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Canonical static concept record

Locally ringed space

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Summary

A ringed space whose stalks are local rings.

Record metadata

Canonical name: Locally ringed space
Concept ID: locally_ringed_space
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)

ringed space $(X,O_X)$

Axioms / constraints

every stalk $O_{X,x}$ is a local ring

Concept sources

Wikipedia — Ringed space — Ringed space · expository · source ID wp-ringed-space
Stacks Project — Locally ringed spaces — Locally ringed spaces · reference · source ID stacks-locally-ringed

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Ringed space Locally ringed space

Permalink to relation

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

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

Authored explanation

Require every stalk of the structure sheaf to be a local ring.

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 — Locally ringed spaces — Locally ringed spaces · reference · source ID stacks-locally-ringed
nLab — Locally ringed space — Locally ringed space · reference · source ID nlab-locally-ringed
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.