DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

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Localisation of a module at a multiplicative subset

Definition

Let $R$ be a commutative ring, let $S \subseteq R$ be a multiplicative subset, and let $M$ be a left $R$ -module. On $M \times S$ , define

(m, s) \sim (n, t) ⟺ u (t m - s n) = 0 for some u \in S .

The localisation of $M$ at $S$ is the set $S^{- 1} M$ of equivalence classes for this relation, and the class of $(m, s)$ is written $m / s$ .

The proposed addition and $S^{- 1} R$ -scalar action are

\frac{m}{s} + \frac{n}{t} : = \frac{t m + s n}{s t}, \frac{a}{u} \cdot \frac{m}{s} : = \frac{a m}{u s} .

The canonical map is

λ_{M} : M ⟶ S^{- 1} M, m ⟼ \frac{m}{1} .

The relation, the addition formula, and the scalar action are justified by The module-fraction relation is an equivalence relation ↗, Addition of localised module fractions is independent of representatives ↗, and The localised scalar action is independent of representatives ↗.

Depends on

Used by

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Section 12 (standard reference, not scraped)
The Stacks Project, Section 10.9: Localization (standard reference, not scraped)