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Canonical static concept record

Localization of a commutative ring

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Summary

For a multiplicative subset $S\subseteq R$ , the ring $S^{-1}R$ obtained by formally inverting every element of $S$ .

Record metadata

Canonical name: Localization of a commutative ring
Concept ID: ring_localization
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)

commutative ring $R$
multiplicative subset $S$

Data

fractions $r/s$ modulo the localization relation

Axioms / constraints

elements of $S$ become units

Canonically induces

Local rings $R_{\mathfrak p}$ and basic open coordinate rings

Concept sources

Wikipedia — Localization (commutative algebra) — Localization (commutative algebra) · encyclopedic reference · source ID wp-ring-localization
nLab — Localization of a commutative ring — Localization of a commutative ring · mathematical reference · source ID nlab-ring-localization
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

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Commutative ring Localization of a commutative ring

Permalink to relation

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

Relation ID: story_commutative_ring_to_ring_localization_canonical_construction
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: invert a multiplicative subset

Authored explanation

Localization universally makes selected ring elements invertible.

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

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