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Canonical static concept record

Prime spectrum $\operatorname{Spec}(R)$

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Summary

The set of prime ideals of a commutative ring $R$ , equipped with the Zariski topology and its structure sheaf.

Record metadata

Canonical name: Prime spectrum $\operatorname{Spec}(R)$
Concept ID: prime_spectrum
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)

commutative ring $R$

Data

prime ideals
Zariski topology
structure sheaf

Canonically induces

An affine scheme

Concept sources

Wikipedia — Spectrum of a ring — Spectrum of a ring · encyclopedic reference · source ID wp-prime-spectrum
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.

Commutative ring Prime spectrum $\operatorname{Spec}(R)$

Permalink to relation

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

Relation ID: story_commutative_ring_to_prime_spectrum_canonical_construction
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: form $\operatorname{Spec}(R)$

Authored explanation

Prime ideals with the Zariski topology and structure sheaf form the affine spectrum.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

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

Prime ideal Prime spectrum $\operatorname{Spec}(R)$

Permalink to relation

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

Relation ID: number_prime_ideal_to_prime_spectrum
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: supply points of $\operatorname{Spec}(R)$

Authored explanation

The prime spectrum of a commutative ring is formed from its prime ideals, equipped with the Zariski topology and its canonical structure sheaf.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text
Stacks Project — Schemes — Schemes · reference · source ID stacks-scheme

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.