{ "schema": "exact-course-prerequisite-map/v1", "course": "GL-DMOD", "consumer": { "lesson_id": "GL-DMOD-07", "source": "docs/courses/GL-DMOD/src/inverse-images.md", "source_sha256": "9f82f82d9c0fc2aefa671c8511318f3cf55f9a209dacf7bdee0bff2c861bc446", "reader": "docs/courses/GL-DMOD/inverse-images.html", "reader_sha256": "1b22c475b7e6ba5e4fbbb01aa1645273a288bfceeb2fee24df0b5868d77a0a24" }, "providers": [ { "lesson_id": "GL-DMOD-01", "source": "docs/courses/GL-DMOD/src/differential-operators-and-the-weyl-algebra.md", "source_sha256": "2590c0e5aa15a1dacc355518b8c8819b3a80717aef6567f5cb4a732bfc24f744", "reader": "docs/courses/GL-DMOD/differential-operators-and-the-weyl-algebra.html", "reader_sha256": "2fc3562d6877ba4f82662b1e1f5640eafa4f389f087a9b32fe1c12c3892136b3", "anchor": "5-polynomial-operators-two-filtrations-one-ring", "proof_locator": "Theorem5.1", "used_for": "Filtered Noetherianity used for the tangent-operator and its Rees algebra." }, { "lesson_id": "GL-DMOD-03", "source": "docs/courses/GL-DMOD/src/good-filtrations-and-the-characteristic-variety.md", "source_sha256": "5a426491e32e478754cd7b3951c19f105ddeca9db0f09d2d7be6d181a63fe7c6", "reader": "docs/courses/GL-DMOD/good-filtrations-and-the-characteristic-variety.html", "reader_sha256": "80b650c6955b0140ca191753df34590cb6917e138d793137cb44c8b0e4901a22", "anchor": "4-submodules-quotients-and-exact-sequences", "proof_locator": "Theorem4.1, with Section3 Rees Noetherian proof", "used_for": "Good induced and quotient filtrations; characteristic support of a submodule. Theorem7.0, return to each minimal prime, proves the involutivity used to exclude vertical characteristic components in Theorem6.5.", "additional_proof_locators": [ { "locator": "Theorem7.0, return to the radical ideal", "anchor": "return-to-the-radical-ideal", "used_for": "Each minimal characteristic prime is involutive; repeated bracket with t contradicts a monic normal-symbol equation on a vertical component." } ] }, { "lesson_id": "GL-DMOD-11", "source": "docs/courses/GL-DMOD/src/preservation-of-holonomicity-and-minimal-extensions.md", "source_sha256": "520ce42daaf8c816988bd6efa5a130828573854a4a0547d468db6f69f5e0861d", "reader": "docs/courses/GL-DMOD/preservation-of-holonomicity-and-minimal-extensions.html", "reader_sha256": "e451689b6afc23b3488ee071926a755c696115fa78949fae861256c9b6d9158f", "anchor": "2-inverse-images-including-characteristic-embeddings", "proof_locator": "Theorem2.1 and Proposition1.1", "used_for": "Algebraic holonomic inverse image for every smooth-variety morphism; all bounded holonomic complexes; used by Exercise7.5." }, { "lesson_id": "GL-DMOD-10", "source": "docs/courses/GL-DMOD/src/adjunctions-base-change-and-the-projection-formula.md", "source_sha256": "2e6cda561fabfa1611bb34e2d27c0c084676e38cf0c0427f23158a086d1a7881", "reader": "docs/courses/GL-DMOD/adjunctions-base-change-and-the-projection-formula.html", "reader_sha256": "afdf61a575593a0236d857742b7569f082efea4bc34d2086208f5ad31ce0fd03", "anchor": "1-localization-and-the-supported-part", "proof_locator": "Lemma1.1", "used_for": "Localization triangle used by GL11 Theorem2.1; the proof does not use the later holonomic adjunction assumptions." }, { "lesson_id": "GL-DMOD-09", "source": "docs/courses/GL-DMOD/src/kashiwaras-equivalence-and-singular-spaces.md", "source_sha256": "8dd09ee51c6a9f9f696de961f0b1dd769787a54a792d725a05ce797028cde5c3", "reader": "docs/courses/GL-DMOD/kashiwaras-equivalence-and-singular-spaces.html", "reader_sha256": "862bacb03c8a08fd73f772301ed0caebdf77419db48b1cfaf732919294acaa08", "anchor": "3-the-equivalence-with-its-intrinsic-inverse", "proof_locator": "Theorem3.1 and coherence/characteristic-support subsection", "used_for": "Supported equivalence, intrinsic determinant, exactness and preservation/reflection of holonomicity used by GL11." }, { "lesson_id": "GL-DMOD-05", "source": "docs/courses/GL-DMOD/src/bernstein-sato-polynomials.md", "source_sha256": "d6d90bfd0182a4e3969e029add3e2da4d51b7ed59dedcda7e1e013750352f0f1", "reader": "docs/courses/GL-DMOD/bernstein-sato-polynomials.html", "reader_sha256": "e8022028ff211f8c3e00ec6b55842c83cd6e57798c955438c6118c82eaae7ff7", "anchor": "lemma-2-1-bounded-holonomic-growth", "proof_locator": "Lemma2.1", "used_for": "Growth criterion proving finite generation in GL11 Proposition1.1; does not use cusp minimality or negative-rational roots." }, { "lesson_id": "GL-DMOD-06", "source": "docs/courses/GL-DMOD/src/holonomic-d-modules-and-duality.md", "source_sha256": "f38a54cdd1c8d71748fa42074b42cb97a7263a7f371b6d5de6c5057322072dc1", "reader": "docs/courses/GL-DMOD/holonomic-d-modules-and-duality.html", "reader_sha256": "8fece522e05fa813e54c9b2f1cf886a502bcdd5d64baffbe63afdc05ccdc9018", "anchor": "analytic-local-operator-resolutions", "proof_locator": "Theorem3.0c, Lemmas3.0c.1–3.0c.4 and the finite-presentation paragraphs", "used_for": "Analytic good filtrations, relative analytification flatness, zero-section detection, one-neighborhood polynomial relations, uniform operator kernels and good induced images. Lemma6.6 treats independent coefficient/symbol counts and missing-derivative parameters.", "additional_proof_locators": [ { "anchor": "analytic-local-algebra-and-relative-completion", "locator": "Lemma3.0c.1", "used_for": "Identical relative completions and faithful-flat descent; extended to any number of auxiliary variables in Lemma6.6." }, { "anchor": "uniform-symbol-relations-and-operator-coherence", "locator": "Lemmas3.0c.3–3.0c.4", "used_for": "Uniform finite symbol and operator relations; coherent quotient filtration, not merely finite stalks." } ] } ], "intra_lesson_providers": [ "Theorem2.1 composition", "Theorem3.1 smooth pullback, including characteristic equality", "Proposition4.1 normal Koszul complex", "Lemmas6.1–6.2 and Proposition6.3 algebraic non-characteristic coherence, vanishing and containment", "Lemma6.4 central specialization without strictness, with exact Rees-torsion length cancellation", "Theorem6.5 full algebraic non-characteristic characteristic equality for arbitrary morphisms", "Lemma6.6 uniform analytic tangent operators, good central-quotient filtrations and holomorphic projection flatness", "Theorem6.7 full analytic non-characteristic coherence, concentration and characteristic equality" ], "source_correspondence": [ { "author": "Pierre Schapira", "title": "An introduction to D-modules", "edition": "Draft v7, March2020", "url": "https://webusers.imj-prg.fr/~pierre.schapira/LectNotes/Dmod.pdf", "sha256": "070e804c4d849706f083d8cc6bc4c152eab8a9e175b7d2b4699ac18bd43df087", "locators": [ "Definition2.3.1", "Theorem2.3.7 pp28–31", "Remark2.3.8 p30", "Definition3.1.10 and Lemma3.1.11 p54" ], "scope": "The theorem proves analytic concentration, coherence and containment; the remark states equality and refers its proof elsewhere. f_D inverse is unshifted Lf*." }, { "author": "Christian Schnell", "title": "D-modules", "url": "https://www.math.stonybrook.edu/~cschnell/pdf/notes/d-modules.pdf", "sha256": "c5f328e82408df398db8ff03d1c7ca2b37b7740d792d14de5f167d530ceb038f", "locators": [ "Theorem16.5 pp79–83", "Theorem18.5(b) p91" ], "scope": "Full algebraic non-characteristic statement; final equality requires a further filtration argument. Preservation statement is supplementary, not the only proof route for Exercise7.5." } ], "remaining_proof_obligations": [], "dependency_order": "GL11 Theorem2.1 uses GL07 smooth pullback/composition, GL05 growth, GL09 supported equivalence and GL10 localization, not GL07 Exercise7.5 or the characteristic equality proved in Theorem6.5.", "mathematical_scope": { "algebraic_coherence_tor_containment_proved": true, "algebraic_characteristic_equality_proved": true, "analytic_coherence_tor_containment_proved": true, "analytic_characteristic_equality_proved": true, "full_noncharacteristic_statement_proved": true }, "algebraic_proof": { "lemma": "6.4", "theorem": "6.5", "scope": "Every algebraic morphism of smooth characteristic-zero varieties and every coherent module satisfying the original non-characteristic condition; not restricted to holonomic modules.", "strictness_assumed": false, "key_inputs": [ "Finite tangent-ring symbol module and Noetherian Rees ring from6.3", "Normal-coordinate injectivity6.2", "Componentwise involutivity GL03 Theorem7.0" ], "key_steps": [ "Exact h-torsion cancellation for central t specialization", "Positive kernel/cokernel length difference on a minimal intersection prime", "No vertical components by iterated normal-symbol derivative of a monic equation", "Successive hypersurfaces and graph factorization retain the full original morphism" ] }, "analytic_proof": { "lemma": "6.6", "theorem": "6.7", "scope": "All holomorphic maps of complex manifolds and every coherent left analytic D-module satisfying6.1; no holonomic/regular/proper restriction.", "key_steps": [ "Relative completion for independent counts of coefficients and auxiliary variables", "Uniform tangent operator relations and actual good quotient filtration through K+tF0", "Monic annihilator on one analytic neighborhood, not only in one fiber", "Normal multiplication injectivity with exact Weyl commutator", "No vertical components using full Noetherian minimal-prime involutivity", "Central-specialization equality followed by all-covector analytic rank tests", "Successive analytic embeddings and flat holomorphic graph projection" ], "strictness_assumed": false, "stalk_Noetherianity_substituted_for_sheaf_coherence": false } }