scheme
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.
Bundle → Layer → Finding → Questions Filled
4 bundles · 8 layers · 8 findings · 16 questions
Understand What a scheme is.
Definition
Definition.
Definition
Definition.
- What is a scheme, and how does it differ from a variety, algebraic space, stack and everyday scheme? definition
- Is the question about algebraic geometry or another sense of scheme? boundary
Properties
Properties and constructions.
Properties
Properties.
- What are spectra, locally Noetherian, reduced and separated schemes, and subschemes? definition
- Which entry fits the specific notion? action
Theory Theory.
Foundations
Foundations.
Foundations
Foundations.
- How are schemes defined via sheaves of rings and gluing? provenance
- Which references are standard? provenance
Morphisms
Morphisms.
Morphisms
Morphisms.
- What are flat, proper and smooth morphisms of schemes? provenance
- Which sources are cited? provenance
Applications Applications.
Number theory
Number theory.
Number theory
Number theory.
- How did schemes contribute to proofs such as the Weil conjectures and Fermat s Last Theorem? provenance
- Which entry fits Weil conjectures? action
Generalisations
Generalisations.
Generalisations
Generalisations.
- How do algebraic spaces and stacks extend schemes? provenance
- Which entry fits algebraic stack? action
Context History.
History
History.
History
History.
- How did Grothendieck and the Paris school develop scheme theory? provenance
- Which entry fits Alexander Grothendieck? action
Learning
Learning.
Learning
Learning.
- What textbooks introduce schemes? provenance
- Which entry fits algebraic geometry? action
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QLT
- Other names and narrower kinds
- 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
What it is Filled
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.
Why it exists Filled
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.
Distinguishing features Filled
- Locally affine
- Allows nilpotents
- Arbitrary base
- Foundational
What robots and AI may and may not do Filled
Must not
- Confuse the mathematical and everyday senses.
- Present properties in the aliases as types of scheme without care.
- State theorems without their hypotheses.
Only with a human decision
- Changing the registry's alias mapping for this sense.
May
- Explain schemes in algebraic geometry, citing mathematical sources.
- Distinguish schemes from varieties.
Moral aspects Filled
- Precise mathematics matters for students and research.
- Errors propagate in teaching.
Who is affected
- Students
- Mathematicians
Owners Filled
Steward
Nobody owns mathematical concepts; they are held in common.
Links to other meta-models Filled
parent
- Q15975724 - registry parent class
- Q4724016 - registry parent class
- Q91438488 - registry parent class
related
- locally ringed space - in registry terms
- algebraic space - in registry terms
- Alexander Grothendieck
- spectrum of a ring
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.scheme
- Wikidata: Q1155772 (https://www.wikidata.org/wiki/Q1155772)
Direct properties not applicable Not applicable
- introduced: late 1950s note - Grothendieck
- EGA: 1960-1967 note - Elements de geometrie algebrique
- affine scheme: Spec of a commutative ring note
Plane XCT: no invented physical properties.
Recognition optional Filled
- 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 and actions required Filled
- explain the concept
- relay definitions and properties
- describe constructions
- distinguish related objects
Hazards and failure modes required Filled
- Confusing schemes with varieties
- Confusing mathematical and everyday senses
- Registry aliases mixing properties and constructions
Standards and interfaces required Filled
- No regulation
Context of use required Filled
- Not applicable; a mathematical object.
- affine schemes
- projective schemes
- locally Noetherian schemes
- reduced and integral schemes
- separated and proper schemes
- subschemes
Sources Filled
- Wikidata item Q1155772: scheme - identity and sense of the item
- Wikipedia: Scheme (mathematics) - general description of the item
Open questions
- 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?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q1155772/spec.json