{
  "schema": "programme-local-freeness-prerequisites/v1",
  "lesson_id": "GL-DMOD-02",
  "title": "Algebra prerequisites for local freeness of a coherent module with connection",
  "licence": "CC0-1.0",
  "consumer": {
    "source": "docs/courses/GL-DMOD/src/d-modules-flat-connections-and-local-systems.md",
    "source_sha256": "7dee2e7f4c167853e999833598844ce8ba851f9973b64c93632fa9a03db07dd6",
    "reader": "docs/courses/GL-DMOD/d-modules-flat-connections-and-local-systems.html",
    "reader_sha256": "d933a78db0c0ff2515972f7d62f1e0b76491b0043e110177756aabcbbb2659e5",
    "anchor": "3-why-finite-modules-cannot-have-relations-at-a-smooth-point",
    "theorem": "Theorem3.1; integrability is not needed for the local-freeness conclusion."
  },
  "providers": [
    {
      "lesson_id": "AG-CA-02",
      "title": "Localization, local properties and support",
      "proof_locator": "Theorem2.1",
      "anchor": "2-relations-survive-localization",
      "reader": "docs/courses/AG-CA/AG-CA-02.html",
      "reader_sha256": "0b10d06729deee42c4b19c165d2377e202925c99e05833aa892c03359ecf5be3",
      "source": "docs/courses/AG-CA/src/localization-local-properties-and-support.md",
      "source_sha256": "d1904f62cf325bffc91809405e2e8ba0979a6f0f4454332ae60b74a626b487e8",
      "conditions_and_use": "Exactness of localization."
    },
    {
      "lesson_id": "AG-CA-02",
      "title": "Localization, local properties and support",
      "proof_locator": "Lemma4.1 and Theorem4.2",
      "anchor": "4-finite-generators-near-a-point",
      "reader": "docs/courses/AG-CA/AG-CA-02.html",
      "reader_sha256": "0b10d06729deee42c4b19c165d2377e202925c99e05833aa892c03359ecf5be3",
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      "source_sha256": "d1904f62cf325bffc91809405e2e8ba0979a6f0f4454332ae60b74a626b487e8",
      "conditions_and_use": "Finite-module determinant trick and Nakayama; no Noetherian hypothesis."
    },
    {
      "lesson_id": "AG-CA-02",
      "title": "Localization, local properties and support",
      "proof_locator": "Theorem5.1",
      "anchor": "5-recovering-global-information",
      "reader": "docs/courses/AG-CA/AG-CA-02.html",
      "reader_sha256": "0b10d06729deee42c4b19c165d2377e202925c99e05833aa892c03359ecf5be3",
      "source": "docs/courses/AG-CA/src/localization-local-properties-and-support.md",
      "source_sha256": "d1904f62cf325bffc91809405e2e8ba0979a6f0f4454332ae60b74a626b487e8",
      "conditions_and_use": "Zero modules detected at maximal localizations."
    },
    {
      "lesson_id": "AG-CA-02",
      "title": "Localization, local properties and support",
      "proof_locator": "Proposition6.1",
      "anchor": "6-where-a-module-is-visible",
      "reader": "docs/courses/AG-CA/AG-CA-02.html",
      "reader_sha256": "0b10d06729deee42c4b19c165d2377e202925c99e05833aa892c03359ecf5be3",
      "source": "docs/courses/AG-CA/src/localization-local-properties-and-support.md",
      "source_sha256": "d1904f62cf325bffc91809405e2e8ba0979a6f0f4454332ae60b74a626b487e8",
      "conditions_and_use": "Finite denominators annihilate a finite module with zero stalk; support calculus."
    },
    {
      "lesson_id": "AG-CA-03",
      "title": "Noetherian and Artinian rings",
      "proof_locator": "Section1",
      "anchor": "1-three-ways-to-recognize-a-noetherian-module",
      "reader": "docs/courses/AG-CA/AG-CA-03.html",
      "reader_sha256": "4cd0b10dfbfb5e16acaaaf75fb3c6145be8f5b70bb0a4b078ea8cfa5b4718b1d",
      "source": "docs/courses/AG-CA/src/noetherian-and-artinian-rings.md",
      "source_sha256": "94b3cd74c201f9ac1daf33aa90ca4218263559884e56ee47c0ad98d82187b771",
      "conditions_and_use": "Submodules of finite modules over a Noetherian ring are finite."
    },
    {
      "lesson_id": "AG-CA-03",
      "title": "Noetherian and Artinian rings",
      "proof_locator": "Theorem2.1 and consequences",
      "anchor": "2-polynomial-rings-and-finite-geometric-descriptions",
      "reader": "docs/courses/AG-CA/AG-CA-03.html",
      "reader_sha256": "4cd0b10dfbfb5e16acaaaf75fb3c6145be8f5b70bb0a4b078ea8cfa5b4718b1d",
      "source": "docs/courses/AG-CA/src/noetherian-and-artinian-rings.md",
      "source_sha256": "94b3cd74c201f9ac1daf33aa90ca4218263559884e56ee47c0ad98d82187b771",
      "conditions_and_use": "Finite-type field algebras and their localizations are Noetherian."
    },
    {
      "lesson_id": "AG-CA-03",
      "title": "Noetherian and Artinian rings",
      "proof_locator": "Theorem5.1",
      "anchor": "5-intersecting-a-filtration-with-a-submodule",
      "reader": "docs/courses/AG-CA/AG-CA-03.html",
      "reader_sha256": "4cd0b10dfbfb5e16acaaaf75fb3c6145be8f5b70bb0a4b078ea8cfa5b4718b1d",
      "source": "docs/courses/AG-CA/src/noetherian-and-artinian-rings.md",
      "source_sha256": "94b3cd74c201f9ac1daf33aa90ca4218263559884e56ee47c0ad98d82187b771",
      "conditions_and_use": "Artin-Rees for a submodule of a finite module over a Noetherian ring."
    },
    {
      "lesson_id": "AG-CA-03",
      "title": "Noetherian and Artinian rings",
      "proof_locator": "Theorem6.1",
      "anchor": "6-what-can-survive-every-power",
      "reader": "docs/courses/AG-CA/AG-CA-03.html",
      "reader_sha256": "4cd0b10dfbfb5e16acaaaf75fb3c6145be8f5b70bb0a4b078ea8cfa5b4718b1d",
      "source": "docs/courses/AG-CA/src/noetherian-and-artinian-rings.md",
      "source_sha256": "94b3cd74c201f9ac1daf33aa90ca4218263559884e56ee47c0ad98d82187b771",
      "conditions_and_use": "Krull intersection with its ideal-in-Jacobson-radical condition; applied to m and A^r."
    },
    {
      "lesson_id": "AG-CA-06",
      "title": "The Nullstellensatz and Jacobson rings",
      "proof_locator": "Theorems1.1,1.3; Lemma1.2",
      "anchor": "1-why-an-algebra-generated-field-is-finite",
      "reader": "docs/courses/AG-CA/AG-CA-06.html",
      "reader_sha256": "0cb7bedbf35af53ac3d06561f76e19fa5b7248c767709637514f9b7ed9fde93c",
      "source": "docs/courses/AG-CA/src/the-nullstellensatz-and-jacobson-rings.md",
      "source_sha256": "9ffb1e5cce54ff2c7188e866860c67d74075e652513c53b6ffe49718961d10e1",
      "conditions_and_use": "Artin-Tate and Zariski lemma supply weak Nullstellensatz."
    },
    {
      "lesson_id": "AG-CA-06",
      "title": "The Nullstellensatz and Jacobson rings",
      "proof_locator": "Theorem2.1",
      "anchor": "2-closed-points-detect-radical-equations",
      "reader": "docs/courses/AG-CA/AG-CA-06.html",
      "reader_sha256": "0cb7bedbf35af53ac3d06561f76e19fa5b7248c767709637514f9b7ed9fde93c",
      "source": "docs/courses/AG-CA/src/the-nullstellensatz-and-jacobson-rings.md",
      "source_sha256": "9ffb1e5cce54ff2c7188e866860c67d74075e652513c53b6ffe49718961d10e1",
      "conditions_and_use": "Closed points of finite-type algebras over an algebraically closed field are rational."
    },
    {
      "lesson_id": "AG-CA-06S",
      "title": "Resolutions, Tor and Ext",
      "proof_locator": "Theorem1.1",
      "anchor": "1-constructing-and-comparing-resolutions",
      "reader": "docs/courses/AG-CA/AG-CA-06S.html",
      "reader_sha256": "c3bea2e9845e8da2edf80bc0a2e01e42ea94eba399c8b840540e9f7e6d7352ab",
      "source": "docs/courses/AG-CA/src/resolutions-tor-and-ext.md",
      "source_sha256": "41edca05da2d3d03825b78f3b5040590cce6e97718c1adac5112caa2f23047cd",
      "conditions_and_use": "Free resolutions exist; comparison maps and homotopy independence."
    },
    {
      "lesson_id": "AG-CA-06S",
      "title": "Resolutions, Tor and Ext",
      "proof_locator": "Section2",
      "anchor": "2-the-connecting-homomorphism",
      "reader": "docs/courses/AG-CA/AG-CA-06S.html",
      "reader_sha256": "c3bea2e9845e8da2edf80bc0a2e01e42ea94eba399c8b840540e9f7e6d7352ab",
      "source": "docs/courses/AG-CA/src/resolutions-tor-and-ext.md",
      "source_sha256": "41edca05da2d3d03825b78f3b5040590cce6e97718c1adac5112caa2f23047cd",
      "conditions_and_use": "Long exact sequence of homology."
    },
    {
      "lesson_id": "AG-CA-06S",
      "title": "Resolutions, Tor and Ext",
      "proof_locator": "Lemma3.1 and Theorem3.2",
      "anchor": "3-balance-and-symmetry-of-tor",
      "reader": "docs/courses/AG-CA/AG-CA-06S.html",
      "reader_sha256": "c3bea2e9845e8da2edf80bc0a2e01e42ea94eba399c8b840540e9f7e6d7352ab",
      "source": "docs/courses/AG-CA/src/resolutions-tor-and-ext.md",
      "source_sha256": "41edca05da2d3d03825b78f3b5040590cce6e97718c1adac5112caa2f23047cd",
      "conditions_and_use": "Either-variable Tor computation and exact sequences."
    },
    {
      "lesson_id": "AG-CA-07",
      "title": "Tor and flat modules",
      "proof_locator": "Theorem2.1",
      "anchor": "2-the-ideal-and-tor-criteria",
      "reader": "docs/courses/AG-CA/AG-CA-07.html",
      "reader_sha256": "72b1050623a1332379ed340395e8d3d9ea2f94545272eedd0d451ea44f132a76",
      "source": "docs/courses/AG-CA/src/tor-and-flat-modules.md",
      "source_sha256": "210b21719f7021478941153585e49e5e97f0d084b047cafe9049666435371336",
      "conditions_and_use": "Flatness iff Tor1 vanishes for every module; unrestricted module."
    },
    {
      "lesson_id": "AG-CA-07",
      "title": "Tor and flat modules",
      "proof_locator": "Proposition3.2 and Theorem3.3",
      "anchor": "3-stability-and-localization",
      "reader": "docs/courses/AG-CA/AG-CA-07.html",
      "reader_sha256": "72b1050623a1332379ed340395e8d3d9ea2f94545272eedd0d451ea44f132a76",
      "source": "docs/courses/AG-CA/src/tor-and-flat-modules.md",
      "source_sha256": "210b21719f7021478941153585e49e5e97f0d084b047cafe9049666435371336",
      "conditions_and_use": "Base change and localization detect flatness."
    },
    {
      "lesson_id": "AG-CA-07",
      "title": "Tor and flat modules",
      "proof_locator": "Theorems5.1-5.3",
      "anchor": "5-relations-and-finite-free-bases",
      "reader": "docs/courses/AG-CA/AG-CA-07.html",
      "reader_sha256": "72b1050623a1332379ed340395e8d3d9ea2f94545272eedd0d451ea44f132a76",
      "source": "docs/courses/AG-CA/src/tor-and-flat-modules.md",
      "source_sha256": "210b21719f7021478941153585e49e5e97f0d084b047cafe9049666435371336",
      "conditions_and_use": "Finite flat over local implies free; finitely presented flat is locally free without Noetherianity."
    },
    {
      "lesson_id": "AG-CA-08",
      "title": "Faithful flatness and the local criterion for flatness",
      "proof_locator": "Section1 and Theorem1.1",
      "anchor": "1-detecting-nonzero-modules",
      "reader": "docs/courses/AG-CA/AG-CA-08.html",
      "reader_sha256": "23ed8dd5998685b58029efbf751020b3d13080e26516a7add1e235fb76420cdb",
      "source": "docs/courses/AG-CA/src/faithful-flatness-and-the-local-criterion-for-flatness.md",
      "source_sha256": "c0dd2827b12cecc779eeea7593b1640991677b2b09aef55ac787357f53fb7a96",
      "conditions_and_use": "Flat tensor commutes with homology; faithful tensor detects zero."
    },
    {
      "lesson_id": "AG-CA-08",
      "title": "Faithful flatness and the local criterion for flatness",
      "proof_locator": "Theorem3.1",
      "anchor": "3-descending-finiteness-and-flatness",
      "reader": "docs/courses/AG-CA/AG-CA-08.html",
      "reader_sha256": "23ed8dd5998685b58029efbf751020b3d13080e26516a7add1e235fb76420cdb",
      "source": "docs/courses/AG-CA/src/faithful-flatness-and-the-local-criterion-for-flatness.md",
      "source_sha256": "c0dd2827b12cecc779eeea7593b1640991677b2b09aef55ac787357f53fb7a96",
      "conditions_and_use": "Faithfully flat descent of flatness, finiteness and finite presentation; alternative route retained."
    },
    {
      "lesson_id": "AG-CA-17",
      "title": "Formally smooth, unramified and étale ring maps",
      "proof_locator": "Lemma1.1 and Theorem1.2",
      "anchor": "1-extending-maps-across-a-square-zero-ideal",
      "reader": "docs/courses/AG-CA/AG-CA-17.html",
      "reader_sha256": "9ccfdbb6b05847044e26a7c7bc01bc18a20507a93794a197772a7225205f2ead",
      "source": "docs/courses/AG-CA/src/formally-smooth-unramified-and-etale-ring-maps.md",
      "source_sha256": "7c743d4ef1eb5c0937dd12af0a0c7764c21bb8b555855ec0295e9f3bccedaf7b",
      "conditions_and_use": "Unique lifts across nilpotent ideals and localization/base-change stability."
    },
    {
      "lesson_id": "AG-CA-17",
      "title": "Formally smooth, unramified and étale ring maps",
      "proof_locator": "Theorem3.1",
      "anchor": "3-existence-is-measured-by-the-conormal-sequence",
      "reader": "docs/courses/AG-CA/AG-CA-17.html",
      "reader_sha256": "9ccfdbb6b05847044e26a7c7bc01bc18a20507a93794a197772a7225205f2ead",
      "source": "docs/courses/AG-CA/src/formally-smooth-unramified-and-etale-ring-maps.md",
      "source_sha256": "7c743d4ef1eb5c0937dd12af0a0c7764c21bb8b555855ec0295e9f3bccedaf7b",
      "conditions_and_use": "Split conormal sequence supplies local standard form."
    },
    {
      "lesson_id": "AG-CA-17",
      "title": "Formally smooth, unramified and étale ring maps",
      "proof_locator": "Theorem4.1",
      "anchor": "4-standard-smooth-presentations",
      "reader": "docs/courses/AG-CA/AG-CA-17.html",
      "reader_sha256": "9ccfdbb6b05847044e26a7c7bc01bc18a20507a93794a197772a7225205f2ead",
      "source": "docs/courses/AG-CA/src/formally-smooth-unramified-and-etale-ring-maps.md",
      "source_sha256": "7c743d4ef1eb5c0937dd12af0a0c7764c21bb8b555855ec0295e9f3bccedaf7b",
      "conditions_and_use": "Invertible Jacobian correction; étale projection when free variables are fixed."
    },
    {
      "lesson_id": "AG-CA-17",
      "title": "Formally smooth, unramified and étale ring maps",
      "proof_locator": "Theorem5.1",
      "anchor": "5-from-a-split-sequence-to-local-coordinates",
      "reader": "docs/courses/AG-CA/AG-CA-17.html",
      "reader_sha256": "9ccfdbb6b05847044e26a7c7bc01bc18a20507a93794a197772a7225205f2ead",
      "source": "docs/courses/AG-CA/src/formally-smooth-unramified-and-etale-ring-maps.md",
      "source_sha256": "7c743d4ef1eb5c0937dd12af0a0c7764c21bb8b555855ec0295e9f3bccedaf7b",
      "conditions_and_use": "Local standard smooth presentation at every prime."
    },
    {
      "lesson_id": "AG-CA-19",
      "title": "Completion",
      "proof_locator": "Section1",
      "anchor": "1-compatible-choices-in-an-inverse-system",
      "reader": "docs/courses/AG-CA/AG-CA-19.html",
      "reader_sha256": "e614e7ce7b4b2de6f6baa16cece3a8217f4a9990474f748b2a3ae34ada8f374c",
      "source": "docs/courses/AG-CA/src/completion.md",
      "source_sha256": "ddbd225b9c5a584a2da5d1ab05f46e7f7e0b03f61dd063a391431602a6e8e2d2",
      "conditions_and_use": "Inverse-limit universal property and compatible maps."
    },
    {
      "lesson_id": "AG-CA-19",
      "title": "Completion",
      "proof_locator": "Section2, Propositions2.2-2.3",
      "anchor": "2-topology-and-the-powers-of-the-ideal",
      "reader": "docs/courses/AG-CA/AG-CA-19.html",
      "reader_sha256": "e614e7ce7b4b2de6f6baa16cece3a8217f4a9990474f748b2a3ae34ada8f374c",
      "source": "docs/courses/AG-CA/src/completion.md",
      "source_sha256": "ddbd225b9c5a584a2da5d1ab05f46e7f7e0b03f61dd063a391431602a6e8e2d2",
      "conditions_and_use": "Completion filtration, separatedness kernel and local residue ring."
    }
  ],
  "context_only": [
    {
      "lesson_id": "AG-CA-18",
      "role": "Smoothness/Jacobian comparison; not an additional proof prerequisite."
    }
  ],
  "comparison_proofs_here": [
    "Compatible inverse quotient maps for rational-point completed coordinates",
    "Continuity and leading-degree lowering for coordinate derivations",
    "Finite-denominator extension of stalk bases and affine-chart coverage",
    "Explicit faithfully flat Tor base-change chain maps and alternative direct descent"
  ],
  "portability": "Provider paths are repository-root-relative. These are linked programme proofs; a standalone lesson download does not embed them. An export omitting them has external proof dependencies."
}