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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

Regular local ring

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Summary

A Noetherian local ring $(R,\mathfrak m)$ whose Krull dimension equals the dimension of its Zariski cotangent space $\mathfrak m/\mathfrak m^2$ over the residue field $R/\mathfrak m$ .

Record metadata

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

Carrier(s)

Noetherian local ring $(R,\mathfrak m)$

Axioms / constraints

$\dim R=\dim_{R/\mathfrak m}(\mathfrak m/\mathfrak m^2)$

Concept sources

Wikipedia — Regular local ring — Regular local ring · expository · source ID wp-regular-local-ring
nLab — Regular local ring — Regular local ring · mathematical reference · source ID nlab-regular-local-ring
Stacks Project — Definition of regular local rings — Regular local rings in dimension theory · specialist mathematical reference · source ID stacks-regular-local-definition
Stacks Project — DVR and regular local ring equivalences — DVR and regular local ring equivalences · reference · source ID stacks-dvr-regular
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.

Noetherian + local regularity(construction junction)Regular local ring

Permalink to relation

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

Relation ID: j_regular_local__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

Use the same ring and require the embedding dimension to equal the Krull dimension.

How to interpret this relation type

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

Relation sources

Stacks Project — DVR and regular local ring equivalences — DVR and regular local ring equivalences · reference · source ID stacks-dvr-regular
Wikipedia — Regular local ring — Regular local ring · expository · source ID wp-regular-local-ring
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.

No direct relations are authored in this direction.

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