{
  "schema": "zariski-complete-unit-routes/v1",
  "licence": "CC0-1.0",
  "units": [
    {
      "tag": null,
      "historical_label": "definition-module-finite-type",
      "historical_anchor": "algebra-definition-module-finite-type",
      "title": "Finite modules and finite presentations",
      "reader": "docs/support/zariski-algebra/definitions-and-dimension.html",
      "anchor": "finite-modules-and-their-presentations",
      "source": "docs/support/zariski-algebra/src/definitions-and-dimension.md",
      "locator": "Finite modules and module presentations",
      "source_sha256": "feb9f5ba55433a0b9140dbad2508feda2256d7ed387fa3bac516c079117e7bef",
      "reader_sha256": "171fff090b428d658feca1f7e7238e7acd7811289db29f9764dec9e7eb8928c7",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "this revision; bytes identified by SHA-256",
      "source_revision": "this revision; bytes identified by SHA-256",
      "reader_revision": "this revision; bytes identified by SHA-256"
    },
    {
      "tag": "00F3",
      "historical_label": "definition-finite-type",
      "historical_anchor": "algebra-definition-finite-type",
      "title": "Finite-type and finitely presented algebras",
      "reader": "docs/support/zariski-algebra/definitions-and-dimension.html",
      "anchor": "finite-type-and-finitely-presented-algebras",
      "source": "docs/support/zariski-algebra/src/definitions-and-dimension.md",
      "locator": "Finite-type and finitely presented algebras",
      "source_sha256": "feb9f5ba55433a0b9140dbad2508feda2256d7ed387fa3bac516c079117e7bef",
      "reader_sha256": "171fff090b428d658feca1f7e7238e7acd7811289db29f9764dec9e7eb8928c7",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "this revision; bytes identified by SHA-256",
      "source_revision": "this revision; bytes identified by SHA-256",
      "reader_revision": "this revision; bytes identified by SHA-256"
    },
    {
      "tag": null,
      "historical_label": "definition-finite-ring-map",
      "historical_anchor": "algebra-definition-finite-ring-map",
      "title": "Finite ring maps",
      "reader": "docs/support/zariski-algebra/definitions-and-dimension.html",
      "anchor": "finite-ring-maps",
      "source": "docs/support/zariski-algebra/src/definitions-and-dimension.md",
      "locator": "Finite ring maps",
      "source_sha256": "feb9f5ba55433a0b9140dbad2508feda2256d7ed387fa3bac516c079117e7bef",
      "reader_sha256": "171fff090b428d658feca1f7e7238e7acd7811289db29f9764dec9e7eb8928c7",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "this revision; bytes identified by SHA-256",
      "source_revision": "this revision; bytes identified by SHA-256",
      "reader_revision": "this revision; bytes identified by SHA-256"
    },
    {
      "tag": "0564",
      "historical_label": "lemma-finite-finitely-presented-extension",
      "historical_anchor": "algebra-lemma-finite-finitely-presented-extension",
      "title": "Finite presentation across a finite algebra",
      "reader": "docs/support/zariski-algebra/index.html",
      "anchor": "algebra-lemma-finite-finitely-presented-extension",
      "source": "docs/support/zariski-algebra/src/zariski-algebra.md",
      "locator": "Finite presentation across a finite algebra",
      "source_sha256": "c62f72d45035d33720d7a67234db65b118ca820134e2b32fd9c495edde9778ab",
      "reader_sha256": "44d226a0e1dc02931c02195e034080c7035764ba659592c2f93e28378db76c81",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "this revision; bytes identified by SHA-256",
      "source_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "reader_revision": "this revision; bytes identified by SHA-256"
    },
    {
      "tag": "00P2",
      "historical_label": "lemma-codimension",
      "historical_anchor": "algebra-lemma-codimension",
      "title": "Codimension and dimension at a point",
      "reader": "docs/support/zariski-algebra/definitions-and-dimension.html",
      "anchor": "dimension-comparison-for-a-surjective-algebra-map",
      "source": "docs/support/zariski-algebra/src/definitions-and-dimension.md",
      "locator": "Quotient dimension comparison",
      "source_sha256": "feb9f5ba55433a0b9140dbad2508feda2256d7ed387fa3bac516c079117e7bef",
      "reader_sha256": "171fff090b428d658feca1f7e7238e7acd7811289db29f9764dec9e7eb8928c7",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "this revision; bytes identified by SHA-256",
      "source_revision": "this revision; bytes identified by SHA-256",
      "reader_revision": "this revision; bytes identified by SHA-256"
    },
    {
      "tag": "00P4",
      "historical_label": "lemma-dimension-at-a-point-preserved-field-extension",
      "historical_anchor": "algebra-lemma-dimension-at-a-point-preserved-field-extension",
      "title": "Dimension at a point under field extension",
      "reader": "docs/support/zariski-algebra/index.html",
      "anchor": "algebra-lemma-dimension-at-a-point-preserved-field-extension",
      "source": "docs/support/zariski-algebra/src/zariski-algebra.md",
      "locator": "Dimension near a point after extension of the ground field",
      "source_sha256": "c62f72d45035d33720d7a67234db65b118ca820134e2b32fd9c495edde9778ab",
      "reader_sha256": "44d226a0e1dc02931c02195e034080c7035764ba659592c2f93e28378db76c81",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "this revision; bytes identified by SHA-256",
      "source_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "reader_revision": "this revision; bytes identified by SHA-256"
    },
    {
      "tag": "0307",
      "historical_label": "lemma-integral-closure-localize",
      "historical_anchor": "algebra-lemma-integral-closure-localize",
      "title": "Integral closure and localization",
      "reader": "docs/courses/AG-QC/the-theorem-on-formal-functions.html",
      "anchor": "z-1-integral-closure-and-localization",
      "source": "docs/courses/AG-QC/src/the-theorem-on-formal-functions.md",
      "locator": "Appendix Z, Lemma Z.1",
      "source_sha256": "4f43273e79df76cdfc715121fb1cef02769c1c9a575b87039a16d0b8485339ad",
      "reader_sha256": "384f16ef3a936d38afcd7524bc328556605bbf46cd20089c59eea8b2c92b5c35",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "source_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "reader_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a"
    },
    {
      "tag": "00PN",
      "historical_label": "lemma-four-rings",
      "historical_anchor": "algebra-lemma-four-rings",
      "title": "Quasi-finiteness through a finite tensor-product map",
      "reader": "docs/courses/AG-QC/the-theorem-on-formal-functions.html",
      "anchor": "z-8-quasi-finiteness-in-a-four-ring-diagram",
      "source": "docs/courses/AG-QC/src/the-theorem-on-formal-functions.md",
      "locator": "Appendix Z, Lemma Z.8",
      "source_sha256": "4f43273e79df76cdfc715121fb1cef02769c1c9a575b87039a16d0b8485339ad",
      "reader_sha256": "384f16ef3a936d38afcd7524bc328556605bbf46cd20089c59eea8b2c92b5c35",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "source_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "reader_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a"
    },
    {
      "tag": "00PT",
      "historical_label": "lemma-make-integral-trick",
      "historical_anchor": "algebra-lemma-make-integral-trick",
      "title": "A monic polynomial integrality argument",
      "reader": "docs/courses/AG-QC/the-theorem-on-formal-functions.html",
      "anchor": "z-2-removing-a-monic-denominator",
      "source": "docs/courses/AG-QC/src/the-theorem-on-formal-functions.md",
      "locator": "Appendix Z, Lemma Z.2",
      "source_sha256": "4f43273e79df76cdfc715121fb1cef02769c1c9a575b87039a16d0b8485339ad",
      "reader_sha256": "384f16ef3a936d38afcd7524bc328556605bbf46cd20089c59eea8b2c92b5c35",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "source_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "reader_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a"
    },
    {
      "tag": "00PV",
      "historical_label": "lemma-combine-lemmas",
      "historical_anchor": "algebra-lemma-combine-lemmas",
      "title": "Clearing a leading coefficient",
      "reader": "docs/courses/AG-QC/the-theorem-on-formal-functions.html",
      "anchor": "z-3-clearing-a-leading-coefficient",
      "source": "docs/courses/AG-QC/src/the-theorem-on-formal-functions.md",
      "locator": "Appendix Z, Lemma Z.3",
      "source_sha256": "4f43273e79df76cdfc715121fb1cef02769c1c9a575b87039a16d0b8485339ad",
      "reader_sha256": "384f16ef3a936d38afcd7524bc328556605bbf46cd20089c59eea8b2c92b5c35",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "source_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "reader_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a"
    },
    {
      "tag": "00PW",
      "historical_label": "situation-one-transcendental-element",
      "historical_anchor": "algebra-situation-one-transcendental-element",
      "title": "The conductor setup",
      "reader": "docs/courses/AG-QC/the-theorem-on-formal-functions.html",
      "anchor": "z-4-the-conductor-setup",
      "source": "docs/courses/AG-QC/src/the-theorem-on-formal-functions.md",
      "locator": "Appendix Z, Situation Z.4",
      "source_sha256": "4f43273e79df76cdfc715121fb1cef02769c1c9a575b87039a16d0b8485339ad",
      "reader_sha256": "384f16ef3a936d38afcd7524bc328556605bbf46cd20089c59eea8b2c92b5c35",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "source_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "reader_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a"
    },
    {
      "tag": "00PX",
      "historical_label": "lemma-leading-coefficient-in-J",
      "historical_anchor": "algebra-lemma-leading-coefficient-in-J",
      "title": "Leading coefficients and the conductor",
      "reader": "docs/courses/AG-QC/the-theorem-on-formal-functions.html",
      "anchor": "z-5-the-leading-coefficient-enters-the-conductor",
      "source": "docs/courses/AG-QC/src/the-theorem-on-formal-functions.md",
      "locator": "Appendix Z, Lemma Z.5",
      "source_sha256": "4f43273e79df76cdfc715121fb1cef02769c1c9a575b87039a16d0b8485339ad",
      "reader_sha256": "384f16ef3a936d38afcd7524bc328556605bbf46cd20089c59eea8b2c92b5c35",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "source_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "reader_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a"
    },
    {
      "tag": "00PY",
      "historical_label": "lemma-all-coefficients-in-J",
      "historical_anchor": "algebra-lemma-all-coefficients-in-J",
      "title": "All coefficients lie in the conductor radical",
      "reader": "docs/courses/AG-QC/the-theorem-on-formal-functions.html",
      "anchor": "z-6-every-coefficient-enters-the-radical",
      "source": "docs/courses/AG-QC/src/the-theorem-on-formal-functions.md",
      "locator": "Appendix Z, Lemma Z.6",
      "source_sha256": "4f43273e79df76cdfc715121fb1cef02769c1c9a575b87039a16d0b8485339ad",
      "reader_sha256": "384f16ef3a936d38afcd7524bc328556605bbf46cd20089c59eea8b2c92b5c35",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "source_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "reader_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a"
    },
    {
      "tag": null,
      "historical_label": "definition-strongly-transcendental",
      "historical_anchor": "algebra-definition-strongly-transcendental",
      "title": "Strong transcendence",
      "reader": "docs/courses/AG-QC/the-theorem-on-formal-functions.html",
      "anchor": "z-7-strong-transcendence-and-the-conductor-quotient",
      "source": "docs/courses/AG-QC/src/the-theorem-on-formal-functions.md",
      "locator": "Appendix Z, Definition Z.7",
      "source_sha256": "4f43273e79df76cdfc715121fb1cef02769c1c9a575b87039a16d0b8485339ad",
      "reader_sha256": "384f16ef3a936d38afcd7524bc328556605bbf46cd20089c59eea8b2c92b5c35",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "source_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "reader_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a"
    },
    {
      "tag": "00Q0",
      "historical_label": "lemma-reduced-strongly-transcendental-minimal-prime",
      "historical_anchor": "algebra-lemma-reduced-strongly-transcendental-minimal-prime",
      "title": "Passing to a minimal prime",
      "reader": "docs/courses/AG-QC/the-theorem-on-formal-functions.html",
      "anchor": "z-9-passage-to-a-minimal-prime",
      "source": "docs/courses/AG-QC/src/the-theorem-on-formal-functions.md",
      "locator": "Appendix Z, Lemma Z.9",
      "source_sha256": "4f43273e79df76cdfc715121fb1cef02769c1c9a575b87039a16d0b8485339ad",
      "reader_sha256": "384f16ef3a936d38afcd7524bc328556605bbf46cd20089c59eea8b2c92b5c35",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "source_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "reader_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a"
    },
    {
      "tag": "00Q1",
      "historical_label": "lemma-domains-transcendental-not-quasi-finite",
      "historical_anchor": "algebra-lemma-domains-transcendental-not-quasi-finite",
      "title": "Transcendence excludes quasi-finite points for domains",
      "reader": "docs/courses/AG-QC/the-theorem-on-formal-functions.html",
      "anchor": "z-10-transcendence-excludes-quasi-finite-points-for-domains",
      "source": "docs/courses/AG-QC/src/the-theorem-on-formal-functions.md",
      "locator": "Appendix Z, Lemma Z.10",
      "source_sha256": "4f43273e79df76cdfc715121fb1cef02769c1c9a575b87039a16d0b8485339ad",
      "reader_sha256": "384f16ef3a936d38afcd7524bc328556605bbf46cd20089c59eea8b2c92b5c35",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "source_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "reader_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a"
    },
    {
      "tag": "00Q2",
      "historical_label": "lemma-reduced-strongly-transcendental-not-quasi-finite",
      "historical_anchor": "algebra-lemma-reduced-strongly-transcendental-not-quasi-finite",
      "title": "Strong transcendence excludes quasi-finite points",
      "reader": "docs/courses/AG-QC/the-theorem-on-formal-functions.html",
      "anchor": "z-11-strong-transcendence-excludes-quasi-finite-points",
      "source": "docs/courses/AG-QC/src/the-theorem-on-formal-functions.md",
      "locator": "Appendix Z, Lemma Z.11",
      "source_sha256": "4f43273e79df76cdfc715121fb1cef02769c1c9a575b87039a16d0b8485339ad",
      "reader_sha256": "384f16ef3a936d38afcd7524bc328556605bbf46cd20089c59eea8b2c92b5c35",
      "licence": "CC0-1.0",
      "scope_check": "Full supplied statement and hypotheses compared against the admitted selected source; no scope reduction.",
      "provider_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "source_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a",
      "reader_revision": "91a87f19b795706fd36fd7f3b309984e5bb0622a"
    }
  ],
  "source_edition": {
    "authors": "The Stacks Project authors",
    "fork": "AI Integrated Stacks",
    "revision": "565b10e987aba5969b21145a0833f42d69f96790",
    "chapter": "algebra.tex",
    "chapter_sha256": "eb0db6ee32deab7b24e055a73cf2cc19904394e516a5b22567e7ef895a7b9cec"
  },
  "preservation": "All seventeen selected units retain exact reader routes. Independent programme expression replaces the old excerpts; the cited source retains its genuine licence.",
  "checking": "Statement/hypothesis and direct proof comparison; existing proof dependencies remain exactly cited. No whole-programme recursive proof certification or independent review is asserted.",
  "new_text_provenance": {
    "model": "gpt-6-astra",
    "reasoning_effort": "ultra",
    "date": "2026-10-10",
    "scope": "Definitions, quotient-dimension corollary and route integration; existing provider authorship unchanged."
  }
}
