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Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Scheme

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Summary

A locally ringed space locally isomorphic to affine schemes.

Record metadata

Canonical name: Scheme
Concept ID: scheme
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebraic geometry
Domains: Algebraic geometry, Topology, Algebra
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

locally ringed space $X$

Axioms / constraints

affine open neighborhood at every point

Concept sources

Stacks Project — Schemes — Schemes · reference · source ID stacks-scheme
Wikipedia — Scheme (mathematics) — Scheme (mathematics) · expository · source ID wp-scheme
nLab — Scheme — Scheme · reference · source ID nlab-scheme
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Locally ringed space Scheme

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_locally_scheme
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + affine open cover

Authored explanation

Require every point to have an affine open neighborhood.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Stacks Project — Schemes — Schemes · reference · source ID stacks-scheme
Wikipedia — Scheme (mathematics) — Scheme (mathematics) · expository · source ID wp-scheme
nLab — Scheme — Scheme · reference · source ID nlab-scheme
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Scheme Affine scheme

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: scheme_to_affine_scheme
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + globally affine

Authored explanation

Require the scheme to be isomorphic to $\operatorname{Spec}(R)$ for some commutative ring $R$ .

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Stacks Project — Affine schemes — Affine schemes · reference · source ID stacks-affine-scheme
Wikipedia — Affine scheme — Affine scheme · expository · source ID wp-affine-scheme
nLab — Affine scheme — Affine scheme · reference · source ID nlab-affine-scheme
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Scheme scheme over a field(construction junction)

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: j_variety__scheme
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: geometric object

Authored explanation

Supply the Scheme structure as one jointly required input.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Wikipedia — Algebraic variety — Algebraic variety · expository · source ID wp-variety
nLab — Variety — Variety · reference · source ID nlab-variety
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Scheme Noetherian scheme

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_scheme_noetherian
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + local Noetherianity and quasi-compactness

Authored explanation

Require Noetherian affine neighborhoods and quasi-compactness.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Stacks Project — Noetherian schemes — Noetherian schemes · reference · source ID stacks-noetherian-scheme
Wikipedia — Noetherian scheme — Noetherian scheme · expository · source ID wp-noetherian-scheme
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Scheme Reduced scheme

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_scheme_reduced
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + reduced local rings

Authored explanation

Require no nonzero nilpotents in local rings.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Wikipedia — Reduced scheme — Reduced scheme · expository · source ID wp-reduced-scheme
nLab — Reduced scheme — Reduced scheme · reference · source ID nlab-reduced-scheme
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Scheme Regular scheme

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_scheme_regular
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: require local Noetherian regularity

Authored explanation

Require $X$ to be locally Noetherian and every local ring $\mathcal O_{X,x}$ to be a regular local ring; quasi-compactness is not required.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Stacks Project — Regular schemes — Regular schemes · reference · source ID stacks-regular-scheme
Wikipedia — Regular scheme — Regular scheme · expository · source ID wp-regular-scheme
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Scheme Morphism of schemes

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: algebraic_geometry_scheme_to_scheme_morphism
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: choose source, target, and local sheaf map

Authored explanation

A scheme morphism adds a continuous map between two schemes and a sheaf homomorphism $\mathcal O_Y\to f_*\mathcal O_X$ whose maps on stalks are local ring homomorphisms.

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

Stacks Project — Schemes — Schemes · reference · source ID stacks-scheme
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Scheme Scheme over a base

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: scheme_to_scheme_over_base
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + base and structure morphism

Authored explanation

Choose a base scheme $S$ and a morphism $X\to S$ .

How to interpret this relation type

Relation sources

Stacks Project — Schemes — Schemes · reference · source ID stacks-scheme
Stacks Project — Projective morphisms — Projective morphisms · reference · source ID stacks-projective-morphisms
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Scheme Separated scheme

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_scheme_separated
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + closed diagonal

Authored explanation

For the absolute node, require $\Delta_{X/\operatorname{Spec}(\mathbb{Z})}\colon X\to X\times_{\operatorname{Spec}(\mathbb{Z})}X$ to be a closed immersion; over a base $S$ use $\Delta_{X/S}$ .

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Stacks Project — Separation axioms — Separation axioms · reference · source ID stacks-separated-scheme