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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

Algebraic variety

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Summary

Under the adopted Stacks Project convention, an integral separated scheme of finite type over a field.

Record metadata

Canonical name: Algebraic variety
Concept ID: algebraic_variety
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebraic geometry
Domains: Algebraic geometry
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

field $k$
scheme $X$ with structure morphism $X\to \operatorname{Spec}(k)$

Data

$k$ -scheme structure

Axioms / constraints

$X$ is integral
$X\to \operatorname{Spec}(k)$ is separated
$X\to \operatorname{Spec}(k)$ is of finite type

Notes

Terminology varies across authors; this atlas fixes the Stacks Project convention rather than mixing reduced, irreducible, or nonseparated variants.

Concept sources

Wikipedia — Algebraic variety — Algebraic variety · expository · source ID wp-variety
nLab — Variety — Variety · reference · source ID nlab-variety
Stacks Project — Varieties — Varieties · reference · source ID stacks-varieties
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.

scheme over a field(construction junction)Algebraic variety

Permalink to relation

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

Relation ID: j_variety__out
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: impose compatibility

Authored explanation

Require $X$ to be integral and require $X\to\operatorname{Spec}(k)$ to be separated and of finite type.

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 — Varieties — Varieties · reference · source ID stacks-varieties
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.