Q[T]\\) is locally constant.",
      "source": "src/euler-characteristics-and-hilbert-polynomials.md",
      "source_sha256": "4a60fc9c4f97065c8bddd8c2ae1c0a4337c04eefa167d5bb8f3586ab33e8a263",
      "source_line": 141,
      "reader": "euler-characteristics-and-hilbert-polynomials.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-08.Proposition-1.1",
      "unit": "AG-QC-08",
      "label": "Proposition 1.1",
      "statement_heading": "Proposition 1.1 (affine morphisms). If \\(f\\) is affine, every \\(\\beta^q\\) is an isomorphism for every base change \\(g\\), without a flatness assumption.",
      "source": "src/base-change-and-the-grothendieck-complex.md",
      "source_sha256": "d74a9c8c4789238fb8bea05cf6de26f0b2fb1df88a357b74cffec1d14243dcf8",
      "source_line": 46,
      "reader": "base-change-and-the-grothendieck-complex.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-08.Theorem-2.1",
      "unit": "AG-QC-08",
      "label": "Theorem 2.1",
      "statement_heading": "Theorem 2.1 (flat base change). If \\(f\\) is quasi-compact and quasi-separated, \\(F\\) is quasi-coherent, and \\(g\\) is flat, then",
      "source": "src/base-change-and-the-grothendieck-complex.md",
      "source_sha256": "d74a9c8c4789238fb8bea05cf6de26f0b2fb1df88a357b74cffec1d14243dcf8",
      "source_line": 58,
      "reader": "base-change-and-the-grothendieck-complex.html#AG-QC-08-flat-base-change",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-08.Lemma-3.1",
      "unit": "AG-QC-08",
      "label": "Lemma 3.1",
      "statement_heading": "Lemma 3.1 (bounded flat complexes). Let \\(A\\) be Noetherian. Let \\(C^\\bullet\\) be a complex of flat \\(A\\)-modules, zero outside \\([a,b]\\), with finite cohomology modules. There is a complex \\(K^\\bullet\\) of finite projective modules, also zero outside \\([a,b]\\), and a quasi-isomorphism",
      "source": "src/base-change-and-the-grothendieck-complex.md",
      "source_sha256": "d74a9c8c4789238fb8bea05cf6de26f0b2fb1df88a357b74cffec1d14243dcf8",
      "source_line": 87,
      "reader": "base-change-and-the-grothendieck-complex.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-08.Theorem-3.2",
      "unit": "AG-QC-08",
      "label": "Theorem 3.2",
      "statement_heading": "Theorem 3.2 (the Grothendieck complex). Let \\(A\\) be Noetherian, \\(f:X\\to\\operatorname{Spec}A\\) proper, and \\(F\\) coherent and flat over \\(A\\). There is a bounded complex \\(K\\) of finite projective \\(A\\)-modules, in nonnegative degrees, such that for every \\(A\\)-algebra \\(B\\),",
      "source": "src/base-change-and-the-grothendieck-complex.md",
      "source_sha256": "d74a9c8c4789238fb8bea05cf6de26f0b2fb1df88a357b74cffec1d14243dcf8",
      "source_line": 131,
      "reader": "base-change-and-the-grothendieck-complex.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-08.Corollary-4.1",
      "unit": "AG-QC-08",
      "label": "Corollary 4.1",
      "statement_heading": "Corollary 4.1. Under Theorem 3.2, the fiber Euler characteristic \\(s\\mapsto\\chi(X_s,F_s)\\) is locally constant on \\(\\operatorname{Spec}A\\).",
      "source": "src/base-change-and-the-grothendieck-complex.md",
      "source_sha256": "d74a9c8c4789238fb8bea05cf6de26f0b2fb1df88a357b74cffec1d14243dcf8",
      "source_line": 149,
      "reader": "base-change-and-the-grothendieck-complex.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-09.Lemma-0.1",
      "unit": "AG-QC-09",
      "label": "Lemma 0.1",
      "statement_heading": "Lemma 0.1 (a finite complex over an arbitrary base). If \\(A\\) is any ring, \\(X\\to\\operatorname{Spec}A\\) is proper of finite presentation and \\(F\\) is finitely presented and \\(A\\)-flat, there is a finite projective complex \\(K\\), in nonnegative degrees, which computes \\(H^q(X_B,F_B)\\) after every \\(A\\)-algebra change \\(B\\).",
      "source": "src/semicontinuity-and-grauerts-theorem.md",
      "source_sha256": "b6fdd5f794d77aedb97b216b27a9cea4edb61538afcd59cd21c715c64c6afac1",
      "source_line": 9,
      "reader": "semicontinuity-and-grauerts-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-09.Theorem-1.1",
      "unit": "AG-QC-09",
      "label": "Theorem 1.1",
      "statement_heading": "Theorem 1.1 (semicontinuity). Under the stated proper flat-sheaf hypotheses, every \\(h^q:S\\to\\mathbf Z_{\\geq0}\\) is upper semicontinuous. Its level sets are locally constructible, and its values commute with arbitrary base change of parameter schemes.",
      "source": "src/semicontinuity-and-grauerts-theorem.md",
      "source_sha256": "b6fdd5f794d77aedb97b216b27a9cea4edb61538afcd59cd21c715c64c6afac1",
      "source_line": 36,
      "reader": "semicontinuity-and-grauerts-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-09.Lemma-2.1",
      "unit": "AG-QC-09",
      "label": "Lemma 2.1",
      "statement_heading": "Lemma 2.1 (a minimal complex near a point). For a fixed \\(s\\), after restricting to an affine neighborhood and trivializing the terms, the complex can be replaced by a finite free complex whose differentials are all zero over \\(\\kappa(s)\\). The replacement preserves its cohomology after every base change.",
      "source": "src/semicontinuity-and-grauerts-theorem.md",
      "source_sha256": "b6fdd5f794d77aedb97b216b27a9cea4edb61538afcd59cd21c715c64c6afac1",
      "source_line": 44,
      "reader": "semicontinuity-and-grauerts-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-09.Lemma-2.2",
      "unit": "AG-QC-09",
      "label": "Lemma 2.2",
      "statement_heading": "Lemma 2.2 (lifting cocycles kills a differential). For a complex as in Lemma 2.1, \\(\\varphi^q(s)\\) is surjective if and only if \\(d^q=0\\) in the local ring at \\(s\\). If it is surjective, \\(d^q\\) is zero on a smaller neighborhood.",
      "source": "src/semicontinuity-and-grauerts-theorem.md",
      "source_sha256": "b6fdd5f794d77aedb97b216b27a9cea4edb61538afcd59cd21c715c64c6afac1",
      "source_line": 54,
      "reader": "semicontinuity-and-grauerts-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-09.Theorem-3.1",
      "unit": "AG-QC-09",
      "label": "Theorem 3.1",
      "statement_heading": "Theorem 3.1 (the surjectivity criterion). Suppose \\(\\varphi^q(s)\\) is surjective. There is an open neighborhood \\(U\\) of \\(s\\) on which formation of \\(R^qf_*F\\) commutes with every base change. In particular the comparison is an isomorphism at every point of \\(U\\). Under this hypothesis, the following are equivalent:",
      "source": "src/semicontinuity-and-grauerts-theorem.md",
      "source_sha256": "b6fdd5f794d77aedb97b216b27a9cea4edb61538afcd59cd21c715c64c6afac1",
      "source_line": 62,
      "reader": "semicontinuity-and-grauerts-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-09.Theorem-4.1",
      "unit": "AG-QC-09",
      "label": "Theorem 4.1",
      "statement_heading": "Theorem 4.1 (Grauert). Suppose \\(S\\) is reduced and \\(h^q\\) is locally constant. Then \\(R^qf_*F\\) is finite locally free and its formation commutes with every base change.",
      "source": "src/semicontinuity-and-grauerts-theorem.md",
      "source_sha256": "b6fdd5f794d77aedb97b216b27a9cea4edb61538afcd59cd21c715c64c6afac1",
      "source_line": 93,
      "reader": "semicontinuity-and-grauerts-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-09.Corollary-5.1",
      "unit": "AG-QC-09",
      "label": "Corollary 5.1",
      "statement_heading": "Corollary 5.1 (vanishing gives local freeness). If \\(R^qf_*F=0\\) for every \\(q>0\\), then \\(f_*F\\) is finite locally free and commutes with arbitrary base change. Its higher direct images also vanish after every base change.",
      "source": "src/semicontinuity-and-grauerts-theorem.md",
      "source_sha256": "b6fdd5f794d77aedb97b216b27a9cea4edb61538afcd59cd21c715c64c6afac1",
      "source_line": 109,
      "reader": "semicontinuity-and-grauerts-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-09.Theorem-5.2",
      "unit": "AG-QC-09",
      "label": "Theorem 5.2",
      "statement_heading": "Theorem 5.2 (constants in a proper flat family). Let \\(f\\) be proper, flat and of finite presentation, with nonempty geometrically reduced and geometrically connected fibers. Then the unit is an isomorphism",
      "source": "src/semicontinuity-and-grauerts-theorem.md",
      "source_sha256": "b6fdd5f794d77aedb97b216b27a9cea4edb61538afcd59cd21c715c64c6afac1",
      "source_line": 113,
      "reader": "semicontinuity-and-grauerts-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-09.Corollary-5.3",
      "unit": "AG-QC-09",
      "label": "Corollary 5.3",
      "statement_heading": "Corollary 5.3 (vanishing on every fiber). In either setting at the start of the lesson, assume \\(H^q(X_s,F_s)=0\\) for every point \\(s\\) and every \\(q>0\\). Then \\(f_*F\\) is finite locally free and commutes with every base change, and all positive higher direct images vanish universally. No reducedness assumption on \\(S\\) is required.",
      "source": "src/semicontinuity-and-grauerts-theorem.md",
      "source_sha256": "b6fdd5f794d77aedb97b216b27a9cea4edb61538afcd59cd21c715c64c6afac1",
      "source_line": 124,
      "reader": "semicontinuity-and-grauerts-theorem.html#AG-QC-09-vanishing-local-freeness",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Lemma-0.1",
      "unit": "AG-QC-10",
      "label": "Lemma 0.1",
      "statement_heading": "Lemma 0.1 (polynomial normality). If \\(R\\) is an integrally closed domain, then \\(R[x_1,\\ldots,x_n]\\) is an integrally closed domain for every finite \\(n\\).",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 13,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Lemma-2.1",
      "unit": "AG-QC-10",
      "label": "Lemma 2.1",
      "statement_heading": "Lemma 2.1 (graded cohomological finiteness). For each \\(q\\geq0\\), the graded module",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 69,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Lemma-2.2",
      "unit": "AG-QC-10",
      "label": "Lemma 2.2",
      "statement_heading": "Lemma 2.2 (bounds on the image and kernel). There are integers \\(c_J,c_K\\geq0\\) such that",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 94,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Proposition-3.1",
      "unit": "AG-QC-10",
      "label": "Proposition 3.1",
      "statement_heading": "Proposition 3.1 (the Mittag–Leffler bound). For fixed \\(q\\), choose \\(c\\) as the kernel bound in degree \\(q+1\\). Then for \\(m\\geq n+c\\),",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 115,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Theorem-4.1",
      "unit": "AG-QC-10",
      "label": "Theorem 4.1",
      "statement_heading": "Theorem 4.1 (formal functions). Under the hypotheses of Section 1, the canonical maps give an isomorphism",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 133,
      "reader": "the-theorem-on-formal-functions.html#AG-QC-10-formal-functions",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Corollary-4.2",
      "unit": "AG-QC-10",
      "label": "Corollary 4.2",
      "statement_heading": "Corollary 4.2 (the stalk form). Let \\(f:X\\to S\\) be proper, \\(S\\) locally Noetherian, \\(F\\) coherent, and \\(s\\in S\\). Put \\(A_s=\\mathcal O_{S,s}\\), with maximal ideal \\(\\mathfrak m_s\\), and",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 145,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Theorem-5.1",
      "unit": "AG-QC-10",
      "label": "Theorem 5.1",
      "statement_heading": "Theorem 5.1 (vanishing above a fiber's dimension). For the morphism of Corollary 4.2,",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 159,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Theorem-5.2",
      "unit": "AG-QC-10",
      "label": "Theorem 5.2",
      "statement_heading": "Theorem 5.2 (proper with finite fibers is finite). A proper morphism over a locally Noetherian base whose fibers have finitely many points is finite.",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 168,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Corollary-5.3",
      "unit": "AG-QC-10",
      "label": "Corollary 5.3",
      "statement_heading": "Corollary 5.3 (a finite fiber has a finite neighborhood). If \\(f:X\\to S\\) is proper, \\(S\\) is locally Noetherian, and \\(X_s\\) is a finite set, then \\(f\\) is finite over an open neighborhood of \\(s\\).",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 180,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Lemma-5.4",
      "unit": "AG-QC-10",
      "label": "Lemma 5.4",
      "statement_heading": "Lemma 5.4 (cartesian nilpotent thickenings). Let \\(Y_0\\hookrightarrow Y\\) be defined by an ideal \\(J\\) with \\(J^N=0\\), let \\(f:X\\to Y\\), and put \\(X_0=X\\times_Y Y_0\\). If \\(f_0:X_0\\to Y_0\\) is of finite type, a closed immersion, or proper, respectively, then so is \\(f\\).",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 184,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Theorem-8.1",
      "unit": "AG-QC-10",
      "label": "Theorem 8.1",
      "statement_heading": "Theorem 8.1. If \\(f:X\\to S\\) is proper with \\(S\\) locally Noetherian and \\(L|_{X_s}\\) is ample, then \\(L\\) is relatively ample over an open neighborhood of \\(s\\).",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 254,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Lemma-Z.1",
      "unit": "AG-QC-10",
      "label": "Lemma Z.1",
      "statement_heading": "Lemma Z.1. Let \\(A\\to B\\) be a ring map, let \\(T\\subset A\\) be multiplicative, and let \\(B^{\\mathrm{int}}\\subset B\\) be the subalgebra of elements integral over \\(A\\). The integral closure of \\(T^{-1}A\\) in \\(T^{-1}B\\) is \\(T^{-1}B^{\\mathrm{int}}\\), viewed as a subring of \\(T^{-1}B\\).",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 276,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Lemma-Z.2",
      "unit": "AG-QC-10",
      "label": "Lemma Z.2",
      "statement_heading": "Lemma Z.2. Let \\(\\varphi:R[X]\\to S\\) be a ring map and let \\(t\\in S\\) be integral over \\(R[X]\\). Suppose that for some monic \\(p\\in R[X]\\) the element \\(t\\varphi(p)\\) belongs to \\(\\operatorname{im}\\varphi\\). There is \\(q\\in R[X]\\) such that \\(t-\\varphi(q)\\) is integral over \\(R\\).",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 299,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Lemma-Z.3",
      "unit": "AG-QC-10",
      "label": "Lemma Z.3",
      "statement_heading": "Lemma Z.3. Let \\(\\varphi:R[X]\\to S\\) be a ring map, let \\(t\\in S\\) be integral over \\(R[X]\\), and let",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 322,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Lemma-Z.5",
      "unit": "AG-QC-10",
      "label": "Lemma Z.5",
      "statement_heading": "Lemma Z.5. In Situation Z.4, let \\(u\\in S\\) and \\(p=\\sum_{i=0}^k a_iX^i\\in R[X]\\). If \\(u\\varphi(p)\\in J\\), there is \\(m\\geq0\\) such that \\(u\\varphi(a_k)^m\\in J\\).",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 353,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Lemma-Z.6",
      "unit": "AG-QC-10",
      "label": "Lemma Z.6",
      "statement_heading": "Lemma Z.6. In Situation Z.4, if \\(u\\varphi(p)\\in\\sqrt J\\), where \\(u\\in S\\) and \\(p=\\sum_{i=0}^k a_iX^i\\), then \\(u\\varphi(a_i)\\in\\sqrt J\\) for every \\(i\\).",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 367,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Lemma-Z.8",
      "unit": "AG-QC-10",
      "label": "Lemma Z.8",
      "statement_heading": "Lemma Z.8. Consider a commutative square of rings",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 401,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Lemma-Z.9",
      "unit": "AG-QC-10",
      "label": "Lemma Z.9",
      "statement_heading": "Lemma Z.9. Let \\(R\\subset S\\) be an inclusion of reduced rings, let \\(x\\in S\\) be strongly transcendental over \\(R\\), and let \\(\\mathfrak q\\) be a minimal prime of \\(S\\). Put \\(\\mathfrak p=R\\cap\\mathfrak q\\). The image of \\(x\\) in \\(S/\\mathfrak q\\) is strongly transcendental over \\(R/\\mathfrak p\\).",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 429,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Lemma-Z.10",
      "unit": "AG-QC-10",
      "label": "Lemma Z.10",
      "statement_heading": "Lemma Z.10. Suppose \\(R\\subset S\\) are domains, \\(x\\in S\\) is transcendental over \\(R\\), and \\(S\\) is finite over \\(R[x]\\). Then \\(R\\to S\\) is not quasi-finite at any prime of \\(S\\).",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 437,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Lemma-Z.11",
      "unit": "AG-QC-10",
      "label": "Lemma Z.11",
      "statement_heading": "Lemma Z.11. Suppose \\(R\\subset S\\) are reduced rings, \\(x\\in S\\) is strongly transcendental over \\(R\\), and \\(S\\) is finite over \\(R[x]\\). Then \\(R\\to S\\) is not quasi-finite at any prime of \\(S\\).",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 458,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-10.Corollary-Z.12",
      "unit": "AG-QC-10",
      "label": "Corollary Z.12",
      "statement_heading": "Corollary Z.12. In Situation Z.4, if \\(A\\to S\\) is quasi-finite at a prime \\(\\mathfrak q\\subset S\\), then \\(J\\not\\subset\\mathfrak q\\).",
      "source": "src/the-theorem-on-formal-functions.md",
      "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3",
      "source_line": 468,
      "reader": "the-theorem-on-formal-functions.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Lemma-1.1",
      "unit": "AG-QC-11",
      "label": "Lemma 1.1",
      "statement_heading": "Lemma 1.1 (idempotents across a thickening). If \\(T_0\\hookrightarrow T\\) is a closed immersion defined by a nilpotent ideal, restriction induces a bijection on global idempotents.",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 19,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Theorem-2.1",
      "unit": "AG-QC-11",
      "label": "Theorem 2.1",
      "statement_heading": "Theorem 2.1. Let \\(f:X\\to S\\) be proper, with \\(S\\) locally Noetherian. If the unit map",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 39,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Theorem-3.1",
      "unit": "AG-QC-11",
      "label": "Theorem 3.1",
      "statement_heading": "Theorem 3.1 (Stein factorization). In (2), \\(\\pi\\) is finite, \\(g\\) is proper and surjective with geometrically connected fibers, and its unit map gives",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 83,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Proposition-4.1",
      "unit": "AG-QC-11",
      "label": "Proposition 4.1",
      "statement_heading": "Proposition 4.1. For \\(s\\in S\\), the points of the finite scheme \\(Y_s\\) correspond bijectively to the connected components of \\(X_s\\). For a geometric point \\(\\overline s\\), the geometric points of \\(Y_{\\overline s}\\) correspond to the connected components of \\(X_{\\overline s}\\).",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 107,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Theorem-5.1",
      "unit": "AG-QC-11",
      "label": "Theorem 5.1",
      "statement_heading": "Theorem 5.1. Let \\(f:X\\to S\\) be a proper birational morphism of integral Noetherian schemes, and suppose \\(S\\) is normal. Then the unit is an isomorphism",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 121,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Theorem-6.1",
      "unit": "AG-QC-11",
      "label": "Theorem 6.1",
      "statement_heading": "Theorem 6.1 (Stein factorization over an arbitrary base). Let \\(f:X\\to S\\) be proper, with \\(S\\) any scheme. Put",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 153,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Lemma-A.2.1",
      "unit": "AG-QC-11",
      "label": "Lemma A.2.1",
      "statement_heading": "Lemma A.2.1. Every qcqs scheme \\(T\\) is an inverse limit of schemes of finite type over \\(\\mathbf Z\\), with affine transition maps. A prescribed finite affine cover of \\(T\\) can be represented eventually by an affine cover of the stages.",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 222,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Lemma-A.3.1",
      "unit": "AG-QC-11",
      "label": "Lemma A.3.1",
      "statement_heading": "Lemma A.3.1 (finite submodules). On a qcqs scheme, every quasi-coherent module is the directed union of its quasi-coherent submodules of finite type. In particular a quasi-coherent ideal is the directed union of finite-type quasi-coherent ideals.",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 259,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Lemma-A.3.2",
      "unit": "AG-QC-11",
      "label": "Lemma A.3.2",
      "statement_heading": "Lemma A.3.2 (finite-type ambient embedding). A separated finite-type morphism \\(X\\to\\operatorname{Spec}A\\) admits a closed immersion into a separated scheme \\(Y\\) of finite presentation over \\(A\\).",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 265,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Lemma-A.4.1",
      "unit": "AG-QC-11",
      "label": "Lemma A.4.1",
      "statement_heading": "Lemma A.4.1 (relative proper approximation). If \\(X\\to\\operatorname{Spec}A\\) is proper, then",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 286,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Lemma-A.4.2",
      "unit": "AG-QC-11",
      "label": "Lemma A.4.2",
      "statement_heading": "Lemma A.4.2 (properness at a Noetherian base stage). Every proper finitely presented \\(Z\\to\\operatorname{Spec}A\\) descends to a proper morphism over a finitely generated \\(\\mathbf Z\\)-subalgebra \\(A_i\\subset A\\).",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 301,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Lemma-B.1",
      "unit": "AG-QC-11",
      "label": "Lemma B.1",
      "statement_heading": "Lemma B.1 (functions are integral). If \\(f:X\\to\\operatorname{Spec}A\\) is proper, then \\(B=\\Gamma(X,\\mathcal O_X)\\) is an integral \\(A\\)-algebra.",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 309,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Lemma-B.2",
      "unit": "AG-QC-11",
      "label": "Lemma B.2",
      "statement_heading": "Lemma B.2 (fiber idempotents lift as functions). For a proper morphism \\(f:X\\to S\\), a point \\(s\\in S\\), and an idempotent \\(e\\in\\Gamma(X_s,\\mathcal O_{X_s})\\), there is an element of the stalk \\((f_*\\mathcal O_X)_s\\) whose restriction to the fiber is \\(e\\). The stalk element itself need not be an idempotent.",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 313,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Lemma-B.3",
      "unit": "AG-QC-11",
      "label": "Lemma B.3",
      "statement_heading": "Lemma B.3 (degree-zero flat base change). For a proper morphism \\(f:X\\to S\\), formation of \\(f_*\\mathcal O_X\\) commutes with flat base change, without a Noetherian or finite-presentation assumption.",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 330,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Lemma-C.1",
      "unit": "AG-QC-11",
      "label": "Lemma C.1",
      "statement_heading": "Lemma C.1. A nonempty proper scheme \\(T\\) over a field \\(k\\) is geometrically connected if it is connected after every finite separable extension of \\(k\\).",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 343,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-11.Theorem-C.2",
      "unit": "AG-QC-11",
      "label": "Theorem C.2",
      "statement_heading": "Theorem C.2 (arbitrary-base connectedness). If \\(f:X\\to S\\) is proper and the unit \\(\\mathcal O_S\\to f_*\\mathcal O_X\\) is an isomorphism, its fibers are nonempty and geometrically connected.",
      "source": "src/zariski-connectedness-and-stein-factorization.md",
      "source_sha256": "e1c87f639af4a21908a9d242de86a5b62eef279a70748664537a357127f76d11",
      "source_line": 353,
      "reader": "zariski-connectedness-and-stein-factorization.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-12.Lemma-1.1",
      "unit": "AG-QC-12",
      "label": "Lemma 1.1",
      "statement_heading": "Lemma 1.1 (the affine description). If \\(X=\\operatorname{Spec}R\\) and \\(J=I\\mathcal O_X\\), this category is equivalent to finite modules over \\(\\widehat R=\\varprojlim R/I^n\\). The correspondence is",
      "source": "src/grothendiecks-existence-theorem.md",
      "source_sha256": "73d2a498391d48873bba0eafe7ec46ab14056c1b8bc60843220cb17e544af77b",
      "source_line": 18,
      "reader": "grothendiecks-existence-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-12.Theorem-2.1",
      "unit": "AG-QC-12",
      "label": "Theorem 2.1",
      "statement_heading": "Theorem 2.1. The category \\(\\operatorname{Coh}(X,J)\\) is abelian, exactness is local on \\(X\\), and the completion functor",
      "source": "src/grothendiecks-existence-theorem.md",
      "source_sha256": "73d2a498391d48873bba0eafe7ec46ab14056c1b8bc60843220cb17e544af77b",
      "source_line": 42,
      "reader": "grothendiecks-existence-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-12.Lemma-3.1",
      "unit": "AG-QC-12",
      "label": "Lemma 3.1",
      "statement_heading": "Lemma 3.1 (completed Hom). For coherent \\(F,G\\), with \\(H=\\mathcal Hom(F,G)\\),",
      "source": "src/grothendiecks-existence-theorem.md",
      "source_sha256": "73d2a498391d48873bba0eafe7ec46ab14056c1b8bc60843220cb17e544af77b",
      "source_line": 78,
      "reader": "grothendiecks-existence-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-12.Theorem-3.2",
      "unit": "AG-QC-12",
      "label": "Theorem 3.2",
      "statement_heading": "Theorem 3.2 (full faithfulness). Completion on coherent sheaves on \\(X\\) is fully faithful.",
      "source": "src/grothendiecks-existence-theorem.md",
      "source_sha256": "73d2a498391d48873bba0eafe7ec46ab14056c1b8bc60843220cb17e544af77b",
      "source_line": 92,
      "reader": "grothendiecks-existence-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-12.Proposition-3.3",
      "unit": "AG-QC-12",
      "label": "Proposition 3.3",
      "statement_heading": "Proposition 3.3 (extension comparison, also with proper supports). If \\(F,G\\) are coherent on a separated finite-type \\(X/A\\), and have proper support, then completion induces isomorphisms",
      "source": "src/grothendiecks-existence-theorem.md",
      "source_sha256": "73d2a498391d48873bba0eafe7ec46ab14056c1b8bc60843220cb17e544af77b",
      "source_line": 98,
      "reader": "grothendiecks-existence-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-12.Lemma-4.1",
      "unit": "AG-QC-12",
      "label": "Lemma 4.1",
      "statement_heading": "Lemma 4.1 (uniform vanishing). There exists \\(d_0\\) such that",
      "source": "src/grothendiecks-existence-theorem.md",
      "source_sha256": "73d2a498391d48873bba0eafe7ec46ab14056c1b8bc60843220cb17e544af77b",
      "source_line": 131,
      "reader": "grothendiecks-existence-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-12.Theorem-5.1",
      "unit": "AG-QC-12",
      "label": "Theorem 5.1",
      "statement_heading": "Theorem 5.1 (projective existence). For projective \\(X\\) over \\(A\\), completion gives an equivalence",
      "source": "src/grothendiecks-existence-theorem.md",
      "source_sha256": "73d2a498391d48873bba0eafe7ec46ab14056c1b8bc60843220cb17e544af77b",
      "source_line": 146,
      "reader": "grothendiecks-existence-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-12.Theorem-6.1",
      "unit": "AG-QC-12",
      "label": "Theorem 6.1",
      "statement_heading": "Theorem 6.1 (Grothendieck existence). Let \\(A\\) be Noetherian and \\(I\\)-adically complete, and let \\(X\\) be separated and of finite type over \\(A\\). Completion gives an equivalence between coherent sheaves on \\(X\\) with support proper over \\(A\\) and coherent formal modules whose first-level support is proper over \\(A\\). In particular, when \\(X\\) is proper, all coherent sheaves and all coherent formal modules participate.",
      "source": "src/grothendiecks-existence-theorem.md",
      "source_sha256": "73d2a498391d48873bba0eafe7ec46ab14056c1b8bc60843220cb17e544af77b",
      "source_line": 170,
      "reader": "grothendiecks-existence-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-12.Proposition-7.1",
      "unit": "AG-QC-12",
      "label": "Proposition 7.1",
      "statement_heading": "Proposition 7.1. Every coherent formal module on \\(\\mathbf A^1_{k[[t]]}\\) is algebraizable.",
      "source": "src/grothendiecks-existence-theorem.md",
      "source_sha256": "73d2a498391d48873bba0eafe7ec46ab14056c1b8bc60843220cb17e544af77b",
      "source_line": 214,
      "reader": "grothendiecks-existence-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-13.Theorem-1.1",
      "unit": "AG-QC-13",
      "label": "Theorem 1.1",
      "statement_heading": "Theorem 1.1. There is a unique closed subscheme \\(Z\\hookrightarrow X\\) whose base changes are the specified \\(Z_n\\), compatibly as subschemes of \\(X_n\\).",
      "source": "src/algebraization-of-formal-schemes.md",
      "source_sha256": "1e9ba22657b922a40253804e023697846fe882e40a82a1626c8546981c0577b5",
      "source_line": 19,
      "reader": "algebraization-of-formal-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-13.Proposition-2.1",
      "unit": "AG-QC-13",
      "label": "Proposition 2.1",
      "statement_heading": "Proposition 2.1. Suppose \\(X\\) is proper over \\(S\\), and \\(Y_n\\to X_n\\) are compatible finite morphisms with cartesian squares. They algebraize to a finite morphism \\(Y\\to X\\), with \\(Y\\) proper over \\(S\\).",
      "source": "src/algebraization-of-formal-schemes.md",
      "source_sha256": "1e9ba22657b922a40253804e023697846fe882e40a82a1626c8546981c0577b5",
      "source_line": 31,
      "reader": "algebraization-of-formal-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-13.Theorem-2.2",
      "unit": "AG-QC-13",
      "label": "Theorem 2.2",
      "statement_heading": "Theorem 2.2 (algebraization of morphisms). If \\(X\\) is proper and \\(Y\\) is separated of finite type over \\(S\\), any compatible system \\(g_n:X_n\\to Y_n\\) comes from a unique \\(S\\)-morphism \\(g:X\\to Y\\).",
      "source": "src/algebraization-of-formal-schemes.md",
      "source_sha256": "1e9ba22657b922a40253804e023697846fe882e40a82a1626c8546981c0577b5",
      "source_line": 37,
      "reader": "algebraization-of-formal-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-13.Lemma-3.1",
      "unit": "AG-QC-13",
      "label": "Lemma 3.1",
      "statement_heading": "Lemma 3.1 (uniform section lifting). There is an integer \\(d_0\\) such that, for \\(d\\geq d_0\\),",
      "source": "src/algebraization-of-formal-schemes.md",
      "source_sha256": "1e9ba22657b922a40253804e023697846fe882e40a82a1626c8546981c0577b5",
      "source_line": 60,
      "reader": "algebraization-of-formal-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-13.Theorem-4.1",
      "unit": "AG-QC-13",
      "label": "Theorem 4.1",
      "statement_heading": "Theorem 4.1. The polarized system of Section 3 comes from a projective scheme \\(T\\to S\\) and an ample invertible \\(L\\), with compatible identifications",
      "source": "src/algebraization-of-formal-schemes.md",
      "source_sha256": "1e9ba22657b922a40253804e023697846fe882e40a82a1626c8546981c0577b5",
      "source_line": 86,
      "reader": "algebraization-of-formal-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-13.Theorem-5.1",
      "unit": "AG-QC-13",
      "label": "Theorem 5.1",
      "statement_heading": "Theorem 5.1 (the flat \\(H^2\\) criterion). Let \\((A,\\mathfrak m,k)\\) be complete Noetherian local. Suppose the cartesian system \\(T_n\\to\\operatorname{Spec}(A/\\mathfrak m^n)\\) is flat at every level, \\(T_1\\) is projective over \\(k\\), and",
      "source": "src/algebraization-of-formal-schemes.md",
      "source_sha256": "1e9ba22657b922a40253804e023697846fe882e40a82a1626c8546981c0577b5",
      "source_line": 122,
      "reader": "algebraization-of-formal-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-14.Proposition-1.1",
      "unit": "AG-QC-14",
      "label": "Proposition 1.1",
      "statement_heading": "Proposition 1.1 (local computation). Let \\(X\\) be Noetherian and let \\(F,G\\) be coherent. Every \\(\\mathcal E xt^i_X(F,G)\\) is coherent, and",
      "source": "src/ext-sheaves-and-serre-duality-on-projective-space.md",
      "source_sha256": "75b477b9ad460398fb168e58d57642c66fad898ebbaf81b4dc33db745910c096",
      "source_line": 23,
      "reader": "ext-sheaves-and-serre-duality-on-projective-space.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-14.Proposition-1.2",
      "unit": "AG-QC-14",
      "label": "Proposition 1.2",
      "statement_heading": "Proposition 1.2 (a vector bundle in the first input). For finite locally free \\(E\\), there are natural isomorphisms",
      "source": "src/ext-sheaves-and-serre-duality-on-projective-space.md",
      "source_sha256": "75b477b9ad460398fb168e58d57642c66fad898ebbaf81b4dc33db745910c096",
      "source_line": 43,
      "reader": "ext-sheaves-and-serre-duality-on-projective-space.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-14.Proposition-2.1",
      "unit": "AG-QC-14",
      "label": "Proposition 2.1",
      "statement_heading": "Proposition 2.1. For sheaves \\(F,G\\) there is a natural first-quadrant spectral sequence",
      "source": "src/ext-sheaves-and-serre-duality-on-projective-space.md",
      "source_sha256": "75b477b9ad460398fb168e58d57642c66fad898ebbaf81b4dc33db745910c096",
      "source_line": 64,
      "reader": "ext-sheaves-and-serre-duality-on-projective-space.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-14.Theorem-4.1",
      "unit": "AG-QC-14",
      "label": "Theorem 4.1",
      "statement_heading": "Theorem 4.1. The map (7) is an isomorphism",
      "source": "src/ext-sheaves-and-serre-duality-on-projective-space.md",
      "source_sha256": "75b477b9ad460398fb168e58d57642c66fad898ebbaf81b4dc33db745910c096",
      "source_line": 121,
      "reader": "ext-sheaves-and-serre-duality-on-projective-space.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-14.Theorem-5.1",
      "unit": "AG-QC-14",
      "label": "Theorem 5.1",
      "statement_heading": "Theorem 5.1 (Serre duality on projective space). For every coherent sheaf \\(F\\) and every integer \\(i\\), there are natural isomorphisms",
      "source": "src/ext-sheaves-and-serre-duality-on-projective-space.md",
      "source_sha256": "75b477b9ad460398fb168e58d57642c66fad898ebbaf81b4dc33db745910c096",
      "source_line": 145,
      "reader": "ext-sheaves-and-serre-duality-on-projective-space.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-14.Corollary-5.2",
      "unit": "AG-QC-14",
      "label": "Corollary 5.2",
      "statement_heading": "Corollary 5.2 (vector-bundle form). If \\(E\\) is finite locally free, then",
      "source": "src/ext-sheaves-and-serre-duality-on-projective-space.md",
      "source_sha256": "75b477b9ad460398fb168e58d57642c66fad898ebbaf81b4dc33db745910c096",
      "source_line": 184,
      "reader": "ext-sheaves-and-serre-duality-on-projective-space.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-15.Proposition-1.1",
      "unit": "AG-QC-15",
      "label": "Proposition 1.1",
      "statement_heading": "Proposition 1.1 (uniqueness). Two dualizing sheaves with traces have a unique isomorphism respecting their representing maps and traces.",
      "source": "src/dualizing-sheaves-and-serre-duality-for-projective-schemes.md",
      "source_sha256": "2eb5d75cf6789870a3868831c7e701c12ab4033a8f6122f3937cb945e1ad5223",
      "source_line": 26,
      "reader": "dualizing-sheaves-and-serre-duality-for-projective-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-15.Lemma-2.1",
      "unit": "AG-QC-15",
      "label": "Lemma 2.1",
      "statement_heading": "Lemma 2.1. The coherent sheaves \\(\\mathcal E xt_P^j(i_*\\mathcal O_X,\\omega_P)\\) vanish for \\(j<c\\). If \\(X\\) is Cohen–Macaulay and equidimensional of dimension \\(n\\), they vanish for every \\(j\\ne c\\).",
      "source": "src/dualizing-sheaves-and-serre-duality-for-projective-schemes.md",
      "source_sha256": "2eb5d75cf6789870a3868831c7e701c12ab4033a8f6122f3937cb945e1ad5223",
      "source_line": 36,
      "reader": "dualizing-sheaves-and-serre-duality-for-projective-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-15.Theorem-4.1",
      "unit": "AG-QC-15",
      "label": "Theorem 4.1",
      "statement_heading": "Theorem 4.1 (existence). Every projective \\(k\\)-scheme \\(X\\) of dimension \\(n\\) has a dualizing sheaf with trace. For an embedding in \\(\\mathbf P^N_k\\), it is",
      "source": "src/dualizing-sheaves-and-serre-duality-for-projective-schemes.md",
      "source_sha256": "2eb5d75cf6789870a3868831c7e701c12ab4033a8f6122f3937cb945e1ad5223",
      "source_line": 103,
      "reader": "dualizing-sheaves-and-serre-duality-for-projective-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-15.Theorem-4.2",
      "unit": "AG-QC-15",
      "label": "Theorem 4.2",
      "statement_heading": "Theorem 4.2 (Serre duality). If \\(X\\) is projective, Cohen–Macaulay and equidimensional of dimension \\(n\\), then for every coherent \\(F\\) and every integer \\(i\\),",
      "source": "src/dualizing-sheaves-and-serre-duality-for-projective-schemes.md",
      "source_sha256": "2eb5d75cf6789870a3868831c7e701c12ab4033a8f6122f3937cb945e1ad5223",
      "source_line": 121,
      "reader": "dualizing-sheaves-and-serre-duality-for-projective-schemes.html#AG-QC-15-CM-duality",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-15.Theorem-5.1",
      "unit": "AG-QC-15",
      "label": "Theorem 5.1",
      "statement_heading": "Theorem 5.1 (adjunction). There is an isomorphism",
      "source": "src/dualizing-sheaves-and-serre-duality-for-projective-schemes.md",
      "source_sha256": "2eb5d75cf6789870a3868831c7e701c12ab4033a8f6122f3937cb945e1ad5223",
      "source_line": 137,
      "reader": "dualizing-sheaves-and-serre-duality-for-projective-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-15.Proposition-8.1",
      "unit": "AG-QC-15",
      "label": "Proposition 8.1",
      "statement_heading": "Proposition 8.1 (the normalized complex, independently of an embedding). For every bounded complex \\(F\\) with coherent cohomology on \\(X\\), evaluation and the ambient trace induce a natural isomorphism",
      "source": "src/dualizing-sheaves-and-serre-duality-for-projective-schemes.md",
      "source_sha256": "2eb5d75cf6789870a3868831c7e701c12ab4033a8f6122f3937cb945e1ad5223",
      "source_line": 260,
      "reader": "dualizing-sheaves-and-serre-duality-for-projective-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-15.Proposition-8.2",
      "unit": "AG-QC-15",
      "label": "Proposition 8.2",
      "statement_heading": "Proposition 8.2 (the smooth canonical bundle). If \\(X/k\\) is smooth of pure dimension \\(n\\), then",
      "source": "src/dualizing-sheaves-and-serre-duality-for-projective-schemes.md",
      "source_sha256": "2eb5d75cf6789870a3868831c7e701c12ab4033a8f6122f3937cb945e1ad5223",
      "source_line": 281,
      "reader": "dualizing-sheaves-and-serre-duality-for-projective-schemes.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Lemma-1.A",
      "unit": "AG-QC-16",
      "label": "Lemma 1.A",
      "statement_heading": "Lemma 1.A (compactness and a finite Hom window). Every perfect object \\(P\\) on a qcqs scheme \\(X\\) is compact in \\(D_{\\mathrm{QCoh}}(X)\\). If \\(P^\\vee\\) has Tor amplitude \\([a,b]\\) and quasi-coherent cohomological dimension of \\(X\\) is at most \\(d\\), then",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 27,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Lemma-1.B",
      "unit": "AG-QC-16",
      "label": "Lemma 1.B",
      "statement_heading": "Lemma 1.B (affine denominators in derived maps). Let \\(V=\\operatorname{Spec}A\\), \\(W=\\bigcup_{i=1}^rD(f_i)\\), and let \\(P\\) be perfect on \\(V\\). Every derived map \\(P|_W\\to E|_W\\), with \\(E\\in D_{\\mathrm{QCoh}}(V)\\), is represented by",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 37,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Lemma-1.C",
      "unit": "AG-QC-16",
      "label": "Lemma 1.C",
      "statement_heading": "Lemma 1.C (lifting a perfect object and its map). Let \\(U\\subset X\\) be quasi-compact open in a qcqs scheme. Given perfect \\(P\\) on \\(U\\), quasi-coherent \\(E\\) on \\(X\\), and \\(\\alpha:P\\to E|_U\\), there is perfect \\(R\\) on \\(X\\) and \\(R\\to E\\) such that \\(R|_U\\) is a finite sum of shifts of \\(P\\), containing \\(P\\) as a summand on which the map is \\(\\alpha\\). If \\(P\\) is supported on \\(T\\cap U\\), where \\(T\\) is closed with quasi-compact complement, \\(R\\) can be chosen supported on \\(T\\).",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 64,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Lemma-1.D",
      "unit": "AG-QC-16",
      "label": "Lemma 1.D",
      "statement_heading": "Lemma 1.D (the affine support detector). On \\(V=\\operatorname{Spec}A\\), let \\(Z=V(f_1,\\ldots,f_r)\\) and \\(j:V\\setminus Z\\hookrightarrow V\\). For the Koszul complex \\(K=K(f_1,\\ldots,f_r)\\),",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 83,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Theorem-1.E",
      "unit": "AG-QC-16",
      "label": "Theorem 1.E",
      "statement_heading": "Theorem 1.E (qcqs compact generation). The category \\(D_{\\mathrm{QCoh}}(X)\\) has a single perfect compact generator. More generally, if \\(T\\subset X\\) is closed with quasi-compact complement, its full subcategory of supported objects has a perfect compact generator, which is compact also in the ambient category.",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 93,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Lemma-1.F",
      "unit": "AG-QC-16",
      "label": "Lemma 1.F",
      "statement_heading": "Lemma 1.F (cellular generation). Let \\(\\mathcal T\\) be a locally small triangulated category with coproducts, and let a set \\(\\mathcal G\\) of compact objects, closed under shifts, detect zero objects. Every object belongs to the smallest triangulated subcategory containing \\(\\mathcal G\\) and closed under coproducts.",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 110,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Theorem-1.G",
      "unit": "AG-QC-16",
      "label": "Theorem 1.G",
      "statement_heading": "Theorem 1.G (Brown representability and adjoints). A contravariant cohomological functor \\(H:\\mathcal T\\to\\mathrm{Ab}\\) which sends coproducts to products, with \\(\\mathcal T\\) as in Lemma 1.F, is represented by an object of \\(\\mathcal T\\). Consequently every coproduct-preserving exact functor from \\(\\mathcal T\\) to a locally small triangulated category has a right adjoint.",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 124,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Proposition-1.1",
      "unit": "AG-QC-16",
      "label": "Proposition 1.1",
      "statement_heading": "Proposition 1.1. The functor \\(a_f\\) is exact, commutes with shifts, and sends bounded-below objects to bounded-below objects. For composable morphisms \\(X\\xrightarrow fY\\xrightarrow gZ\\), there is a canonical isomorphism",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 146,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Proposition-2.1",
      "unit": "AG-QC-16",
      "label": "Proposition 2.1",
      "statement_heading": "Proposition 2.1. Applying \\(DQ_Y\\) to (4) gives an isomorphism. Taking global sections of (4) gives",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 185,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Lemma-2.2",
      "unit": "AG-QC-16",
      "label": "Lemma 2.2",
      "statement_heading": "Lemma 2.2 (finite factorization and killing a supported map). If \\(C\\) is compact and \\(E\\) is built by the cellular construction of Lemma 1.F from compact objects \\(G\\), every map \\(C\\to E\\) factors through a finite sequence of sums, shifts and cones of the \\(G\\)'s. Consequently, if \\(T\\subset Y\\) is closed with quasi-compact complement \\(V\\), \\(P\\) is pseudo-coherent, and \\(Q\\in D^+_{\\mathrm{QCoh}}(Y)\\) is supported on \\(T\\), every map \\(P\\to Q\\) is killed after precomposition with",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 213,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Theorem-2.3",
      "unit": "AG-QC-16",
      "label": "Theorem 2.3",
      "statement_heading": "Theorem 2.3 (proper restriction, with a verified Noetherian case). If \\(f\\) is proper and \\(Y\\) is Noetherian, (6), for its quasi-compact open \\(V\\), is an isomorphism for every \\(K\\in D^+_{\\mathrm{QCoh}}(Y)\\). More generally, the same proof applies whenever \\(Rf_*P\\) is pseudo-coherent for every perfect \\(P\\) on \\(X\\).",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 226,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Lemma-2.4",
      "unit": "AG-QC-16",
      "label": "Lemma 2.4",
      "statement_heading": "Lemma 2.4 (proper flat finite presentation). If \\(f:X\\to Y\\) is proper, flat and of finite presentation, with \\(Y\\) qcqs, then \\(Rf_*P\\) is perfect for every perfect \\(P\\) on \\(X\\). In particular it is pseudo-coherent, and Theorem 2.3 proves (6) for this case as well, for every quasi-compact open \\(V\\) and every bounded-below input.",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 247,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Theorem-3.1",
      "unit": "AG-QC-16",
      "label": "Theorem 3.1",
      "statement_heading": "Theorem 3.1. For \\(K\\in D^+_{\\mathrm{QCoh}}(Y)\\), the object \\(a_fK\\), expressed as a \\(B\\)-complex on \\(Y\\), is",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 275,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Theorem-4.1",
      "unit": "AG-QC-16",
      "label": "Theorem 4.1",
      "statement_heading": "Theorem 4.1. For \\(\\pi:P=\\mathbf P^n_k\\to\\operatorname{Spec}k\\),",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 313,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Theorem-4.2",
      "unit": "AG-QC-16",
      "label": "Theorem 4.2",
      "statement_heading": "Theorem 4.2 (the relative projective bundle). Let \\(E\\) be locally free of rank \\(n+1\\geq1\\) on a qcqs scheme \\(Y\\), and use the quotient convention for \\(\\pi:\\mathbf P(E)\\to Y\\). Then",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 330,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Theorem-5.1",
      "unit": "AG-QC-16",
      "label": "Theorem 5.1",
      "statement_heading": "Theorem 5.1. For every \\(K\\in D_{\\mathrm{QCoh}}(X)\\) and every integer \\(i\\), the trace induces",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 372,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Theorem-5.2",
      "unit": "AG-QC-16",
      "label": "Theorem 5.2",
      "statement_heading": "Theorem 5.2 (the complex on an arbitrary proper scheme). For every proper \\(k\\)-scheme \\(X\\), the complex \\(\\omega_X^\\bullet=a_p(k)\\) is bounded coherent, is locally dualizing, and has cohomology only in \\([-\\dim X,0]\\).",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 397,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Proposition-6.1",
      "unit": "AG-QC-16",
      "label": "Proposition 6.1",
      "statement_heading": "Proposition 6.1 (independence of compactification). Formula (14) is independent of the compactification, with canonical comparisons satisfying the cocycle condition.",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 460,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Proposition-6.2",
      "unit": "AG-QC-16",
      "label": "Proposition 6.2",
      "statement_heading": "Proposition 6.2 (composition). For composable separated finite-type maps of Noetherian schemes, there is a canonical isomorphism \\((gf)^!\\simeq f^!g^!\\). These comparisons are associative.",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 472,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Lemma-N.A.1",
      "unit": "AG-QC-16",
      "label": "Lemma N.A.1",
      "statement_heading": "Lemma N.A.1 (the blowup calculations). Let \\(T\\) be Noetherian and \\(I\\subset\\mathcal O_T\\) coherent. The blowup \\(b:T'\\to T\\) is proper. Its pulled-back ideal is invertible as an ideal; the complement of its zero locus is schematically dense in \\(T'\\). Over the locus where \\(I=\\mathcal O_T\\), the map is an isomorphism. It has the following additional properties.",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 552,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Lemma-N.A.2",
      "unit": "AG-QC-16",
      "label": "Lemma N.A.2",
      "statement_heading": "Lemma N.A.2 (separating closures). If \\(T_1,T_2\\) are disjoint closed subsets of an open \\(D\\subset T\\), there is a \\(D\\)-admissible blowup after which their closures are disjoint.",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 585,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Lemma-N.A.3",
      "unit": "AG-QC-16",
      "label": "Lemma N.A.3",
      "statement_heading": "Lemma N.A.3 (isomorphism descent used here). If an existing map becomes an isomorphism under an étale covering of its target, it is an isomorphism.",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 589,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Lemma-N.A.4",
      "unit": "AG-QC-16",
      "label": "Lemma N.A.4",
      "statement_heading": "Lemma N.A.4 (the quasi-affine input, proved without the nonaffine Zariski Main Theorem). A separated quasi-finite finite-type morphism \\(q:Z\\to T\\) of Noetherian schemes is quasi-affine.",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 593,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Theorem-N.B.1",
      "unit": "AG-QC-16",
      "label": "Theorem N.B.1",
      "statement_heading": "Theorem N.B.1. Let \\(T\\) be Noetherian, let \\(W\\subset V\\subset T\\) be opens with \\(W\\) schematically dense in \\(T\\), and let \\(p:Y\\to T\\) be separated and finite type. Let \\(s:W\\to Y\\) be a section whose graph is closed in \\(Y\\times_T V\\). There is a finite \\(V\\)-admissible sequence \\(T'\\to T\\) for which the closure of the lifted \\(s(W)\\) in \\(Y\\times_T T'\\) maps isomorphically to an open of \\(T'\\). This open contains \\(W\\).",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 607,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Corollary-N.B.2",
      "unit": "AG-QC-16",
      "label": "Corollary N.B.2",
      "statement_heading": "Corollary N.B.2 (the elimination form). Let \\(p:Z\\to T\\) be separated and finite type between Noetherian schemes, and let \\(D\\subset T\\) be open with \\(p^{-1}D\\to D\\) an isomorphism. After a finite \\(D\\)-admissible sequence, the strict transform of \\(Z\\) is an open subscheme of the modified \\(T\\). If \\(p\\) is proper, that open is the whole modified \\(T\\).",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 662,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Lemma-N.C.1",
      "unit": "AG-QC-16",
      "label": "Lemma N.C.1",
      "statement_heading": "Lemma N.C.1 (improvement near a closure). Let \\(p:Z\\to T\\) be proper between Noetherian schemes, \\(V\\subset T\\) open, and \\(F\\subset V\\) closed. Suppose \\(p\\) is an isomorphism over an open neighbourhood of \\(F\\) in \\(V\\). There is a finite \\(V\\)-admissible sequence after which its strict transform is an isomorphism over a neighbourhood of the closure of \\(F\\).",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 668,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Lemma-N.C.2",
      "unit": "AG-QC-16",
      "label": "Lemma N.C.2",
      "statement_heading": "Lemma N.C.2 (separated common enlargement). Let \\(S\\) be Noetherian, and let \\(D\\to T_1,T_2\\) be open embeddings of separated finite-type \\(S\\)-schemes. Finite \\(D\\)-admissible modifications \\(T_i'\\to T_i\\) can be embedded openly into one separated finite-type \\(S\\)-scheme, identifying their copies of \\(D\\).",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 672,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Lemma-N.C.3",
      "unit": "AG-QC-16",
      "label": "Lemma N.C.3",
      "statement_heading": "Lemma N.C.3 (valuative tests from a dense open suffice in this Noetherian setting). Let \\(M\\to S\\) be separated and finite type with \\(S\\) Noetherian. Let \\(D\\subset M\\) be dense and open. If every valuation diagram whose generic map lands in \\(D\\) has an extension to \\(M\\), then \\(M\\to S\\) is proper.",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 678,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Theorem-N.D.1",
      "unit": "AG-QC-16",
      "label": "Theorem N.D.1",
      "statement_heading": "Theorem N.D.1. Let \\(X\\to S\\) be separated and finite type with \\(S\\) Noetherian. Suppose \\(X=U_1\\cup U_2\\), the intersection \\(D=U_1\\cap U_2\\) is dense in \\(X\\), and each \\(U_i\\) has a proper compactification over \\(S\\). Then \\(X\\) has a proper compactification over \\(S\\).",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 688,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-16.Theorem-N.E.1",
      "unit": "AG-QC-16",
      "label": "Theorem N.E.1",
      "statement_heading": "Theorem N.E.1. Every separated finite-type morphism \\(f:X\\to S\\) between Noetherian schemes factors as an open immersion \\(X\\hookrightarrow\\overline X\\) followed by a proper morphism \\(\\overline X\\to S\\).",
      "source": "src/the-right-adjoint-of-derived-pushforward.md",
      "source_sha256": "b865af6c9c9f5a0ac261bdfaec525bbb1e8ea334af229ea9838428f93590133c",
      "source_line": 731,
      "reader": "the-right-adjoint-of-derived-pushforward.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-17.Proposition-1.1",
      "unit": "AG-QC-17",
      "label": "Proposition 1.1",
      "statement_heading": "Proposition 1.1 (coherence on quotient models). The structure sheaf of every quotient model is coherent. Its coherent modules are exactly its locally finitely presented modules. Kernels, images, cokernels, extensions, tensor products and internal Hom of coherent modules are coherent. If \\(i:A\\hookrightarrow W\\) is the model inclusion, a module on the model is coherent exactly when its pushforward is coherent over \\(\\mathcal O_W\\).",
      "source": "src/complex-analytic-spaces-and-analytification.md",
      "source_sha256": "3d2aea96f2b5d621bdd3de89d70464d36607274527963d3d18100b6cd33a3487",
      "source_line": 52,
      "reader": "complex-analytic-spaces-and-analytification.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-17.Lemma-2.1",
      "unit": "AG-QC-17",
      "label": "Lemma 2.1",
      "statement_heading": "Lemma 2.1. The ring \\(H_n\\) is Noetherian, and its maximal-adic completion is \\(\\mathbf C[[z_1,\\ldots,z_n]]\\).",
      "source": "src/complex-analytic-spaces-and-analytification.md",
      "source_sha256": "3d2aea96f2b5d621bdd3de89d70464d36607274527963d3d18100b6cd33a3487",
      "source_line": 132,
      "reader": "complex-analytic-spaces-and-analytification.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-17.Lemma-2.2",
      "unit": "AG-QC-17",
      "label": "Lemma 2.2",
      "statement_heading": "Lemma 2.2 (holomorphic inverse and implicit functions). A holomorphic map between open subsets of \\(\\mathbf C^n\\) whose derivative is invertible at a point has a holomorphic inverse on smaller neighbourhoods. If \\(r\\) equations have Jacobian rank \\(r\\), their zero set is locally a complex manifold of dimension \\(n-r\\).",
      "source": "src/complex-analytic-spaces-and-analytification.md",
      "source_sha256": "3d2aea96f2b5d621bdd3de89d70464d36607274527963d3d18100b6cd33a3487",
      "source_line": 146,
      "reader": "complex-analytic-spaces-and-analytification.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-17.Lemma-3.1",
      "unit": "AG-QC-17",
      "label": "Lemma 3.1",
      "statement_heading": "Lemma 3.1. For a radical algebraic ideal \\(J\\) and a complex point \\(x\\), its extension to the ambient convergent local ring is radical and equals the analytic vanishing ideal of its zero germ.",
      "source": "src/complex-analytic-spaces-and-analytification.md",
      "source_sha256": "3d2aea96f2b5d621bdd3de89d70464d36607274527963d3d18100b6cd33a3487",
      "source_line": 163,
      "reader": "complex-analytic-spaces-and-analytification.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-17.Theorem-3.2",
      "unit": "AG-QC-17",
      "label": "Theorem 3.2",
      "statement_heading": "Theorem 3.2. Every scheme locally of finite type over \\(\\mathbf C\\) has an analytification, uniquely characterized by these finite type affine quotient charts. Scheme morphisms induce analytic morphisms, compatibly with identities and composition. Analytification commutes with products over \\(\\mathbf C\\), open immersions and closed immersions, including their full defining ideals. It therefore commutes with locally closed immersions. A separated scheme has Hausdorff analytification; a finite type scheme has second countable analytification.",
      "source": "src/complex-analytic-spaces-and-analytification.md",
      "source_sha256": "3d2aea96f2b5d621bdd3de89d70464d36607274527963d3d18100b6cd33a3487",
      "source_line": 221,
      "reader": "complex-analytic-spaces-and-analytification.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-17.Corollary-3.3",
      "unit": "AG-QC-17",
      "label": "Corollary 3.3",
      "statement_heading": "Corollary 3.3 (reduction). There is a canonical identification",
      "source": "src/complex-analytic-spaces-and-analytification.md",
      "source_sha256": "3d2aea96f2b5d621bdd3de89d70464d36607274527963d3d18100b6cd33a3487",
      "source_line": 258,
      "reader": "complex-analytic-spaces-and-analytification.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-17.Theorem-4.1",
      "unit": "AG-QC-17",
      "label": "Theorem 4.1",
      "statement_heading": "Theorem 4.1. Let \\(X\\) be any scheme locally of finite type over \\(\\mathbf C\\), and let \\(x\\in X(\\mathbf C)\\). The local map",
      "source": "src/complex-analytic-spaces-and-analytification.md",
      "source_sha256": "3d2aea96f2b5d621bdd3de89d70464d36607274527963d3d18100b6cd33a3487",
      "source_line": 272,
      "reader": "complex-analytic-spaces-and-analytification.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-17.Proposition-4.2",
      "unit": "AG-QC-17",
      "label": "Proposition 4.2",
      "statement_heading": "Proposition 4.2 (finite isomorphisms are detected analytically). Let \\(f:Z\\to X\\) be a finite morphism between schemes locally of finite type over \\(\\mathbf C\\). If \\(f^{\\mathrm{an}}\\) is an analytic isomorphism, then \\(f\\) is an isomorphism.",
      "source": "src/complex-analytic-spaces-and-analytification.md",
      "source_sha256": "3d2aea96f2b5d621bdd3de89d70464d36607274527963d3d18100b6cd33a3487",
      "source_line": 313,
      "reader": "complex-analytic-spaces-and-analytification.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-17.Proposition-5.1",
      "unit": "AG-QC-17",
      "label": "Proposition 5.1",
      "statement_heading": "Proposition 5.1. For every scheme \\(X\\) locally of finite type over \\(\\mathbf C\\), analytification is exact and faithful on coherent modules, preserves coherence, and commutes with tensor products and internal Hom of coherent modules. It commutes with pullback along scheme morphisms and with pushforward along closed immersions. If \\(\\mathcal I\\subset\\mathcal O_X\\) is a coherent ideal, then \\(\\mathcal I^{\\mathrm{an}}\\) identifies with the ideal it generates in \\(\\mathcal O_{X^{\\mathrm{an}}}\\), and",
      "source": "src/complex-analytic-spaces-and-analytification.md",
      "source_sha256": "3d2aea96f2b5d621bdd3de89d70464d36607274527963d3d18100b6cd33a3487",
      "source_line": 337,
      "reader": "complex-analytic-spaces-and-analytification.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-17.Proposition-5.2",
      "unit": "AG-QC-17",
      "label": "Proposition 5.2",
      "statement_heading": "Proposition 5.2. If \\(X\\) is a projective scheme over \\(\\mathbf C\\), then \\(X^{\\mathrm{an}}\\) is compact, whether or not \\(X\\) is reduced.",
      "source": "src/complex-analytic-spaces-and-analytification.md",
      "source_sha256": "3d2aea96f2b5d621bdd3de89d70464d36607274527963d3d18100b6cd33a3487",
      "source_line": 375,
      "reader": "complex-analytic-spaces-and-analytification.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-17.Proposition-5.3",
      "unit": "AG-QC-17",
      "label": "Proposition 5.3",
      "statement_heading": "Proposition 5.3 (properness and compactness). A separated scheme \\(X\\) of finite type over \\(\\mathbf C\\) is proper if and only if \\(X^{\\mathrm{an}}\\) is compact.",
      "source": "src/complex-analytic-spaces-and-analytification.md",
      "source_sha256": "3d2aea96f2b5d621bdd3de89d70464d36607274527963d3d18100b6cd33a3487",
      "source_line": 379,
      "reader": "complex-analytic-spaces-and-analytification.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Lemma-2.1",
      "unit": "AG-QC-18",
      "label": "Lemma 2.1",
      "statement_heading": "Lemma 2.1. The analytic structure sheaf of projective space has cohomology \\(\\mathbf C\\) in degree zero and zero in every positive degree. The natural comparison from algebraic cohomology is an isomorphism.",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 40,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Lemma-2.2",
      "unit": "AG-QC-18",
      "label": "Lemma 2.2",
      "statement_heading": "Lemma 2.2. Comparison is an isomorphism for \\(\\mathcal O(d)\\) on \\(\\mathbf P^n\\), for every integer \\(d\\) and every degree.",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 65,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Theorem-3.1",
      "unit": "AG-QC-18",
      "label": "Theorem 3.1",
      "statement_heading": "Theorem 3.1 (cohomological GAGA). If \\(X\\) is projective over \\(\\mathbf C\\) and \\(F\\) is coherent, (2) is an isomorphism for every \\(q\\geq0\\).",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 79,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Theorem-4.1",
      "unit": "AG-QC-18",
      "label": "Theorem 4.1",
      "statement_heading": "Theorem 4.1 (full faithfulness). For coherent \\(F,G\\) on a projective complex scheme,",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 100,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Lemma-5.1",
      "unit": "AG-QC-18",
      "label": "Lemma 5.1",
      "statement_heading": "Lemma 5.1 (analytic generation). Every coherent analytic module \\(M\\) on \\(P^{\\mathrm{an}}\\) has \\(M(d)\\) generated by finitely many global sections for all sufficiently large \\(d\\).",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 120,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Theorem-5.2",
      "unit": "AG-QC-18",
      "label": "Theorem 5.2",
      "statement_heading": "Theorem 5.2 (essential surjectivity). Every coherent analytic module on \\(X^{\\mathrm{an}}\\), for projective \\(X\\), is isomorphic to \\(F^{\\mathrm{an}}\\) for some coherent algebraic \\(F\\). Together with Theorem 4.1, analytification is an equivalence of coherent categories.",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 149,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Theorem-6.1",
      "unit": "AG-QC-18",
      "label": "Theorem 6.1",
      "statement_heading": "Theorem 6.1 (Chow). Every closed analytic subset of a projective complex scheme is the analytification of a unique reduced closed algebraic subscheme.",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 165,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Theorem-6.2",
      "unit": "AG-QC-18",
      "label": "Theorem 6.2",
      "statement_heading": "Theorem 6.2. Every holomorphic morphism between the analytifications of projective complex schemes is the analytification of a unique algebraic morphism.",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 171,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Proposition-6.3",
      "unit": "AG-QC-18",
      "label": "Proposition 6.3",
      "statement_heading": "Proposition 6.3. For \\(n\\geq1\\), every analytic line bundle on \\(\\mathbf P^n(\\mathbf C)\\) is \\(\\mathcal O(d)^{\\mathrm{an}}\\) for a unique \\(d\\in\\mathbf Z\\).",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 179,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Lemma-6.4",
      "unit": "AG-QC-18",
      "label": "Lemma 6.4",
      "statement_heading": "Lemma 6.4. A reduced proper complex scheme \\(X\\) admits a projective surjection \\(p:Y\\to X\\), with \\(Y\\) reduced and projective, which is an isomorphism over an open \\(U\\) containing every generic point of \\(X\\). Every proper complex scheme has compact analytification.",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 197,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Lemma-6.5",
      "unit": "AG-QC-18",
      "label": "Lemma 6.5",
      "statement_heading": "Lemma 6.5 (twists with holomorphic parameters). Let \\(D\\subset\\mathbf C^m\\) be a polydisc and let \\(r:\\mathbf P^n(\\mathbf C)\\times D\\to D\\) be the projection. For every \\(d\\in\\mathbf Z\\),",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 207,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Proposition-6.6",
      "unit": "AG-QC-18",
      "label": "Proposition 6.6",
      "statement_heading": "Proposition 6.6. Let \\(f:Y\\to X\\) be a projective morphism of complex schemes, and let \\(F\\) be coherent. Then the natural maps",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 238,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Theorem-6.7",
      "unit": "AG-QC-18",
      "label": "Theorem 6.7",
      "statement_heading": "Theorem 6.7. If \\(X\\) is proper over \\(\\mathbf C\\) and \\(F\\) is coherent, then",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 262,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Lemma-6.8",
      "unit": "AG-QC-18",
      "label": "Lemma 6.8",
      "statement_heading": "Lemma 6.8. For coherent \\(F,G\\) on a complex scheme, the sheaves \\(\\mathcal Ext^q_X(F,G)\\) are coherent and",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 286,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Proposition-6.9",
      "unit": "AG-QC-18",
      "label": "Proposition 6.9",
      "statement_heading": "Proposition 6.9. For proper \\(X\\) and coherent \\(F,G\\), the natural maps",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 298,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Lemma-6.10",
      "unit": "AG-QC-18",
      "label": "Lemma 6.10",
      "statement_heading": "Lemma 6.10 (a uniform support power). Let \\(X\\) be proper, \\(Z\\subset X\\) a closed reduced subscheme with coherent ideal \\(J\\), and \\(M\\) a coherent analytic module whose support is contained in \\(Z^{\\mathrm{an}}\\). Some power \\((J^{\\mathrm{an}})^r\\) annihilates \\(M\\).",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 320,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    },
    {
      "id": "AG-QC-18.Theorem-6.11",
      "unit": "AG-QC-18",
      "label": "Theorem 6.11",
      "statement_heading": "Theorem 6.11 (proper GAGA, with nilpotents). For every proper complex scheme \\(X\\), analytification is an equivalence",
      "source": "src/serres-comparison-theorems-and-chows-theorem.md",
      "source_sha256": "852676cee660e6c2fa1e48c75e94d12b890f2d935a6763d66f6fac0e51eb1d7b",
      "source_line": 324,
      "reader": "serres-comparison-theorems-and-chows-theorem.html",
      "proof_status": "argument_present_with_stated_prerequisites",
      "prerequisites": "prerequisites.json",
      "independent_review": false
    }
  ],
  "exact_downstream_providers": "PROVIDER_HANDOFF.json",
  "proof_boundaries": "prerequisites.json",
  "recursive_proof_closure": false
}