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graded Noetherian implies left/right Noetherian; exact Bernstein degree for nonzero operators." }, { "lesson_id": "AG-CA-03", "source": "docs/courses/AG-CA/src/noetherian-and-artinian-rings.md", "source_sha256": "94b3cd74c201f9ac1daf33aa90ca4218263559884e56ee47c0ad98d82187b771", "reader": "docs/courses/AG-CA/AG-CA-03.html", "reader_sha256": "4cd0b10dfbfb5e16acaaaf75fb3c6145be8f5b70bb0a4b078ea8cfa5b4718b1d", "anchor": "1-three-ways-to-recognize-a-noetherian-module", "proof_locator": "Theorem1.1 and Proposition1.2", "conditions_and_use": "Finite modules and their submodules over Noetherian rings." }, { "lesson_id": "AG-CA-03", "source": "docs/courses/AG-CA/src/noetherian-and-artinian-rings.md", "source_sha256": "94b3cd74c201f9ac1daf33aa90ca4218263559884e56ee47c0ad98d82187b771", "reader": "docs/courses/AG-CA/AG-CA-03.html", "reader_sha256": "4cd0b10dfbfb5e16acaaaf75fb3c6145be8f5b70bb0a4b078ea8cfa5b4718b1d", "anchor": "2-polynomial-rings-and-finite-geometric-descriptions", "proof_locator": "Theorem2.1", "conditions_and_use": "Hilbert basis and localization for Noetherian coefficient rings." }, { "lesson_id": "AG-CA-03", "source": "docs/courses/AG-CA/src/noetherian-and-artinian-rings.md", "source_sha256": "94b3cd74c201f9ac1daf33aa90ca4218263559884e56ee47c0ad98d82187b771", "reader": "docs/courses/AG-CA/AG-CA-03.html", "reader_sha256": "4cd0b10dfbfb5e16acaaaf75fb3c6145be8f5b70bb0a4b078ea8cfa5b4718b1d", "anchor": "3-measuring-a-module-by-simple-pieces", "proof_locator": "Theorems3.2-3.3", "conditions_and_use": "Composition length independence and additivity for finite-length modules." }, { "lesson_id": "AG-CA-05", "source": "docs/courses/AG-CA/src/integral-extensions-lying-over-going-up-and-going-down.md", "source_sha256": "60311c428775940a13383d70b4947fecc9c9b1dab48f87fb790e55364513d396", "reader": "docs/courses/AG-CA/AG-CA-05.html", "reader_sha256": "2d8855aaf194bf9fe2626b246d95720b2cee6a971feffb08cdefa00c33b7b4f1", "anchor": "1-a-monic-equation-is-finite-control", "proof_locator": "Theorems1.1-1.2, Proposition1.3", "conditions_and_use": "Finite implies integral; integral equations persist under quotients and localization." }, { "lesson_id": "AG-CA-05", "source": "docs/courses/AG-CA/src/integral-extensions-lying-over-going-up-and-going-down.md", "source_sha256": "60311c428775940a13383d70b4947fecc9c9b1dab48f87fb790e55364513d396", "reader": "docs/courses/AG-CA/AG-CA-05.html", "reader_sha256": "2d8855aaf194bf9fe2626b246d95720b2cee6a971feffb08cdefa00c33b7b4f1", "anchor": "3-lifting-points-and-specializations", "proof_locator": "Theorems3.2-3.3", "conditions_and_use": "Lying over for inclusions, going up and incomparability." }, { "lesson_id": "AG-CA-05", "source": "docs/courses/AG-CA/src/integral-extensions-lying-over-going-up-and-going-down.md", "source_sha256": "60311c428775940a13383d70b4947fecc9c9b1dab48f87fb790e55364513d396", "reader": "docs/courses/AG-CA/AG-CA-05.html", "reader_sha256": "2d8855aaf194bf9fe2626b246d95720b2cee6a971feffb08cdefa00c33b7b4f1", "anchor": "5-chains-and-dimension", "proof_locator": "Theorem5.1", "conditions_and_use": "Dimension preserved by an integral inclusion, not an arbitrary noninjective map." }, { "lesson_id": "AG-CA-09", "source": "docs/courses/AG-CA/src/krull-dimension-and-noether-normalization.md", "source_sha256": "5b0969051396f29b75c0a7466acfdac4cc2457893a73fc4f3f312e3c0d434de4", "reader": "docs/courses/AG-CA/AG-CA-09.html", "reader_sha256": "a7db947490ce1eed7862660b870a5e0037f3e5e2c3eae4895b0ee56bbdce8298", "anchor": "2-making-one-equation-monic", "proof_locator": "Lemma2.1", "conditions_and_use": "Polynomial coordinate monicization over any field." }, { "lesson_id": "AG-CA-09", "source": "docs/courses/AG-CA/src/krull-dimension-and-noether-normalization.md", "source_sha256": "5b0969051396f29b75c0a7466acfdac4cc2457893a73fc4f3f312e3c0d434de4", "reader": "docs/courses/AG-CA/AG-CA-09.html", "reader_sha256": "a7db947490ce1eed7862660b870a5e0037f3e5e2c3eae4895b0ee56bbdce8298", "anchor": "3-normalization-that-respects-ideals", "proof_locator": "Theorem3.1 and Corollary3.2", "conditions_and_use": "Finite polynomial normalization of every nonzero finite-type field algebra, including zero parameters." }, { "lesson_id": "AG-CA-09", "source": "docs/courses/AG-CA/src/krull-dimension-and-noether-normalization.md", "source_sha256": "5b0969051396f29b75c0a7466acfdac4cc2457893a73fc4f3f312e3c0d434de4", "reader": "docs/courses/AG-CA/AG-CA-09.html", "reader_sha256": "a7db947490ce1eed7862660b870a5e0037f3e5e2c3eae4895b0ee56bbdce8298", "anchor": "4-parameters-measure-dimension-and-height", "proof_locator": "Lemma4.1 and Theorem4.2", "conditions_and_use": "Dimension equals parameter count, with integral dimension preservation." }, { "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": "1-finite-generation-with-controlled-orders", "proof_locator": "Proposition1.1 and Theorem1.2", "conditions_and_use": "Bounded-below good filtrations exist and are locally shift-comparable." }, { "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": "2-a-support-that-does-not-depend-on-order-choices", "proof_locator": "Theorem2.1", "conditions_and_use": "Reduced characteristic support is filtration independent." }, { "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": "3-multiplicity-and-a-deformation-parameter", "proof_locator": "Lemma3.1 and Theorem3.2", "conditions_and_use": "Positive generic lengths and filtration independence, preserving nilpotent information." }, { "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", "conditions_and_use": "Strict good-filtration exactness; support union and top-dimensional cycle additivity only." }, { "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": "7-1-an-independent-whole-dimension-bound", "proof_locator": "Theorem7.2", "conditions_and_use": "Nonzero coherent left D-module, quasi-coherent over O, on smooth separated finite-type pure-dimensional variety in characteristic zero; whole-support bound independent of coisotropy." } ], "same_release_providers": [ { "consumer": "GL-DMOD-05", "provider": "GL-DMOD-04", "source_sha256": "abf6742d179d68593b86e5233787d9056ac0a85bf187930ae7c9090506b9c57c", "reader_sha256": "a1b545769ba2ef9237b18f0107153f375f7827b5e9b122a8381ecdf1a0660f9b", "proofs": [ "Theorem1.1: growth and positive integral multiplicity", "Proposition1.2: exact sequence growth and leading coefficients", "Theorem3.1: Joseph/Bernstein inequality", "Theorem5.1: finite length and multiplicity bound" ] } ], "related_analysis_readings": [ { "lesson": "AN01-U016", "reader": "docs/courses/AN-01/reader/AN01-U016.html", "reader_sha256": "0f3e4a793de0011671aca1e79bfabb15a7d59eeaff36fc2363209a37feae4cf0", "scope": "One-sided powers and finite-part models; 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no partial-dimensional weakening." }, { "lesson_id": "AG-CA-12", "source": "docs/courses/AG-CA/src/regular-sequences-depth-and-cohen-macaulay-modules.md", "source_sha256": "eeeff0205fcf94eb6421f1590970382145a6f0cfe1cc5e15de583068298c44c4", "reader": "docs/courses/AG-CA/regular-sequences-depth-and-cohen-macaulay-modules.html", "reader_sha256": "300459db5236d2b07a2c17f6fde00f5f7f48f38146722d938e6c7052a3f6a6a2", "anchor": "1-equations-that-act-injectively", "proof_locator": "Proposition1.2", "conditions_and_use": "Local permutation only; this lesson separately proves the required graded descent to R." } ], "internal_weighted_homogeneous_proof_in_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": "the-general-weighted-homogeneous-formula-and-its-proof", "legacy_anchor": "the-general-weighted-homogeneous-formula-remains-separate", "heading_anchors": [ { "line": 348, "ids": [ "the-cusp-explicit-equation-and-minimality", "the-cusp-a-stated-computation" ] }, { "line": 510, "ids": [ "the-general-weighted-homogeneous-formula-and-its-proof", "the-general-weighted-homogeneous-formula-remains-separate" ] } ], "proof_locator": "Lemmas5.1-5.4 and Theorem5.5; equations(5.22)-(5.47)", "conditions": "Polynomial over C, positive rational weights, weighted degree one, isolated critical point at the origin. Formula uses distinct weighted degrees of the full Jacobian algebra.", "proof_inputs": [ "Section1 formal-power module and derivative rule", "Section3 Bernstein ideal and Proposition3.2", "Six weighted_homogeneous_providers entries with precise hypotheses, revisions and read proofs" ], "supplied_here": [ "Isolated-to-global critical set and graded descent of regularity", "Linear relations of a permutable regular sequence by two inductions", "Order-symbol descent computing Ann_D(f^s)", "Explicit vector quotient and right-E action", "Reduced-polynomial divisibility including the double root -1", "Descending-weight admissible equation" ], "not_used": "Cusp residues, distributional continuation, general negative-rational-root theorem, or an external annihilator theorem." }, "rational_root_providers": [ { "source": "docs/courses/GL-DMOD/src/differential-operators-and-the-weyl-algebra.md", "reader": "docs/courses/GL-DMOD/differential-operators-and-the-weyl-algebra.html", "source_sha256": "2590c0e5aa15a1dacc355518b8c8819b3a80717aef6567f5cb4a732bfc24f744", "reader_sha256": "2fc3562d6877ba4f82662b1e1f5640eafa4f389f087a9b32fe1c12c3892136b3", "anchor": "3-why-étale-coordinates-suffice", "proof_locator": "Theorems3.2,4.1,5.1; sections3-5", "conditions_and_use": "Algebraic PBW, order pieces, symbols and filtered Noetherianity; countability of affine operator ring." }, { "source": "docs/courses/GL-DMOD/src/good-filtrations-and-the-characteristic-variety.md", "reader": "docs/courses/GL-DMOD/good-filtrations-and-the-characteristic-variety.html", "source_sha256": "5a426491e32e478754cd7b3951c19f105ddeca9db0f09d2d7be6d181a63fe7c6", "reader_sha256": "80b650c6955b0140ca191753df34590cb6917e138d793137cb44c8b0e4901a22", "anchor": "1-finite-generation-with-controlled-orders", "proof_locator": "Proposition1.1 and Theorem1.2; sections1-3", "conditions_and_use": "Good filtration and Rees generation/noetherianity; no assumption that ordinary specialization commutes with cohomology." }, { "source": "docs/courses/GL-DMOD/src/good-filtrations-and-the-characteristic-variety.md", "reader": "docs/courses/GL-DMOD/good-filtrations-and-the-characteristic-variety.html", "source_sha256": "5a426491e32e478754cd7b3951c19f105ddeca9db0f09d2d7be6d181a63fe7c6", "reader_sha256": "80b650c6955b0140ca191753df34590cb6917e138d793137cb44c8b0e4901a22", "anchor": "4-submodules-quotients-and-exact-sequences", "proof_locator": "Theorem4.1", "conditions_and_use": "Coherent quotient characteristic containment with actual induced good filtrations." }, { "source": "docs/courses/GL-DMOD/src/holonomic-d-modules-and-duality.md", "reader": "docs/courses/GL-DMOD/holonomic-d-modules-and-duality.html", "source_sha256": "f38a54cdd1c8d71748fa42074b42cb97a7263a7f371b6d5de6c5057322072dc1", "reader_sha256": "8fece522e05fa813e54c9b2f1cf886a502bcdd5d64baffbe63afdc05ccdc9018", "anchor": "2-a-connection-on-a-dense-open-set", "proof_locator": "Theorem2.2", "conditions_and_use": "Finite length of holonomic algebraic modules on finite-type smooth varieties; not a holomorphic-germ statement." }, { "source": "docs/courses/GL-DMOD/src/adjunctions-base-change-and-the-projection-formula.md", "reader": "docs/courses/GL-DMOD/adjunctions-base-change-and-the-projection-formula.html", "source_sha256": "2e6cda561fabfa1611bb34e2d27c0c084676e38cf0c0427f23158a086d1a7881", "reader_sha256": "afdf61a575593a0236d857742b7569f082efea4bc34d2086208f5ad31ce0fd03", "anchor": "3-2-proper-direct-images-and-coherence", "proof_locator": "Lemmas3.2-3.3, Theorem3.4", "conditions_and_use": "Coherent coefficient generators, full Spencer differential, finite transfer amplitude and proper-coherence proof; Rees extension supplied in Lemma6.4." }, { "source": "docs/courses/AG-QC/src/proper-morphisms-and-coherent-direct-images.md", "reader": "docs/courses/AG-QC/proper-morphisms-and-coherent-direct-images.html", "source_sha256": "c423ee6103ec53a08e320dd27c72531730b8da297e43351f758f50a752508dd3", "reader_sha256": "3da1393549ac05f3672f39cac8245e57d7722fcc3730749b015c66cfca181beb", "anchor": "4-a-projective-modification-supplies-the-witness", "proof_locator": "Theorem4.1", "conditions_and_use": "Ordinary coherent higher direct images for proper maps of locally Noetherian schemes, applied also after product with affine h-line." }, { "source": "docs/courses/resolution-of-singularities-in-characteristic-zero/src/principalization-and-resolution.md", "reader": "docs/courses/resolution-of-singularities-in-characteristic-zero/principalization-and-resolution.html", "source_sha256": "01e1ed44d0851959103cff3796b08400896a4d7be0452f1c27e16980d07e8cf7", "reader_sha256": "396f11d6f9193a38ef3278ec88c3ab285f5c9ccaf939010a2558c0ba4417f4fd", "anchor": "3-principalization", "proof_locator": "Theorem3.1 and Corollary3.3", "conditions_and_use": "Principalization of (f) on a smooth complex variety with empty boundary; projective smooth blowups, isomorphism outside V(f), total divisor SNC. Not analytic principalization." } ], "internal_polynomial_root_proof_in_05": { "source": "docs/courses/GL-DMOD/src/bernstein-sato-polynomials.md", "reader": "docs/courses/GL-DMOD/bernstein-sato-polynomials.html", "source_sha256": "d6d90bfd0182a4e3969e029add3e2da4d51b7ed59dedcda7e1e013750352f0f1", "reader_sha256": "e8022028ff211f8c3e00ec6b55842c83cd6e57798c955438c6118c82eaae7ff7", "anchor": "65-negative-rational-roots-for-every-polynomial", "proof_locator": "Lemmas6.2-6.4 and Theorems6.5-6.6; equations(6.16)-(6.43)", "conditions": "All nonconstant complex polynomials, then all characteristic-zero fields by the existing field-invariance argument. Holomorphic germs retained as a separate unclosed claim.", "supplied_here": [ "Shift nilpotence from a polynomial annihilator", "Finite endomorphisms for finite-length modules over countable complex algebras", "Full algebraic proper characteristic containment via Rees transfer", "SNC coherence, characteristic bounds and an explicit unit correction", "Canonical transfer section and holonomic quotient", "Functorial factorization b=t phi without post-pushforward injectivity", "Critical-value shrink and explicit global Bernstein equation descent" ], "sources_read": [ { "url": "https://people.math.harvard.edu/~mpopa/notes/Bernstein-Sato-notes.pdf", "sha256": "2dcc4b69f359c2359320c257f233f0360a359131fb7bc40ae0ec0f88c452e952", "physical_pages": [ 11, 12, 13, 14, 15 ] }, { "url": "https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/Bfunct.pdf", "sha256": "c3faf7cd0f863a26a30a4c4ef4b8e929c1fc4d32b8a7400520b34e63ffc61143", "physical_pages": [ 16, 17, 18, 19 ] } ] }, "analytic_local_hom_providers": [ { "source": "docs/courses/GL-DMOD/src/holonomic-d-modules-and-duality.md", "reader": "docs/courses/GL-DMOD/holonomic-d-modules-and-duality.html", "source_sha256": "f38a54cdd1c8d71748fa42074b42cb97a7263a7f371b6d5de6c5057322072dc1", "reader_sha256": "8fece522e05fa813e54c9b2f1cf886a502bcdd5d64baffbe63afdc05ccdc9018", "anchor": "uniform-symbol-relations-and-operator-coherence", "proof_locator": "Theorem3.0c; Lemmas3.0c.1–3.0c.4 and analytic dimension argument", "conditions_and_use": "Analytic coherent operator presentations, good filtrations, faithful symbol extension and componentwise characteristic dimension bound; not algebraic Section2 finite length." }, { "source": "docs/courses/GL-DMOD/src/kashiwaras-equivalence-and-singular-spaces.md", "reader": "docs/courses/GL-DMOD/kashiwaras-equivalence-and-singular-spaces.html", "source_sha256": "8dd09ee51c6a9f9f696de961f0b1dd769787a54a792d725a05ce797028cde5c3", "reader_sha256": "862bacb03c8a08fd73f772301ed0caebdf77419db48b1cfaf732919294acaa08", "anchor": "2-recovering-every-coefficient", "proof_locator": "Lemma2.1 and equation2.6", "conditions_and_use": "Finite normal projectors and coefficient recovery. Holomorphic coefficient action and analytic coherence are proved in Lemma6.7." }, { "source": "docs/courses/GL-DMOD/src/kashiwaras-equivalence-and-singular-spaces.md", "reader": "docs/courses/GL-DMOD/kashiwaras-equivalence-and-singular-spaces.html", "source_sha256": "8dd09ee51c6a9f9f696de961f0b1dd769787a54a792d725a05ce797028cde5c3", "reader_sha256": "862bacb03c8a08fd73f772301ed0caebdf77419db48b1cfaf732919294acaa08", "anchor": "3-the-equivalence-with-its-intrinsic-inverse", "proof_locator": "Theorem3.1", "conditions_and_use": "Intrinsic transfer evaluation, inverse maps and determinant conormal twist; analytic extension proved in Lemma6.7." }, { "source": "docs/courses/GL-DMOD/src/d-modules-flat-connections-and-local-systems.md", "reader": "docs/courses/GL-DMOD/d-modules-flat-connections-and-local-systems.html", "source_sha256": "7dee2e7f4c167853e999833598844ce8ba851f9973b64c93632fa9a03db07dd6", "reader_sha256": "d933a78db0c0ff2515972f7d62f1e0b76491b0043e110177756aabcbbb2659e5", "anchor": "3-why-finite-modules-cannot-have-relations-at-a-smooth-point", "proof_locator": "Theorem3.1", "conditions_and_use": "Coherent connection local-freeness mechanism; convergent-germ version proved directly in Lemma6.8." }, { "source": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/src/analytic-germs-local-parametrization-and-the-nullstellensatz.md", "reader": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/analytic-germs-local-parametrization-and-the-nullstellensatz.html", "source_sha256": "dbbffc2f2c578a92596ed59067d9318e1a629878ba2f232a173f610868084201", "reader_sha256": "256975d3181569b6ec49f1ac5b7ffad7ab0df887cd1f05841a62af928dfce2a2", "anchor": "4-the-local-parametrization-theorem", "proof_locator": "Lemma3.2, Theorem4.1, Theorems5.1 and5.4, Proposition5.5", "conditions_and_use": "Finite integral projection, minimal-polynomial coefficients, graph locus, cone bound, analytic Nullstellensatz and strict dimension drop." }, { "source": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/src/holomorphic-functions-of-several-variables.md", "reader": "docs/courses/complex-analytic-spaces-and-coherent-sheaves/holomorphic-functions-of-several-variables.html", "source_sha256": "26713fd1704e78ca923541e2590b7bf2dbe2c732d0503151cd9b9f596f0e00dd", "reader_sha256": "5d02f604532e2640090cf08884e4495513176b06eb27072b3764c6c9f6c4462e", "anchor": "4-the-riemann-extension-theorem", "proof_locator": "Theorem4.2 and Corollary4.3", "conditions_and_use": "Connected complement of a nonzero holomorphic discriminant-times-norm; exact proof read." } ], "analytic_local_hom_figure": { "path": "docs/courses/GL-DMOD/assets/GL-DMOD05/ANALYTIC-LOCAL-HOM-A0972C3DA594491A80E11F4222084282080E5731B39044DACBBFC4BAC4DED358.svg", "sha256": "a0972c3da594491a80e11f4222084282080e5731b39044dacbbfc4bac4ded358", "license": "CC0-1.0", "authorship": { "kind": "original", "credit": "GPT-6 Astra (OpenAI), October 2026", "statement": "Original CC0 proof-route diagram for Lemmas 6.7–6.8 and Theorem 6.9; not a geometric plot." }, "source_attribution": [ "GL-DMOD05, Lemmas 6.7–6.8 and Theorem 6.9; exact cited programme prerequisites; no reproduced third-party artwork." ] }, "internal_analytic_local_hom_proof_in_05": { "source": "docs/courses/GL-DMOD/src/bernstein-sato-polynomials.md", "reader": "docs/courses/GL-DMOD/bernstein-sato-polynomials.html", "source_sha256": "d6d90bfd0182a4e3969e029add3e2da4d51b7ed59dedcda7e1e013750352f0f1", "reader_sha256": "e8022028ff211f8c3e00ec6b55842c83cd6e57798c955438c6118c82eaae7ff7", "anchor": "66-local-morphisms-for-holomorphic-differential-operators", "proof_locator": "Lemmas6.7–6.8, Theorem6.9; equations6.44–6.55", "conditions": "Any two coherent holonomic analytic differential-operator module germs on a complex manifold; no regular-singularity or algebraicity restriction.", "supplied_here": [ "Analytic finite length", "Analytic supported normal form with determinant twist", "Finite-minor exceptional-set construction", "Connection piece accumulating at original point", "Scalar endomorphisms of simple germs", "Finite local Hom with composition-length bound" ] }, "analytic_contact_provider_routes": [ { "source": "docs/courses/AG-QC/src/complex-analytic-spaces-and-analytification.md", "reader": "docs/courses/AG-QC/complex-analytic-spaces-and-analytification.html", "source_sha256": "c2ff6a4abab4222170069158ce12cd022180bcf07767b81ced57a9adb177f780", "reader_sha256": "71bc9c87fd32608f5e43c367b58aa83c7ff1ef8acc7c4f9a0a688700a9e03bdf", "anchor": "2-local-analytic-algebra", "proof_locator": "Section2: preparation/division and Lemmas2.1–2.2", "role": "analytic foundation", "conditions_and_use": "Noetherian convergent local ring, formal completion and actual holomorphic inverse; complete consumed arguments read." }, { "source": "docs/courses/AG-CA/src/completion.md", "reader": "docs/courses/AG-CA/completion.html", "source_sha256": "ddbd225b9c5a584a2da5d1ab05f46e7f7e0b03f61dd063a391431602a6e8e2d2", "reader_sha256": "e614e7ce7b4b2de6f6baa16cece3a8217f4a9990474f748b2a3ae34ada8f374c", "anchor": "3-noetherian-exactness-tensoring-and-flatness", "proof_locator": "Sections1–3; Theorems3.1–3.3", "role": "ring-theoretic foundation", "conditions_and_use": "Exact finite-module completion, faithful ideal contraction and separated finite quotients; valid for the analytic Noetherian ring." }, { "source": "docs/courses/resolution-of-singularities-in-characteristic-zero/src/derivative-ideals-under-blowing-up.md", "reader": "docs/courses/resolution-of-singularities-in-characteristic-zero/derivative-ideals-under-blowing-up.html", "source_sha256": "4baa12d87578ebf383be20ea0e5b0fcb393e9ab80c2b57da3cb36fa070faca48", "reader_sha256": "66c652808d48530c9d00bfdd07a3fac8c5da4c6e2ec1fce2d6a349d889888f65", "anchor": "2-chart-formulas", "proof_locator": "Sections1–3, Proposition2.1 and Theorem3.1", "role": "algebraic comparison", "conditions_and_use": "Analytic chain-rule and controlled-transform argument supplied independently in Lemma6.10." }, { "source": "docs/courses/resolution-of-singularities-in-characteristic-zero/src/hypersurfaces-of-maximal-contact.md", "reader": "docs/courses/resolution-of-singularities-in-characteristic-zero/hypersurfaces-of-maximal-contact.html", "source_sha256": "bc0b86a82e04776194ecdbe196cbc4183ba028247421df2d48e22452336e7ea2", "reader_sha256": "21f880e97f153b8dc3a76b1718c11f16fbf85ea062a4a763584ad9de7c59fb90", "anchor": "2-maximal-contact", "proof_locator": "Theorem2.3 and proof", "role": "algebraic comparison", "conditions_and_use": "Analytic existence and persistence supplied in Section6.7 and Lemma6.10; no algebraic-to-analytic assertion by citation." }, { "source": "docs/courses/resolution-of-singularities-in-characteristic-zero/src/uniqueness-of-maximal-contact.md", "reader": "docs/courses/resolution-of-singularities-in-characteristic-zero/uniqueness-of-maximal-contact.html", "source_sha256": "e1c03053663deef7ff1552a5961ba5b80776e8088445c304d94c85d10189ba34", "reader_sha256": "c0001f1391ed437de6eb9b4d185591e415115b87bc6d03872ebbed162466daae", "anchor": "2-automorphisms-close-to-the-identity", "proof_locator": "Sections1–4, Proposition2.1, Theorems3.3 and4.1", "role": "algebraic comparison", "conditions_and_use": "Actual convergent coordinate map and all exceptional-chart comparisons supplied in Lemma6.11 and Theorem6.12." }, { "source": "docs/courses/resolution-of-singularities-in-characteristic-zero/src/order-reduction-for-ideals.md", "reader": "docs/courses/resolution-of-singularities-in-characteristic-zero/order-reduction-for-ideals.html", "source_sha256": "0883ffb41c2dcf69c136054c121856972197bb8c0544fd0b56a509c2645f9616", "reader_sha256": "f0d83224ed33949647a79d4daae4b346b1cea63334908b4fd339c7c882d9cdb9", "anchor": "3-order-reduction-for-ideals-in-dimension-n", "proof_locator": "Sections1–3, Theorem3.1", "role": "unfinished analytic transfer", "conditions_and_use": "Read StepA, StepB, choice independence and gluing; analytic tuning, going-up and finite termination not claimed." }, { "source": "docs/courses/resolution-of-singularities-in-characteristic-zero/src/order-reduction-for-marked-ideals.md", "reader": "docs/courses/resolution-of-singularities-in-characteristic-zero/order-reduction-for-marked-ideals.html", "source_sha256": "7db21f891ba307c727bc35b78522b9e1b66f2b4c0e7acc470d2818942fa68c52", "reader_sha256": "019472620e56a4b950008108cf1e4866b5d8b3bcd1cfdeaa00e11aee3f67e0dc", "anchor": "2-the-construction", "proof_locator": "Sections1–3, Theorem2.1 and Corollary3.2", "role": "unfinished analytic transfer", "conditions_and_use": "Read nonmonomial/monomial construction and termination proof; compact analytic application remains to be established." } ], "analytic_contact_figure": { "path": "docs/courses/GL-DMOD/assets/GL-DMOD05/ANALYTIC-MAXIMAL-CONTACT-D526E0B2A72C87ECC74CF14A4D2F15E6286780A409D741E43155BF8FAE09F907.svg", "sha256": "d526e0b2a72c87ecc74cf14a4d2f15e6286780a409d741e43155bf8fae09f907", "license": "CC0-1.0", "authorship": { "kind": "original", "credit": "GPT-6 Astra (OpenAI), October 2026", "statement": "Original exact real-slice diagram for the complex example in Section6.7; Lemmas6.10–6.11, Theorem6.12 and equation6.70." } }, "internal_analytic_contact_proof_in_05": { "source": "docs/courses/GL-DMOD/src/bernstein-sato-polynomials.md", "reader": "docs/courses/GL-DMOD/bernstein-sato-polynomials.html", "source_sha256": "d6d90bfd0182a4e3969e029add3e2da4d51b7ed59dedcda7e1e013750352f0f1", "reader_sha256": "e8022028ff211f8c3e00ec6b55842c83cd6e57798c955438c6118c82eaae7ff7", "anchor": "67-analytic-maximal-contact-and-equality-of-centres", "proof_locator": "Lemmas6.10–6.11, Theorem6.12 and Corollary6.13; equations6.56–6.70", "conditions": "Complex manifolds of arbitrary dimension, coherent ideals and finite smooth-centre controlled-transform sequences. Holomorphic equivalence requires MC-invariance and both contact hypersurfaces transverse to the labelled SNC boundary. Equality theorem retains its two-covering-maps and equal-pullback-sequences hypotheses.", "supplied_here": [ "Analytic controlled-derivative comparison", "Persistence of maximal contact", "Holomorphic equivalence of invariant contact choices", "Exact equality of permissible centres, including exceptional directions" ], "not_supplied": [ "Synchronized descent and compact-neighbourhood finite termination", "Analytic proper characteristic estimate" ], "mathematical_scope": { "full_analytic_principalization_proved": false }, "tuning_and_going_up": "Proved in Section6.8; the earlier contact proof alone does not establish them." }, "analytic_tuning_provider_routes": [ { "source": "docs/courses/resolution-of-singularities-in-characteristic-zero/src/tuning-of-ideals.md", "reader": "docs/courses/resolution-of-singularities-in-characteristic-zero/tuning-of-ideals.html", "source_sha256": "fca972876204953dfe43e4618c5cb959045ae49214786ced9c79787da09da23c", "reader_sha256": "01afcc5b591e470f2b7fadb501628024d35e26fefe0ed81b098ca6f9da9585f9", "anchor": "1-maximal-coefficient-ideals", "proof_locator": "Sections 1–2, all arguments", "conditions_and_use": "Algebraic programme comparison read; analytic statements proved in Section6.8 using holomorphic charts, finite weighted generators and ordered controlled transforms.", "role": "Independent CC0 programme comparison, not an external analytic proof substitution." }, { "source": "docs/courses/resolution-of-singularities-in-characteristic-zero/src/logarithmic-derivatives-and-going-up.md", "reader": "docs/courses/resolution-of-singularities-in-characteristic-zero/logarithmic-derivatives-and-going-up.html", "source_sha256": "972b43549de7fa073ba8f8a4e9df01481c76e4ad78d36aaaf9893d877b65415a", "reader_sha256": "d646cbab2a3c01b0d2d0140ac3b10909602e1f32ef23af7a18646cbfae3d6242", "anchor": "2-logarithmic-derivations-along-a-hypersurface", "proof_locator": "Sections 1–5, all arguments", "conditions_and_use": "Algebraic programme comparison read; analytic statements proved in Section6.8 using holomorphic charts, finite weighted generators and ordered controlled transforms.", "role": "Independent CC0 programme comparison, not an external analytic proof substitution." }, { "source": "docs/courses/resolution-of-singularities-in-characteristic-zero/src/smooth-blowups-and-transforms-of-ideals.md", "reader": "docs/courses/resolution-of-singularities-in-characteristic-zero/smooth-blowups-and-transforms-of-ideals.html", "source_sha256": "940a4ed190d462bbfd8f6e4a2193c4cee2fcded187595ec0a75682cd72050d72", "reader_sha256": "c0de900674c5ac6663e67b72ec4bc861061fae6f8b2b603350af3f359e4dee31", "anchor": "4-transforms-of-ideals-and-of-marked-ideals", "proof_locator": "Sections 3–4: charts, boundary, nonincrease and restriction", "conditions_and_use": "Algebraic programme comparison read; analytic statements proved in Section6.8 using holomorphic charts, finite weighted generators and ordered controlled transforms.", "role": "Independent CC0 programme comparison, not an external analytic proof substitution." } ], "analytic_tuning_figure": { "path": "docs/courses/GL-DMOD/assets/GL-DMOD05/ANALYTIC-TUNING-AC1AD2467C6D6F74A99833D6CFBF6CEB43AEB52F4A1F9C60185F1800743CE6D7.svg", "sha256": "ac1ad2467c6d6f74a99833d6cfbf6ceb43aeb52f4a1f9c60185f1800743ce6d7", "license": "CC0-1.0", "authorship": { "kind": "original", "credit": "GPT-6 Astra (OpenAI), October 2026", "statement": "Original CC0 exact proof diagram; preceding two diagrams retained without changes." } }, "internal_analytic_tuning_proof_in_05": { "source": "docs/courses/GL-DMOD/src/bernstein-sato-polynomials.md", "reader": "docs/courses/GL-DMOD/bernstein-sato-polynomials.html", "source_sha256": "d6d90bfd0182a4e3969e029add3e2da4d51b7ed59dedcda7e1e013750352f0f1", "reader_sha256": "e8022028ff211f8c3e00ec6b55842c83cd6e57798c955438c6118c82eaae7ff7", "anchor": "68-analytic-tuning-and-lifting-a-hypersurface-sequence", "proof_locator": "Lemma6.14, Theorems6.15–6.16; equations6.71–6.85", "conditions": "Arbitrary finite-dimensional complex manifolds; coherent ideals with finite maximum order for tuning; finite sequences of closed smooth centres. Going-up requires derivative balance initially and nonzero restriction on every hypersurface component; boundary compatibility and ordering are explicit.", "mathematical_scope": { "full_analytic_principalization_proved": false }, "boundary_reduction": "Section6.9 supplies the zero-restriction divisor branch and finite monomial termination. It does not close synchronized general principalization." }, "analytic_boundary_reduction_provider_routes": [ { "source": "docs/courses/resolution-of-singularities-in-characteristic-zero/src/order-reduction-for-ideals.md", "reader": "docs/courses/resolution-of-singularities-in-characteristic-zero/order-reduction-for-ideals.html", "source_sha256": "0883ffb41c2dcf69c136054c121856972197bb8c0544fd0b56a509c2645f9616", "reader_sha256": "f0d83224ed33949647a79d4daae4b346b1cea63334908b4fd339c7c882d9cdb9", "anchor": "2-moving-the-high-order-locus-off-a-divisor", "proof_locator": "Sections1–3: transversality, divisorial separation and conditional dimension descent", "role": "Algebraic programme comparison; analytic versions and convergence/divisibility arguments supplied in Section6.9. No theorem of algebraic geometry silently relabelled analytic." }, { "source": "docs/courses/resolution-of-singularities-in-characteristic-zero/src/order-reduction-for-marked-ideals.md", "reader": "docs/courses/resolution-of-singularities-in-characteristic-zero/order-reduction-for-marked-ideals.html", "source_sha256": "7db21f891ba307c727bc35b78522b9e1b66f2b4c0e7acc470d2818942fa68c52", "reader_sha256": "019472620e56a4b950008108cf1e4866b5d8b3bcd1cfdeaa00e11aee3f67e0dc", "anchor": "2-the-construction", "proof_locator": "Section2 Step3: all monomial-phase selection, preservation and termination arguments", "role": "Algebraic programme comparison; analytic versions and convergence/divisibility arguments supplied in Section6.9. No theorem of algebraic geometry silently relabelled analytic." }, { "source": "docs/courses/resolution-of-singularities-in-characteristic-zero/src/blow-up-sequences-and-the-main-theorems.md", "reader": "docs/courses/resolution-of-singularities-in-characteristic-zero/blow-up-sequences-and-the-main-theorems.html", "source_sha256": "117ec1f1cb49bd206fabd3649c8c64c1d00691409aceb8e55a9e5cabefd72447", "reader_sha256": "ddc2541b63ea4f6db61993e062cbbc1efcd182c93d50e42b0177c796406a0bc6", "anchor": "5-from-local-to-global", "proof_locator": "Section5 Theorem5.1, its proof, Remark5.2 and Exercise6.2: indexed-sequence descent, not just final-space agreement", "role": "Algebraic programme comparison; analytic versions and convergence/divisibility arguments supplied in Section6.9. No theorem of algebraic geometry silently relabelled analytic." } ], "analytic_boundary_reduction_figure": { "path": "docs/courses/GL-DMOD/assets/GL-DMOD05/ANALYTIC-BOUNDARY-REDUCTION-38FBA66EBD55977AA8C4A3A4DFD743B0FA094C409C43892437BA166FE85CF44F.svg", "sha256": "38fba66ebd55977aa8c4a3a4dfd743b0fa094c409c43892437ba166fe85cf44f", "license": "CC0-1.0", "authorship": { "kind": "original", "credit": "GPT-6 Astra (OpenAI), October 2026", "statement": "Original CC0 exact chart and coefficient diagram, not a geometric scale drawing." } }, "internal_analytic_boundary_reduction_proof_in_05": { "source": "docs/courses/GL-DMOD/src/bernstein-sato-polynomials.md", "reader": "docs/courses/GL-DMOD/bernstein-sato-polynomials.html", "source_sha256": "d6d90bfd0182a4e3969e029add3e2da4d51b7ed59dedcda7e1e013750352f0f1", "reader_sha256": "e8022028ff211f8c3e00ec6b55842c83cd6e57798c955438c6118c82eaae7ff7", "anchor": "69-analytic-divisorial-removal-and-finite-boundary-reduction", "proof_locator": "Lemma6.17, Proposition6.18, Theorem6.19, Lemma6.20; equations6.86–6.96", "conditions": "Derivative balance and finite maximum bound in divisorial removal; lower-dimensional sequence supplied in Proposition6.18; monomial ideal with bounded coefficients on a finite ordered SNC boundary in Theorem6.19. Empty labels and Cartier centres retained.", "mathematical_scope": { "divisorial_removal_proved": true, "monomial_stage_termination_proved": true, "full_analytic_principalization_proved": false } } }