{ "schema": "exact-course-proof-providers/v1", "course": "AG-QC", "language": "en", "consumer": "AG-RG", "requested_units": [ "AG-QC-01", "AG-QC-03", "AG-QC-05", "AG-QC-08", "AG-QC-09", "AG-QC-10", "AG-QC-15" ], "providers": [ { "id": "AG-QC-01-Leray", "course_unit": "AG-QC-01", "name": "Leray spectral sequence", "source": "src/cohomology-of-sheaves-on-ringed-spaces.md", "source_sha256": "90c34f08aa0b7cf16ac7ba2bf950f049a8458df6c45feff866231af50780b977", "source_line": 130, "proof_label": "Theorem 4.1", "proof_locus": "Section 4; current sheaf-operations course, derived pullback and pushforward, Proposition 3.1 and Exercise 3.", "conditions": "Any morphism of ringed spaces and any module sheaf in degree zero; first quadrant, finite filtration in every total degree. No finite dimension, separation or properness assumption.", "reader": "cohomology-of-sheaves-on-ringed-spaces.html#AG-QC-01-Leray", "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/AG-QC/cohomology-of-sheaves-on-ringed-spaces.html#AG-QC-01-Leray", "aliases": [ "AG-QC-01-Leray" ], "proof_status": "argument_present_with_stated_prerequisites", "independent_review": false, "formalization": "not_formalized" }, { "id": "AG-QC-03-affine-vanishing", "course_unit": "AG-QC-03", "name": "Affine quasi-coherent vanishing", "source": "src/affine-cohomology-and-serres-criterion.md", "source_sha256": "9113f6aa822a41c9f58e454bf0a86e0e440f71ccee048c2cf94cae34685af2db", "source_line": 49, "proof_label": "Theorem 2.2", "proof_locus": "Lemma 2.1 and Theorem 2.2; Čech lesson, Theorem 4.1.", "conditions": "Every ring R and every R-module M; H^p(Spec R, tilde M)=0 for p>0. No Noetherian or finite-generation assumption.", "reader": "affine-cohomology-and-serres-criterion.html#AG-QC-03-affine-vanishing", "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/AG-QC/affine-cohomology-and-serres-criterion.html#AG-QC-03-affine-vanishing", "aliases": [ "AG-QC-03-affine-vanishing" ], "proof_status": "argument_present_with_stated_prerequisites", "independent_review": false, "formalization": "not_formalized" }, { "id": "AG-QC-03-Serre-affineness", "course_unit": "AG-QC-03", "name": "Serre’s affine criterion", "source": "src/affine-cohomology-and-serres-criterion.md", "source_sha256": "9113f6aa822a41c9f58e454bf0a86e0e440f71ccee048c2cf94cae34685af2db", "source_line": 156, "proof_label": "Theorem 5.2", "proof_locus": "Lemma 5.1 and Theorem 5.2; affine vanishing in Theorem 2.2.", "conditions": "X quasi-compact; affineness is equivalent to vanishing of all positive quasi-coherent cohomology, and to H^1(X,I)=0 for every quasi-coherent ideal. The statement does not require quasi-separatedness.", "reader": "affine-cohomology-and-serres-criterion.html#AG-QC-03-Serre-affineness", "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/AG-QC/affine-cohomology-and-serres-criterion.html#AG-QC-03-Serre-affineness", "aliases": [ "AG-QC-03-Serre-affineness" ], "proof_status": "argument_present_with_stated_prerequisites", "independent_review": false, "formalization": "not_formalized" }, { "id": "AG-QC-05-Serre-projective", "course_unit": "AG-QC-05", "name": "Serre’s theorems on projective space", "source": "src/serres-theorems-on-projective-schemes.md", "source_sha256": "ddac40fd2bf24adfb7184702da4b3adcf5596f2739bbd9fb2d81bf9941f3f116", "source_line": 52, "proof_label": "Theorem 2.1", "proof_locus": "Proposition 1.2; descending induction in Theorem 2.1; preceding projective-space lesson, Theorem 2.2.", "conditions": "R Noetherian, F coherent on P_R^N; finite cohomology, eventual higher-twist vanishing and eventual global generation.", "reader": "serres-theorems-on-projective-schemes.html#AG-QC-05-Serre-projective", "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/AG-QC/serres-theorems-on-projective-schemes.html#AG-QC-05-Serre-projective", "aliases": [ "AG-QC-05-Serre-projective" ], "proof_status": "argument_present_with_stated_prerequisites", "independent_review": false, "formalization": "not_formalized" }, { "id": "AG-QC-05-Serre-ample", "course_unit": "AG-QC-05", "name": "Serre’s theorems for a proper scheme with an ample line bundle", "source": "src/serres-theorems-on-projective-schemes.md", "source_sha256": "ddac40fd2bf24adfb7184702da4b3adcf5596f2739bbd9fb2d81bf9941f3f116", "source_line": 83, "proof_label": "Theorem 2.2", "proof_locus": "Lemma 1.3 proves the closed embedding; Theorems 2.1–2.2, with the finite residue-class argument.", "conditions": "X proper over a Noetherian affine base R, F coherent and L ample; finiteness, and vanishing and global generation for every sufficiently large integer twist. Thresholds on a merely locally Noetherian non-quasi-compact base are only local.", "reader": "serres-theorems-on-projective-schemes.html#AG-QC-05-Serre-ample", "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/AG-QC/serres-theorems-on-projective-schemes.html#AG-QC-05-Serre-ample", "aliases": [ "AG-QC-05-Serre-ample" ], "proof_status": "argument_present_with_stated_prerequisites", "independent_review": false, "formalization": "not_formalized" }, { "id": "AG-QC-08-flat-base-change", "course_unit": "AG-QC-08", "name": "Flat base change", "source": "src/base-change-and-the-grothendieck-complex.md", "source_sha256": "d74a9c8c4789238fb8bea05cf6de26f0b2fb1df88a357b74cffec1d14243dcf8", "source_line": 58, "proof_label": "Theorem 2.1", "proof_locus": "Section 1 constructs the canonical map; Section 2 proves the separated case and the finite spectral-sequence comparison for quasi-separated X.", "conditions": "f quasi-compact and quasi-separated, F quasi-coherent, g flat; every higher direct-image comparison is an isomorphism. F need not be flat. Includes localization and nonseparated X.", "reader": "base-change-and-the-grothendieck-complex.html#AG-QC-08-flat-base-change", "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/AG-QC/base-change-and-the-grothendieck-complex.html#AG-QC-08-flat-base-change", "aliases": [ "AG-QC-08-flat-base-change" ], "proof_status": "argument_present_with_stated_prerequisites", "independent_review": false, "formalization": "not_formalized" }, { "id": "AG-QC-09-vanishing-local-freeness", "course_unit": "AG-QC-09", "name": "Fiber vanishing gives locally free H^0 and arbitrary base change", "source": "src/semicontinuity-and-grauerts-theorem.md", "source_sha256": "b6fdd5f794d77aedb97b216b27a9cea4edb61538afcd59cd21c715c64c6afac1", "source_line": 124, "proof_label": "Corollary 5.3", "proof_locus": "Lemma 2.1 and Corollary 5.3 prove the fiber-vanishing case. Corollary 5.1 proves the higher-direct-image-vanishing case. Lesson 8 proves the Noetherian finite model; Lemma 0.1 of lesson9 writes the general-base finite model using exact earlier finite-presentation and flatness proofs. Constant fiber dimension alone on a nonreduced base is insufficient; Theorem 4.1 assumes a reduced base.", "conditions": "Noetherian base: f proper, F coherent and base-flat. Arbitrary base: f proper of finite presentation, F finitely presented and base-flat. All positive fiber cohomology vanishes at every point. Then f_*F is finite locally free, all positive higher direct images vanish, and both assertions commute with arbitrary base change. No reducedness is required. A line bundle on a flat proper finitely presented family satisfies the sheaf conditions.", "reader": "semicontinuity-and-grauerts-theorem.html#AG-QC-09-vanishing-local-freeness", "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/AG-QC/semicontinuity-and-grauerts-theorem.html#AG-QC-09-vanishing-local-freeness", "aliases": [ "AG-QC-09-vanishing-local-freeness" ], "proof_status": "argument_present_with_stated_prerequisites", "independent_review": false, "formalization": "not_formalized" }, { "id": "AG-QC-10-formal-functions", "course_unit": "AG-QC-10", "name": "Formal functions", "source": "src/the-theorem-on-formal-functions.md", "source_sha256": "76e9b47b2804b9be424b617db2ebdb84b769242e95a81b488c08e2c29e88ddb3", "source_line": 133, "proof_label": "Theorem 4.1", "proof_locus": "Sections 2–4, especially Lemma 2.2, Proposition 3.1 and Theorem 4.1. Corollary 4.2 gives the locally Noetherian stalk form using lesson 8 localization.", "conditions": "A Noetherian, I any ideal, X proper over Spec A, F coherent; the canonical I-adic completed cohomology is the inverse limit of cohomology on X_n. F need not be flat and A need not be complete.", "reader": "the-theorem-on-formal-functions.html#AG-QC-10-formal-functions", "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/AG-QC/the-theorem-on-formal-functions.html#AG-QC-10-formal-functions", "aliases": [ "AG-QC-10-formal-functions" ], "proof_status": "argument_present_with_stated_prerequisites", "independent_review": false, "formalization": "not_formalized" }, { "id": "AG-QC-15-CM-duality", "course_unit": "AG-QC-15", "name": "Cohen–Macaulay projective Serre duality", "source": "src/dualizing-sheaves-and-serre-duality-for-projective-schemes.md", "source_sha256": "2eb5d75cf6789870a3868831c7e701c12ab4033a8f6122f3937cb945e1ad5223", "source_line": 121, "proof_label": "Theorem 4.2", "proof_locus": "Sections 2–4; local Ext vanishing, ambient projective-space duality from lesson 14 and Theorem 4.2. Local algebra inputs are recorded separately.", "conditions": "X projective over a field k, Cohen–Macaulay and equidimensional of dimension n, F coherent; Ext_X^(n-i)(F,omega_X) is naturally dual to H^i(X,F), for every integer i. For a smooth projective equidimensional curve and invertible L this gives H^1(L)^vee=H^0(omega_X tensor L^-1). No non-Cohen–Macaulay sheaf-duality extension is asserted.", "reader": "dualizing-sheaves-and-serre-duality-for-projective-schemes.html#AG-QC-15-CM-duality", "reader_url": "https://kokunoyumeto.github.io/open-math-courses/courses/AG-QC/dualizing-sheaves-and-serre-duality-for-projective-schemes.html#AG-QC-15-CM-duality", "aliases": [ "AG-QC-15-CM-duality" ], "proof_status": "argument_present_with_stated_prerequisites", "independent_review": false, "formalization": "not_formalized" } ], "prerequisites": "prerequisites.json", "verification": "Writing-AI statement and source checks; exact loci and conditions, with unresolved prerequisite proofs recorded. No independent review or recursive proof closure is claimed." }