{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.scheme","code":"thing-q1155772","url":"https://ver.cy/models/thing/q1155772/","name":"scheme","alternateNames":["spectrum of a ring","locally Noetherian scheme","open subscheme","closed subscheme","reduced scheme","separated scheme","torsor","Quot scheme","Picard group","Gorenstein scheme","arithmetic surface","equidimensional scheme"],"kind":"thing","status":"published","version":"0.2.0-wave.4","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QLT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"In algebraic geometry, a locally ringed space that is locally isomorphic to the spectrum of a commutative ring, introduced by Alexander Grothendieck, generalising algebraic varieties to allow nilpotents and arbitrary base rings, with properties and constructions such as locally Noetherian, reduced and separated schemes and open and closed subschemes; schemes are objects of a category and special cases of algebraic spaces.","purpose":"Let an agent explain schemes and their basic properties, relay definitions and constructions from algebraic geometry references, describe the notions the registry aliases name, and distinguish schemes from varieties, algebraic spaces, stacks and the everyday senses of scheme.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Locally affine","Allows nilpotents","Arbitrary base","Foundational"],"structure":{"bundles":[{"id":"understand","name":"Understand","description":"What a scheme is.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is a scheme, and how does it differ from a variety, algebraic space, stack and everyday scheme?","kind":"definition"},{"text":"Is the question about algebraic geometry or another sense of scheme?","kind":"boundary"}]}]},{"id":"properties","name":"Properties","description":"Properties and constructions.","findings":[{"id":"properties-finding","name":"Properties","description":"Properties.","questions":[{"text":"What are spectra, locally Noetherian, reduced and separated schemes, and subschemes?","kind":"definition"},{"text":"Which entry fits the specific notion?","kind":"action"}]}]}]},{"id":"theory","name":"Theory","description":"Theory.","layers":[{"id":"foundations","name":"Foundations","description":"Foundations.","findings":[{"id":"foundations-finding","name":"Foundations","description":"Foundations.","questions":[{"text":"How are schemes defined via sheaves of rings and gluing?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"morphisms","name":"Morphisms","description":"Morphisms.","findings":[{"id":"morphisms-finding","name":"Morphisms","description":"Morphisms.","questions":[{"text":"What are flat, proper and smooth morphisms of schemes?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]}]},{"id":"applications","name":"Applications","description":"Applications.","layers":[{"id":"number","name":"Number theory","description":"Number theory.","findings":[{"id":"number-finding","name":"Number theory","description":"Number theory.","questions":[{"text":"How did schemes contribute to proofs such as the Weil conjectures and Fermat s Last Theorem?","kind":"provenance"},{"text":"Which entry fits Weil conjectures?","kind":"action"}]}]},{"id":"generalisations","name":"Generalisations","description":"Generalisations.","findings":[{"id":"generalisations-finding","name":"Generalisations","description":"Generalisations.","questions":[{"text":"How do algebraic spaces and stacks extend schemes?","kind":"provenance"},{"text":"Which entry fits algebraic stack?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"History.","layers":[{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How did Grothendieck and the Paris school develop scheme theory?","kind":"provenance"},{"text":"Which entry fits Alexander Grothendieck?","kind":"action"}]}]},{"id":"learning","name":"Learning","description":"Learning.","findings":[{"id":"learning-finding","name":"Learning","description":"Learning.","questions":[{"text":"What textbooks introduce schemes?","kind":"provenance"},{"text":"Which entry fits algebraic geometry?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Explain schemes in algebraic geometry, citing mathematical sources.","Distinguish schemes from varieties."],"mustNot":["Confuse the mathematical and everyday senses.","Present properties in the aliases as types of scheme without care.","State theorems without their hypotheses."],"requiresHuman":["Changing the registry's alias mapping for this sense."]},"ethics":{"considerations":["Precise mathematics matters for students and research.","Errors propagate in teaching."],"affectedParties":["Students","Mathematicians"]},"owners":{"steward":"Nobody owns mathematical concepts; they are held in common.","roles":[],"masterSystems":[]},"relations":[{"target":"Q15975724","type":"parent","note":"registry parent class"},{"target":"Q4724016","type":"parent","note":"registry parent class"},{"target":"Q91438488","type":"parent","note":"registry parent class"},{"target":"locally ringed space","type":"related","note":"in registry terms"},{"target":"algebraic space","type":"related","note":"in registry terms"},{"target":"Alexander Grothendieck","type":"related"},{"target":"spectrum of a ring","type":"related"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.scheme","Wikidata: Q1155772 (https://www.wikidata.org/wiki/Q1155772)"]},"properties":{"applicability":"not-applicable","items":["introduced: late 1950s note - Grothendieck","EGA: 1960-1967 note - Elements de geometrie algebrique","affine scheme: Spec of a commutative ring note"]},"recognition":{"applicability":"optional","items":["Space locally modelled on spectra of rings","Spectrum of a ring, locally Noetherian, reduced and separated schemes, open and closed subschemes","Varieties are reduced and of finite type over a field; algebraic spaces and stacks generalise schemes; plans and plots are everyday senses","Not a visible object; an abstract geometric structure."]},"capabilities":{"applicability":"required","items":["explain the concept","relay definitions and properties","describe constructions","distinguish related objects"]},"hazards":{"applicability":"required","items":["Confusing schemes with varieties","Confusing mathematical and everyday senses","Registry aliases mixing properties and constructions"]},"interfaces":{"applicability":"required","items":["No regulation"]},"context":{"applicability":"required","items":["Not applicable; a mathematical object.","affine schemes","projective schemes","locally Noetherian schemes","reduced and integral schemes","separated and proper schemes","subschemes"]}},"sources":[{"title":"Wikidata item Q1155772: scheme","url":"https://www.wikidata.org/wiki/Q1155772","note":"identity and sense of the item"},{"title":"Wikipedia: Scheme (mathematics)","url":"https://en.wikipedia.org/wiki/Scheme_%28mathematics%29","note":"general description of the item"}],"openQuestions":["Should spectrum of a ring be a separate primary entry?","How should algebraic geometry references be linked?","The registry entry has merged aliases naming properties; should they be grouped?"],"resources":{"spec":"/models/things/publications/thing-q1155772/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q1155772/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-13T18:31:19Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-06T19:51:53+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}