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Canonical static concept record

Regular scheme

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Summary

A locally Noetherian scheme whose local rings are regular local rings.

Record metadata

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

Carrier(s)

scheme $X$

Axioms / constraints

locally Noetherian
every local ring $O_{X,x}$ is regular

Notes

Regularity is local Noetherianity plus regularity of every local ring. It does not imply quasi-compactness, so a regular scheme need not be a Noetherian scheme.

Concept sources

Wikipedia — Regular scheme — Regular scheme · expository · source ID wp-regular-scheme
nLab — Regular scheme — Regular scheme · mathematical reference · source ID nlab-regular-scheme
Stacks Project — Regular schemes — Regular schemes · reference · source ID stacks-regular-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.

Scheme Regular scheme

Permalink to relation

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

Outgoing relations (arrows from this concept)

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No direct relations are authored in this direction.

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